H. HAOUES et al.: CORRELATION BETWEEN ELECTRICAL AND OPTICAL PROPERTIES OF DOPED SILICON NANOCRYSTALS 169–173 CORRELATION BETWEEN ELECTRICAL AND OPTICAL PROPERTIES OF DOPED SILICON NANOCRYSTALS KORELACIJA MED ELEKTRI^NIMI IN OPTI^NIMI LASTNOSTMI DOPIRANIH SILICIJEVIH NANOKRISTALOV Hakim Haoues 1,2 , Hachemi Bouridah 1,2 , Amir Ferrah 1 , Nidal Beltas 1 , Mahmoud Riad Beghoul 1,2 , Riad Remmouche 1,2 1 Department of Electronics, University of Jijel, Ouled Aissa 18000, Jijel, Algeria 2 Laboratory of Materials Studies, University of Jijel, Ouled Aissa 18000, Jijel, Algeria Prejem rokopisa – received: 2022-12-21; sprejem za objavo – accepted for publication: 2023-02-15 doi:10.17222/mit.2022.720 We propose through this work a correlation method leading to a determination of a semi-empirical relationship between optical and electrical properties in terms of refractive index and dark conductivity of doped silicon nanocrystals based on experimental data published in literature. First, an analytical model relating the conductivity and bandgap of doped silicon nanocrystals was derived. Using an empirical expression relating the refractive index to the bandgap energy, we correlated the electrical and opti- cal parameters of N-type nanocrystalline silicon with a semi-empirical expression. The semi-empirical relationship was found to account correctly for the experimental results and yield a reasonably good agreement in an interval of the bandgap energy varia- tion of N-type silicon nanocrystal films. The values of the fitting parameters were calculated for the N-type silicon nanocrystal films having their bandgap energy between 1. 7 eV and 2.2 eV. Keywords: silicon nanocystals, doping, dark conductivity, refractive index V ~lanku je predstavljeno delo v katerem so s pomo~jo korelacijske metode dolo~ili polempiri~no povezavo med opti~nimi in elektri~nimi lastnostmi dopiranih silicijevih nanokristalov. Dejansko podana korelacija predstavlja zvezo med lomnim koli~nikom in temno prevodnostjo, ki so jo dobili s pomo~jo eksperimentalnih podatkov najdenih v literaturi. Na za~etku so postavili analiti~ni model, ki se nana{a na prevodnost in energijski prevodni pas elektronov dopiranih nanokristalov. S postavitvijo empiri~nega izraza, ki povezuje lomni koli~nik in energijo v prepovedanem energijskem pasu so s pomo~jo polempiri~nega izraza povezali elektri~ne in opti~ne parametre N-tipa nanokristalini~nega silicija. Ugotovili so, da se polempiri~na zveza to~no ujema z eksperimentalnimi rezultati in dokaj dobro ujema v intervalu v katerem se nahaja energija prepovedanega pasu N-tipa silicijevih nanokristalini~nih filmov (tankih plasti). Vrednosti parametrov usklajevanja so izra~unali za N-tip nanokristalini~nih filmov z energijo med 1,7 eV in 2,2 eV. Klju~ne besede: silicijevi nanokristali, dopiranje, temna prevodnost, lomni koli~nik 1 INTRODUCTION Silicon nanocrystals (Si-NCs) have attracted a great deal of interest due to their several potential applications in photovoltaic, photonic and nanoeletronic devices. 1–4 Indeed, the small size, a few nanometer ranges of Si-NCs, causes different properties with respect to crys- talline bulk silicon. The wide application fields of Si-NCs are due to their tunable optoelectronic properties obtained by controlling their size, shape and doping level and due to the fact that Si-NCs show quantum size ef- fects and can be produced with processes compatible with a complementary metal oxide semiconductor (CMOS). 5 Si-NCs are generally formed in an insulating matrix such as silicon nitride, 6–8 silicon oxide, 9–12 silicon carbide 12–15 or hydrogenated silicon. 16–19 The high resis- tivity of the Si-NCs embedded in insulating matrices limits their applications, thus N- or P-type impurity dop- ing is crucial for the manufacturing of devices based on Si-NCs. 20–23 Controlling both electrical and optical prop- erties of devices based on silicon nanocrystals remains a challenge for the required applications. The photo re- sponse of Si-NCs is directly related to the light absorp- tion efficiency and electrical properties of silicon nano- crystals. In fact, light is absorbed in silicon nanocrystals by free carriers that diffuse in the insulating matrix by different conduction mechanisms, 13–15,24 inducing the ap- pearance of both the photoconductivity effect and photo- voltaic effect. Thus, tailoring and tuning the conductivity and the refractive index are crucial for photovoltaic and optoelectronic applications. The electro-optic effect, which changes a material’s index of refraction via an ap- plied electric field, has been observed in a silicon nanocrystal material as it makes this material a good can- didate for nonlinear optic applications. 5,25 For photovol- taic applications, the electrical and optical properties of Si-NCs used as the base material of solar cells, are criti- cal for the photo-carrier generation and transport, strongly affecting the cell efficiency. 9,26 As the relation- ship expression between the dark conductivity and the refractive index remains important, we try, via this work, Materiali in tehnologije / Materials and technology 57 (2023) 2, 169–173 169 UDK 661.66/.68:537.874.31 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 57(2)169(2023) *Corresponding author's e-mail: hakim.haoues@univ-jijel.dz (Hakim Haoues) to establish a semi-empirical expression relating these two important parameters in doped Si-NC films. 2 DERIVATION OF AN ANALYTICAL MODEL OF ELECTRICAL CONDUCTIVITY OF DOPED SILICON NANOCRYSTALS In this section, we derive an analytical model includ- ing the dark conductivity as a function of the bandgap energy E g of silicon nanocrystals. We consider a thin layer of doped nanocrystalline silicon as a set of silicon nanocrystallites of the same size, dc, separated by an in- sulating matrix with thickness d in (intercrystallite re- gions) and barrier height energy B . The Fermi-Dirac statistical distribution of electrons is given by Equation (1): nN EE KT = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ C fn c B exp (1) where N c is the effective density of states in the conduc- tion band (BC), which is expressed by Equation (2): N h mKT C n B = 2 2 3 32 () π (2) N c :incm –3 ; E fn : electron quasi Fermi level; E c : conduction band level; K B : Boltzmann constant in eV/K; T: temperature in Kelvin; m n : effective mass of electron in kg; h: Planck constant in J s. The intrinsic density of electrons is given by Equa- tion (3): nN EE KT i i = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ C f c B exp (3) E fi : intrinsic Fermi level. Thus the free carrier density (electrons) can be ex- pressed as a function of the intrinsic density as: nn EE KT i i = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ exp fn f B (4) The dark conductivity of the N-type silicon nano- crystal film is given by Equation (5): n eff = qn (5) q: electronic charge in coulomb (C); n: density of free electrons in cm –3 ; μ eff : effective mobility of electrons in cm 2 /Vs. By integrating Equation (4) into Equation (5) and knowing that the intrinsic Fermi level lies in the middle of the bandgap, E g =2.E fi (where Eg is the bandgap en- ergy), we obtain: n eff fn B g B = − qn E KT E KT i exp exp 2 (6) The intrinsic density n i can be expressed by Equation (7): nN N E KT i i =⋅ − () e x p cv f B (7) N v : effective density of states in the valence band (VB) expressed as: () N h mKT v hB / = 2 2 3 32 π (8) N v :incm –3 ; m h : effective mass of hole in kg. By integrating Equation (7) into Equation (6) we ob- tain: n eff cv fn f B g B = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ − qN N EE KT E KT i () e x p e x p 2 (9) The ratio between the free carrier density (electrons) and the intrinsic density deduced from Equation (4) is: n n EE KT i i ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ exp fn f B (10) Integrating Equation (10) into Equation (9) gives: n eff cv g B = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ − qN N n n E KT i ()e x p 2 (11) Considering R n =( n/n i ), the ratio between the free carrier density and the intrinsic density, the dark conduc- tivity expression of N-type silicon nanocrystals can be rewritten as: n eff cvn g B = − qN N R E KT ()e x p 2 (12) For P-type silicon nanocrystal films term R n is re- placed by R p , given by R p =( P/n i ), where p represents the free hole density. The effective carrier mobility μ eff can be calculated using the following expression: 27 eff c c in c in B B 1+ = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ F d dK T exp (13) d c : average crystallite size in nanometer (nm); d in : intercrystallite distance in nanometer (nm); F: factor given by F=1–( d c /L) where L is the total length of the film in nanometer; μ in : mobility in intercrystalline regions; B : barrier height energy of intercrystalline regions; μ c : mobility in crystalline regions taken as the mobility in monocrystalline silicon given by Equation (14): 28 c max min ref min 1+ = − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ + N N (14) N: free carrier density. H. HAOUES et al.: CORRELATION BETWEEN ELECTRICAL AND OPTICAL PROPERTIES OF DOPED SILICON NANOCRYSTALS 170 Materiali in tehnologije / Materials and technology 57 (2023) 2, 169–173 The values of μ max , μ min , N ref and are tabulated in Reference 28 . 3 MODEL VALIDATION Our study is based on the experimental results by W. He et al. 29 concerning films of phosphorus doped silicon nanocrystals embedded in amorphous silicon matrix. These results show the dependence of free carrier den- sity, dark conductivity and free carrier mobility on the silicon nanocrystal size. First, we examine the validity of Equation (13) to de- scribe the experimental effective carrier mobility values. Figure 1 depicts the variation in the carrier mobility as a function of free carrier density. We can clearly observe a good agreement between experimental and theoretical results. Thus, Equation (13) describes well the effective carrier mobility in doped silicon nanocrystals. The best adjustment parameters are summarized in Table 1 in- cluding several crystallite sizes taken from Ref. 29 The intercrystallite distance d in values are taken to be around 1 nm. 30 The optimal value of μ in that allows the best su- perposition between the experimental and theoretical curves is 0.08 cm 2 /V·s. Table 1: Adjustment parameters as a function of crystallite size d c (nm) F B (eV) 6.76 0.986 0.011 5.59 0.989 0.012 4.67 0.991 0.018 4.31 0.9914 0.027 3.59 0.9929 0.0275 Figure 2 represents the comparison between the ex- perimental and theoretical dark conductivities as a func- tion of bandgap E g . It is worth noticing that the experi- mental values from Ref. 29 are given as a function of the Si-NC size. To be able to plot the experimental curve of the conductivity as a function of E g , we proceed to deter- mine the values of E g as a function of the silicon nanocrystallite size using the expression below. 31 Ed E d d gc g c c () () .. () =∞ + + 3 4382 11483 2 (15) E g (d c ) is the silicon nanocrystallites bandgap as a function of the nanocrystallite size, E g ( ) is the bulk bandgap energy of silicon. E g is expressed in eV and d c in nanometers (nm). The best agreement between the two curves from Figure 2 is obtained for an average value of R n .μ eff equal to 4.2.10 15 cm 2 /(V·s). 4 RELATIONSHIP BETWEEN DARK CONDUCTIVITY AND REFRACTIVE INDEX From Equation (12) and the empirical expression giv- ing the refractive index as a function of E g , we try to H. HAOUES et al.: CORRELATION BETWEEN ELECTRICAL AND OPTICAL PROPERTIES OF DOPED SILICON NANOCRYSTALS Materiali in tehnologije / Materials and technology 57 (2023) 2, 169–173 171 Figure 3: Variation in the dark conductivity and refractive index as a function of bandgap energy Figure 1: Variation in the calculated and experimental 29 carrier mobil- ity as a function of free carrier density Figure 2: Comparison between the experimental 29 and theoretical dark conductivities as a function of bandgap energy determine the relationship between the electrical conduc- tivity and the refractive index. Indeed, the refractive index can be determined using Equation (16): 32 2 2 1 −= + A EB () g (16) where A=25E g + 212 eV and B = 0.21 E g + 4.25 eV Figure 3 represents the variation in the dark conduc- tivity and the refractive index as a function of bandgap E g . The curve giving the variation in the dark conductiv- ity as a function of the refractive index can therefore be deduced. Indeed, we use the theoretical data from Fig- ure 2 giving the variation in the dark conductivity as a function of bandgap energy that agrees well with the ex- perimental results from Ref. 29 In Figure 4, we superpose the results obtained with our approach, their fit and the experimental results published in Ref. 33 for comparison. We can clearly see that the experimental results agree well with the calculated results. The best fit of our re- sults, giving the dark conductivity as a function of refrac- tive index, can be expressed with the following equation: +=⋅ − '" e x p 0 (17) where is expressed in S·cm –1 , 0 = 2.56 , = 0.016, ' = 8.29.10 –4 S·cm –1 , ''=46.8×10 –4 S·cm –1 . The fit parameters are calculated for the bandgap en- ergy values of N-type Si-NC films between 1.7 eV and 2.2 eV. 5 CONCLUSIONS A correlation method providing a relationship between the dark conductivity and refractive index of N-type silicon nanocrystal films was developed based on experimental data. We first derived an an- alytical expression giving the dark conductivity as a function of bandgap energy. This expression was combined with an empirical expression relating the refractive index to the bandgap energy and thus a semi-empirical relationship between the dark con- ductivity and refractive index was established. 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HAOUES et al.: CORRELATION BETWEEN ELECTRICAL AND OPTICAL PROPERTIES OF DOPED SILICON NANOCRYSTALS Materiali in tehnologije / Materials and technology 57 (2023) 2, 169–173 173