Optical properties and electronic density of states

Abstract

T he fundam e nta l ab sorption s pectrum of a so lid yie lds information about criti c al points in the opti· cal d e nsity of states. This inform ati on ca n b e used to adjust parameters of the band stru ct ure. Once th e adju s te d band s tru c ture is kn own , the opt ica l prope rti es a nd th e de ns it y of states can be ge nerated by num e ri ca l integration . W e re vi e w in thi s paper the pa ra me trizati on tec hniques used for obtaining band structures s uitable for de ns ity of s tates calc ulation s. The calc ulate d optical cons ta nts a re compared with expe rim e nt a l res ult s. Th e e ne rgy d e riv ative of th ese op ti ca l co ns tan ts is di sc usse d in co nn ection with res ults of modulate d re fl ecta nce meas ure me nts. It is a lso s hown th a t inform at ion abou t de ns ity of e mpty s tates ca n be obt ain e d fro m opti ca l expe rim e nt s in vo lvin g excit ati on from d ee p core le ve ls to th e co nduc tion band.A de ta il e d co mpari son of th e ca lc ul ate d one·e lec tron opti ca l line s hapes with expe rim e nt revea ls dev ia tion s whi ch ca n be int er pre ted as e xciton effects. The acc umul atin g ex pe rim e nt al e vid e nce poin t· in g in this d irectio n is reviewed to ge the r with th e ex is tin g th eo ry of these e ffects .A numb e r of simpl e mode ls for th e co mpli cate d inte rband de ns it y of s ta tes of a n in s ul ator ha ve been proposed. We re vi e w in pa rti c u lar th e P e nn mod e l, whi c h ca n be use d to acco unt for res po nse function s a t zero frequen cy, a nd t he parabo li c mode l, whi c h ca n be use d to accoun t for the di s pers ion of res pon se fun c tion s in th e imm e di ate vi c inity of th e fundam e ntal a bso rpti on e dge. Key word s: C riti ca l po int s; de ns ity of sta tes; d ie lec tri c co ns tant; modul ated re Aect a nce; op ti ca l abso rpti on.1. Optical Properties and One-Electron Density of States osc illa tor s tre ngth ten so r F ef is rela te d to th e ma trix ele me nts of p throu gh F e/= 2 < flp le > < el p lf > w eI-I. Th e Bloc h fun c ti ons are normaliz ed over unitThe opti cal behavior of se mi co nductors and in s ula· tors in the n ear infrared , visible, and ultraviolet is determined by electronic interband transition s. An ad· diti onal intraba nd or free elec tron co ntributi o n to th e optical properties has to be considered for me tals. W e shall di sc uss here th e relationship between th e inte rband contribution and the de n sity of states. The inter· band co ntribution to th e imaginary part of th e diele c tri c con stant can be written as (in atomic units, Ii = 1, m = I ,e= I):(1) wher e w eF w e-WI is th e differen ce in e nergy be tw een the e mpty bands (e) and the fille d bands (E). The spin multiplicity mu s t be include d expli citly in eq (1). The volume. Degenera te statis ti cs has bee n ass ume d in e q (1) and spatial di s persion effects have bee n neglecte d.It is c ustomary to take th e slowly varyin g oscillator stre ngth out of the integral sign in e q (1...

From Green Remediation to Polymer Hybrid Fabrication with Improved Optical Band Gaps

“…Hence, the expression of the Penn model can be formulated in relation to the refractive index [72]. This model makes use of dielectric constant of long wavelength because it is constant and almost plateaus [73]. The optical dielectric constant is illustrated in Figure 11 against the filler fraction, while the optical band gap against filler concentration is shown in the inset of that same figure.…”

Synthesis of Polymer Nanocomposites Based on [Methyl Cellulose](1−x):(CuS)x (0.02 M ≤ x ≤ 0.08 M) with Desired Optical Band Gaps

“…The imaginary part ɛ 2 stands for the optical absorption in the material, which is strongly associated with the valence (occupied) and conduction (unoccupied) bands and is given by [44]:ε2false(ωfalse)=2e2πnormalΩεotrue∑K,V,C|ψKC|trueU→·truer→|ψKV|2δ(EKC−EKV−ℏω) where ω is the incident photon frequency, Ω is the crystal volume, e is the electron charge, ɛ o is the free space permittivity, r→ is the position vector, u→ is a vector defined as the incident electromagnetic wave polarization, and ψkc and ψkv are the conduction and valence band wave functions at k, respectively. Based on the theoretical models, the optical dielectric constant is described by a complex function of frequency, which requires a large-scale computational effort to be calculated [44,45,46]. Experimentally, from the obtained refractive index and extinction coefficient data, the imaginary part of te optical dielectric function (ɛ 2 ) can be estimated using the following relation [31], ɛ 2 = 2 n k where n is the refractive index and k is the extinction coefficient.…”

A Comprehensive Review on Optical Properties of Polymer Electrolytes and Composites

“…The theoretical principle of optical dielectric constant is based on a complex function of frequency. Thus, large-scale computational endeavor is needed to calculate dielectric constants [ 288 , 289 , 290 , 291 ]. On the other hand, experimental estimation of dielectric constant from the optical dielectric function on imaginary part ( ε 2 ) is easily achieved using refractive index and extinction coefficient in the calculation as shown in the following relationship [ 78 , 281 , 282 , 283 , 284 , 285 , 286 ]: where K and n are the extinction coefficient and refractive index, respectively.…”

“…This model relates the refractive index ( n ) to the dielectric constant in the form of ε = n [ 322 ]. According to this model, dielectric constant keeps constant over a long wavelength range that can be used to estimate energy states [ 290 ]. Consequently, as the optical dielectric constant increases, the density of states increases and the band gap energy drops.…”