Absorptive loss and band non-parabolicity as a physical origin of large nonlinearity in epsilon-near-zero materials

For decades, nonlinear optics has been used to control the frequency and propagation of light in unique ways enabling a wide range of applications such as ultrafast lasing, sub-wavelength imaging, and novel sensing methods. Through this, a key thread of research in the field has always been the development of new and improved nonlinear materials to empower these applications. Recently, epsilon-near-zero (ENZ) materials have emerged as a potential platform to enhanced nonlinear interactions, bolstered in large part due to the extreme refractive index tuning ({\Delta}n~ 0.1 - 1) of sub-micron thick films that has been demonstrated in literature. Despite this experimental success, the theory has lagged and is needed to guide future experimental efforts. Here, we construct a theoretical framework for the intensity-dependent refractive index of the most popular ENZ materials, heavily doped semiconductors. We demonstrate that the nonlinearity when excited below bandgap, is due to the modification of the effective mass of the electron sea which produces a shift in the plasma frequency. We discuss trends and trade-offs in the optimization of excitation conditions and material choice (such material loss, band structure, and index dispersion), and provide a figure of merit through which the performance of future materials may be evaluated. By illuminating the framework of the nonlinearity, we hope to propel future applications in this growing field.

Physically, this condition may be satisfied in bulk materials near resonances or through free carriers, as well as in nanostructured materials as an effective property by mixing both metals and dielectrics. [13] For low-loss ENZ materials, the real index is also less than unity -a condition termed near-zero index (NZI) -and such materials have demonstrated an enhancement of nonlinear processes including the intensity-dependent refractive index (IDRI) [20,[22][23][24][25] and frequency conversion [14,[26][27][28] through electric field confinement and slow light effects [29,30], thereby opening a breadth of applications in light manipulation [25]. It should be noted that an effective NZI condition may also be achieved in all-dielectric nanostructures with ε > 1, however, we do not consider this case here as the nonlinearities in this structure are the same as in the constituent dielectrics.
A prominent subset of ENZ materials is the transparent conducting oxides (TCOs) [31][32][33][34], such as indium tin oxide (ITO) or aluminum-doped zinc oxide (AZO). These materials provide a carrier concentration of up to 10 20 -10 21 (cm -3 ) due to the high fraction of donor atoms In 3+ in ITO or Al 3+ in AZO, and produce an ENZ region in the telecommunication spectrum (1.3 -1.5 (µm)). Such materials have exhibited refractive index modulation that is comparable to the linear index (Δn ≈ 0.2 -1) [22][23][24]35,36] while being ultrathin, industrially friendly [37], and CMOS compatible [33]. As a result, index modulation in NZI materials is of interest for potential applications in alloptical switching. Fig. 1: a) Schematic of the intraband nonlinearity in ENZ materials, where the reflectivity (permittivity) of the material is changed through the application of a pump beam. b) The change in permittivity occurs due to a modification of the effective mass of the electron sea as the absorbed pump energy elevates electrons to higher energy, higher mass states. c) The resulting change in effective mass red-shifts the plasma frequency of the material, producing the modulation of permittivity at a fixed frequency.
The continuation of these studies necessitates effective modeling to guide experimental and material synthesis efforts.
Through previous works, the IDRI of NZI materials has been modeled using the Kerr-process description [24]. While this approach has been shown to provide a reasonable quantitative description, it implicitly assumes a polarization driven effect while the nonlinearity has been experimentally shown to be due to free-carrier effects [35,38]. Therefore, to further explore the mechanisms of the nonlinearity, predict ideal regimes of operation, and optimize materials, a more physically accurate model is worth investigation.
This work aims to develop a deterministic (fit-parameter free) physical model to describe the large nonlinearities in Drude NZI materials, such as TCOs, to gain physical insight into the underlying mechanisms as well as predict the optimal material and experimental requirements. Through this, we emphasize that the driving force of the effect is due to the non-parabolic dispersion of the conduction band and the resulting effect on the average effective mass under intraband excitation.

Nonparabolicity of the Band as the Cause of Nonlinearity
In ENZ materials, the intensity-dependent refractive index is generally induced through an intraband absorption process whose effects are schematically shown in Fig. 1 (although other methods are possible such as interband excitation). In this case, energy from the pump is absorbed by free carriers ( abs U ), thereby raising their kinetic energy and the electron temperature e T , Fig. 1b depicted by the smeared occupied state distribution ( ) ( ) is the density of states and ( ) f E is the Fermi-Dirac probability function. As the distribution of carriers shifts to higher energy states, the average effective mass of the electron sea tends to increase since the band is non-parabolic (illustrated by the color gradient for free-carriers in the conduction band). This, in turn, causes a decrease in the plasma frequency and a corresponding increase in dielectric constant (Fig. 1c). The process relaxes as the kinetic energy of the hot electrons is transferred to the lattice (i.e. through acoustic phonons) with the characteristic electron-lattice relaxation time el τ . From this picture, it is clear that several parameters play a key role in the strength of the process: 1) the absorbed energy, 2) the band nonparabolicity, and 3) the dispersion in the index, which is quite different from typical polarization-driven nonlinearities.
To provide more detail into the IDRI in ENZ materials, we begin by defining the momentum-dependent mass of the electron. This can be determined by considering how the momentum of electrons at the Fermi surface change due to an applied electric field F . Here, the field shifts the Fermi-surface in k-space (see Fig. 2a), producing a change in momentum that for some state k is given by: For a field applied along the z-direction, the velocity associated with a particular momentum state becomes: 2 2 , and taking into account that the field is harmonic, we obtain: is the momentum dependent effective mass. Note that this definition is different from the more conventional definition of effective mass as the radius of curvature in k-space, , and the less common transport effective mass [40,41] . In fact, the above formula is a weighted average of the two, simplifying to the first description in the case of a perfectly parabolic band, and to the latter in the case of a perfectly linear band. For a non-parabolic ZnO band, as shown in Fig. 2b (energy band data from ref. [39]), the definitions of the effective mass are compared in Fig. 2c, where the difference between the definitions is apparent.
Given a general description for the effective mass of a state, we can then describe the optical response of the material under an applied field of arbitrary frequency by summing the induced displacements for all electrons in the gas, or for a continuous energy band as: where N is the carrier density, ( ) with chemical potential µ and electron temperature e T .
The effective mass of the electron sea is thus given by the geometric average of the occupied states, whose distribution in energy (e.g. described by the Fermi-Dirac probability) can be modified by absorbing the energy of an optical pump through the intraband absorption process. In this case, the change in permittivity can be found by differentiating Eq.
6 with respect to the incident pump intensity, I , where we find it is proportional to the change in effective mass: Ultimately, this expression can be interpreted as a change of the screened plasma frequency, resulting in a red-shift of the entire dispersion curve as shown in Fig.1c.

Estimating the Strength of the Nonlinearity
We can provide an estimate for the strength and dependence of the nonlinearity by taking a few assumptions in order to obtain an analytical expression for the nonlinear coefficients in the ENZ region. For AZO with an ENZ point near the telecom wavelengths, the carrier density must be approximately 21 3 0.8 10 ( ) N cm − × [42]. In this case, the Fermi level resides well above the bottom of the conduction band (by as much as 1 ( ) eV ) [43]. As a result, we may assume that the average effective mass of the electron gas is approximately equal to the effective mass at the Fermi Level,  Fig.   2b). This simplification allows us to directly write the permittivity of the material using Eq. 6 and provides a good estimate of the nonlinear effect without making any assumption on the distribution of hot electrons. Furthermore, it does not require the introduction of an electron temperature! A similar assumption can be used for many other heavily doped semiconductors where the band dispersion in quasi-linear at the Fermi level. Now, because the number of carriers in the band must be conserved during the intraband process, the change in permittivity is then proportional to the overall variation in the average effective mass due to the absorption of the pump photons. When electrons absorb a pump photon, their energy changes by some amount i E δ , which for electrons near the Fermi level produces a change of momentum for the electron of / Fig. 1b). Given our momentum dependent effective mass, the change in the effective mass due to the intraband absorption process is: Although i E δ , and therefore i k δ , is different for each electron, the sum of the energy change over all of the electrons within a given volume is equal to is the absorbed energy density of the pump beam.
We can then find the total change to the permittivity by summing over the small modifications in the effective mass induced by the excitation of hot electrons which while neglecting loss becomes: and when using the relation at the screened plasma frequency. In general, the average absorbed energy density over the sample thickness d can be written as: where i I is the pump intensity just inside the sample and related to the external intensity I as (1 ) from which, the imaginary part of the susceptibility can found as: Now we can also calculate the IDRI coefficient and nonlinear absorption as in ref.

Predicting Experiments
To move beyond the assumptions inherent in Eq. 19, we retain the material loss and incorporate the complete dispersion of the effective mass which can be determined from a given material band structure. In this case, we directly determine the average effective mass of the electron sea using Eq. 8. Therefore, calculating the nonlinearity reduces to determining the distribution of the electrons within the band, and thus, the modification of The second states that the total change in energy of the electron gas must be equal to the absorbed energy such that: where 0 µ is initial chemical potential before pumping, found for 0 e T T = , and τ is either el τ or p τ , chosen as described for Eq. 15.
Using these constraints, the chemical potential and electron temperature can be uniquely defined for a given pump intensity I, and the average effective mass can then be found then as In general, the Drude scattering rate Γ will also be dependent upon the incident pump intensity (see Appendix A), but in the case of heavily doped TCOs such as AZO and ITO, which are known to contain a large density of defects [43][44][45], Γ is assumed to remain constant, essentially being limited by the defect and ionized donor scattering rate.
Furthermore, a variation in Γ produces a change to the real permittivity of 2 '~2 ( / ) δε ε δ ω ∞ Γ Γ , which is negligibly small compared to the modified effective mass. From Eq. 23, we can then determine the change in permittivity and index as a difference between the properties of the steady-state and the pumped material.
Examining the limits of the nonlinearity, we find that the max variation in the refractive index is achieved when the change in effective mass is also maximized, Furthermore, in contrast to polarization driven nonlinearities this change occurs over an "integration time" of τ leading to 1) an increased nonlinear effect, since the pump energy feeding the nonlinearity continues to build throughout the interaction and 2) a sub-picosecond rise and fall time. Although not instantaneous, this response time is sufficient for numerous potential ultrafast applications and enables the use of the larger nonlinear phenomena while maintaining a strong overlap to many modern ultrafast laser systems with pulse-widths on the order of 10's to 100's ( ) fs .
To verify the model, experimental data from literature studying IDRI in AZO films at ENZ is used [24]. In this case, ) is comparable to the electron-phonon relaxation rate such that the peak temperature of the electron gas is reduced by the competing absorption and phonon scattering processes.  at short wavelengths due to the discrepancy of the permittivity of the approximated and measured permittivity of the sample [24]. Breaking this down we can extract the average effective mass induced in the AZO film and peak change in the index within the ENZ region, Fig. 3b, as a function of the pumping intensity. Here, the average effective mass is shown to vary linearly with the pump intensity, producing an increase of ~20% for 2 1 ( / ) TW cm . However, the peak change of the refractive index in the ENZ region is observed to be nonlinear with intensity. This is expected since the change in the index is inversely proportional to the change in * avg m . However, this demonstrates that a single 2 n value is a limited description of the nonlinearity in ENZ materials. . Ultimately, an increased ε ∞ requires a larger contribution from the free-electrons to produce an ENZ region at a fixed wavelength, which results in a steeper index dispersion. However, this is generally accomplished by increasing the carrier density of the material. As a result, more electrons must undergo an interband transition (i.e. more pump energy must be absorbed) to produce the same shift in permittivity, leading to a factor 1 / N . It is important to note that the nonlinearity also depends on accumulation time τ (e.g. through the absorbed energy as seen in Eq. 15), but the bandwidth of the nonlinear effect is inversely proportional to this time. As a result, we can define a FOM by combining these quantities which form an 2 n -strength-bandwidth product that can be used as an intrinsic material FOM for intraband nonlinearities:

Discussion
To explore this, the band structures of indium tin oxide (ITO), cadmium oxide (CdO), gallium nitride (GaN), and gallium oxide (Ga2O3) were approximated from literature and the FOM for each material was calculated versus mobility. The results are summarized in Fig. 4 while the values used for the calculation are shown in Table 1 where the listed mobility ranges correspond to published values for each material. The films are considered to be under the same experimental conditions as in [24]. As a reference, the peak FOM of each material within the range of mobility The first general observation that can be made from this comparison is that vastly different conductive oxides and nitrides have a FOM that differs by just more than a factor of three, as can be expected based on the values as reported in Table 1. The second observation is that based on Fig. 4 the peak FOM is achieved for moderate mobility films in      Table 1. Although a similar result has been noted in literature [12], an exact comparison is strongly dependent upon the material growth conditions as the effective mass varies with factors such as defects and grain size [41] while the band structure varies with stress/strain and doping density among others.
Beyond these more explored films, Ga2O3 shows the highest FOM within the range of published mobilities; however, it has not been doped into the range necessary for consideration in the telecommunications window and the performance may vary with heavy doping. Yet, Ga2O3 is a uniaxial crystal which may prove useful in exploring nonlinear phenomena that manipulate optical polarization [58]. The performance of CdO and GaN suffers due to their high mobilities which reduce their ability to use the pump energy supplied. However, if the materials are allowed to take any mobility (i.e. considering values outside of the grey dashed regions in Fig. 4

Conclusions
Through analysis of free carrier dynamics, a model has been derived to explain and fit the large nonlinear response of Drude ENZ materials, such as TCOs. We demonstrate that the nonlinearity is due to free-carrier absorption and the subsequent modification of the average electron effective mass through a smeared population in a non-parabolic conduction band. Specifically, the optimization of loss was considered as the balance between efficient absorption of the pump and a steep index dispersion is needed to maximize the performance of a material. In fact, the performance of multiple materials is evaluated using the proposed FOM, and all achieved their peak performance for moderate mobility ranges in the range of 2 20 ( / ) cm V s − , corresponding to " ε~ 0.5. This demonstrates that unlike other applications in plasmonics, pursuing exceptionally low loss ENZ materials is not necessary to maximize the IDRI effect, and can, in fact, be detrimental to the performance. However, this should be balanced with other factors, such as the need for efficient transmission of the probe, based on a given application. Ultimately, ENZ materials represent a unique platform for nonlinear interactions that is ultrathin and can be realized without complex nanofabrication while making use of intraband nonlinearities that are slow enough to enable a good temporal overlap with ultrafast laser systems yet fast enough for many ultrafast optical applications. By enabling a higher degree of predictive power and a platform for evaluating and optimizing new materials, practical devices for ultrafast, compact applications may move closer to reality.

Appendix A: Energy Absorbed in a Thin Sample
The absorbed energy density can be calculated simply as the difference between the energy density just inside the sample and after some distance dz: ( ) n v v dt α = (27) The peak in the absorbed energy density ( abs U δ ) is then found by integrating over a time τ noting that there is a competing relaxation process of electron scattering ( where τ is selected based on the criteria above.
Here the optical mobility is the value as determined through optical measurements, such as ellipsometry, as opposed to the Hall effect. Depending upon the material's structural quality the optical mobility can differ from the Hall mobility by as much as an order of magnitude [41], [59].

Appendix B: Band Structure calculations
The conduction band of zinc oxide [39] is taken as the basis for band calculations. To calculate effective mass and density of states, the band is fit by a hyperbola for simplicity. The hyperbola is defined by the parameters a and b such that For comparison of materials, this hyperbolic approximation can be taken for many doped semiconductors.

Appendix C: Electron Relaxation
A correction factor is needed to calculate the maximum absorbed energy in the case when the electron relaxation rate is comparable to the pulse width, p el τ τ  . Since the electron scattering times are on the order 10s of (fs), the recovery time for the nonlinear effect in AZO was measured to be approximately 170 (fs) and is comparable to the pump pulse width, 100 (fs). This limits the peak temperature reached by the electron gas. In this scenario, we follow a similar logic as shown in Appendix A, where the pump pulse is now taken as a Gaussian in time with pulse width p τ , and our integration time is p τ τ = . However, we need a correction factor to account for the electron relaxation within the pulse envelope, defined as: This model for relaxation provides us with a correction factor of To keep the ENZ wavelength constant, the number of carriers, N , must compensate; this increases the average effective mass resulting in a significant decrease in FOM at low mobilities. For simplicity, the values for each of these parameters given a high mobility are shown in Table 1, while the values for calculation include this shift.