Parameterized optical properties of monolayer MoSe 2

We report a model dielectric function, ε = ε 1 + i ε 2 , of MoSe 2 from 1 to 6.42 eV with which the optical property of a MoSe 2 monolayer can be calculated at arbitrary temperatures from 31 to 300 K for potential application for device designs based on this material. Analytic representations, performed with the dielectric-function parametric model, allow interpolation with respect to both energy and temperature. We used reported spectrum data [Park et al. , Sci. Rep. 8 (1), 3173 (2018)] as the basis of our approach, verifying that the parameterized model dielectric function can reproduce the experimental data at various temperatures and can also produce the dielectric function (and the refractive index) at arbitrary temperatures.


I. INTRODUCTION
Like many transition metal dichalcogenides (TMDCs), molybdenum diselenide (MoSe 2 ) has a layered structure of strong inplane bonding and weak out-of-plane interactions. Because of these interactions, exfoliation is possible to realize two-dimensional layers of one unit cell thickness. MoSe 2 is well known as a TMDC that is a potential substitute for silicon or organic semiconductors in high-technology transistors, sensors, waveguides, and photodetectors. [1][2][3] Dielectric functions, ε = ε 1 + iε 2 , of MoSe 2 , including temperature dependence, are needed to design devices for these applications. 4 As a result, several researchers reported the dielectric function of a MoSe 2 monolayer at low temperatures and room temperature by using spectroscopic ellipsometry (SE). [5][6][7] However, in order to be applied properly for device applications, the dielectric function of the monolayer MoSe 2 should be available at arbitrary temperatures. We note that the surface current model was applied to obtain optical properties of monolayers (and ultrathin films with some atomic layer thickness). 8 However, the purpose of this work is not to extract dielectric function values from experimental data but to utilize previously reported dielectric function spectra for further modeling to get optical properties at arbitrary temperatures.
Here, we meet this need by determining an analytic representation of ε of MoSe 2 over the energy range of 1-6.42 eV for temperatures from 31 to 300 K. We chose the experimental data published in Ref. 7 as the starting point since it covers a broad range of both spectrum and temperature, where obtaining critical-point (CP) energies was pursued from derivative spectra. The purpose of the current work is to reconstruct the original ε spectrum at arbitrary temperatures.
We follow the same approach applied for obtaining the dielectric function of the MoSe 2 monolayer 9 and the InP bulk sample 10 at any arbitrary temperature. It could also be applied to get optical properties of InAlAs 11 and InAsSb alloys 12 at arbitrary compositions, to mention a few. In brief, to represent basic asymmetric 13 features of ε, we used a dielectric-function parametric model (DFPM) that describes ε as a sum of asymmetric oscillators. 14,15 In this work, the parameters were extracted by fitting the reported spectra with 11 dispersive oscillators and a pole ARTICLE scitation.org/journal/adv at each datum. Obtained model parameters at each temperature were fit to polynomials for achieving their temperature dependence to determine the optical properties of MoSe 2 at any temperature. These optical property values can be useful in designing devices based on this material and also in optical monitoring of the growth.

II. MODELING
The dielectric function spectrum in the DFPM is described as the sum of m energy-bounded polynomials within the accessible spectral range and P poles that represent outside contributions. 14,15 The general expression is where Wj(E) is the joint density of states of the jth CP, and Φ is a function to describe the broadening which is usually with either a Lorentzian or Gaussian function. The detailed explanation with a calculation program of the DFPM is given in Refs. 14 and 15. Figure 1 shows the description of each CP by nine parameters. All the parameters are well defined in Refs. 14 and 15, but, in brief, EC is the CP energy with EL and EU as its end points. ELM and EUM play an important role in controlling asymmetric characteristics of the CP structure. For energy regions (I, IV) and (II, III), the second-and fourth-order polynomials were used to construct the structure of the CP, respectively. The real part of ε is calculated by a Kramers-Kronig relation.  To clarify the temperature dependence of the dielectric functions, an offset of 10 is added to every spectrum relative to that for temperature at 31 K. DFPM, which is obtained by the WVASE software (J. A. Woollam Co.) with respect to these data. The contributions from each component CP are well shown by the dashed lines. We reduced the number of data points appropriately to ensure fitting quality. We could successfully reconstruct the spectrum with only 11 CPs. We note that 12 CPs were reported in Ref. 7 where the 2nd derivative spectrum enhanced the resolution of the CP structures. However, this work of dealing only original nonderivative spectra could not resolve small E and F CP structures whose energy difference is predicted to be only about 0.3 eV. 16 The obtained parameters are listed in Table I. Here, the CL and CU values mean the lower and upper energy positions of the respective CPs. ELM and EUM are difference energies relative to EC. Γ is the full width at half maximum. To reduce number of free parameters that fit, parameters with asterisks are fixed. This procedure was repeated for each spectrum at reported temperatures.

III. RESULTS AND DISCUSSION
To obtain ε values for arbitrary temperatures, we used a thirdorder polynomial equation, to fit all parameters in Table I. The best-fit parameters are shown in Table II. With these parameter values, we can obtain dielectric functions for arbitrary temperatures from 31 to 300 K. Figure 4 shows temperature dependence of Ec parameter values. The open dots show parameter values at each reported temperature with the solid lines as the best fits to Eq. (2). Blue shifts of all Ec parameters which represent the CP energy positions in the DFPM are shown, which can be explained phenomenologically by reduced electron-phonon interaction and a reduced lattice constant at low temperatures. 17 However, it should be emphasized that this Ec value is a parameter to construct dielectric function spectrum values at arbitrary temperatures and not to obtain CP energy values from the measured dielectric function spectrum. For the latter purpose, the 2nd derivative method of the spectrum is traditionally used, and precise CP energy values with their temperature dependence are reported in detail in Ref. 7.   Figure 5 compares the reported original experimental data (black dashed lines) to the final reconstruction of this work (gray solid lines) at 31 and 250 K. The reconstructions agree well with the data on this scale along with a slight discrepancy near 1.5-2.5 eV in ε 1 . However, the overall agreement supports the validity of our work. We can now calculate both real and imaginary parts of ε of MoSe 2 for any arbitrary temperature, as shown in Figs. 6(a) and (b), respectively. The spectra are offset by increments of 15 relative to that for 40 K. Careful comparison shows that noise can be well observed, as shown in Fig. 2, of experimental data, while no noise exists in the constructed model values, as shown in Fig. 6. By using the definition of the refractive index, We can also find real and imaginary refractive index values at any arbitrary temperature for device applications, as shown in Figs. 7(a) and 7(b), respectively.   Tables I and II.  Tables I and II.

IV. SUMMARY AND CONCLUSIONS
We obtain a model dielectric function of the MoSe 2 monolayer in a spectral range of 1-6.42 eV for a temperature range of 31-300 K. The DFPM successfully describes the data with 11 physically meaningful CPs including the sharp trionic feature A − . The temperature dependence of the necessary parameters was obtained by fitting the model parameters with 3rd order polynomial equations. Hence, dielectric functions of MoSe 2 for continuous ARTICLE scitation.org/journal/adv temperatures are determined for a spectral range of 1.0-6.42 eV. We believe that these results will be useful especially in technological applications such as monitoring of the growth and the design of MoSe 2 related devices.