S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD 3–9 POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD GRAFI^NA OCENA POTENCIALA PREGRADE NA MEJAH KRISTALNIH ZRN Souad Merabet 1,* , Bilal Djellil 2 1 University of Jijel, Department of Electronics, Laboratory of Renewable Energy, Jijel, Algeria 2 University of Jijel, Department of Electronics, Laboratory of the Materials Studies, Jijel, Algeria Prejem rokopisa – received: 2022-09-23; sprejem za objavo – accepted for publication: 2022-11-17 doi:10.17222/mit.2022.628 In this work, a representative graphical method was used to estimate the potential barrier at grain boundaries in polycrystalline materials. The used equation for the determination of trapping density states is closely related to the doping concentration in the layer. The obtained results showed that this density is a crucial parameter for estimating the maximum barrier value. This pa- rameter strongly depends on the grain size and, consequently, on the grain boundary width. The conduction (or transport) prop- erties represented by the thermionic current and the effective mobility also prove this dependence. Also, the obtained results are in good agreement with the experientially measured values from previous works. Keywords: traps, charge carrier, dopant concentration, grain size V ~lanku je predstavljena izbrana grafi~na metoda za oceno potenciala pregrade na mejah kristalnih zrn v polikristalnih materialih. Za oceno stanja gostote pasti so uporabili ena~bo, ki je tesno povezana s koncentracijo dopanta v plasti. Dobljeni rezultati so pokazali, da je gostota pasti kriti~ni parameter za oceno maksimalne vrednosti potenciala pregrade. Ta parameter je mo~no odvisen od velikosti kristalnih zrn in posledi~no tudi od debeline ({irine) kristalnih mej. Prevodne oziroma transportne lastnosti predstavljene s tokom elektri~no nabitih delcev(termioni~nim tokom) in efektivna mobilnost prav tako dokazujejo to odvisnost. Poleg tega se dobljeni rezultati dobro ujemajo z eksperimentalno izmerjenimi vrednostmi, podanimi v predhodnih raziskavah. Klju~ne besede: pasti, nosilci naboja, koncentracija dopiranja, velikost kristalnih zrn 1 INTRODUCTION The effect of the grain boundaries on the conduction properties of the polycrystalline films used in various ap- plications, including discrete devices and integrated cir- cuits, has prompted several studies of the carrier trans- port. 1–7 In the photovoltaic field, an inexpensive polycrystalline solar cell has also received considerable attention by various research groups. 8–12 The grain boundaries in polycrystalline silicon act as carrier migra- tion routes. 13 These boundaries can modify the undesired trap rate, which is considered as one of the most impor- tant challenges in these materials. A lot of defects found at grains boundaries due to in- complete atomic bonds (dangling bonds) 13,14 are sup- posed to act as traps for the doping atoms as a result of their segregation in the latter. Consequently, trap states are formed, able to trap and immobilize charge carriers. The number of free carriers available for electrical con- duction is reduced. As soon as the free carriers freeze, traps become electrically charged and a potential barrier appears 15–22 , preventing the movement of a grain carrier to its neighbor, thus, limiting their mobility. 23 This dop- ant segregation reduces the number of dopant atoms in the grains and the number of active charge carriers de- creases. It should be noted that, in terms of grain size and grain boundaries, the crystalline structure of poly- silicon films significantly affects the conduction proper- ties even when individual grains are physically small. 23 Understanding the diffusion of dopants in grains and also the grain-boundary segregation is necessary to control the fabrication of devices containing polysilicon layers. Thus, in this study a graphical method is suggested to determine the potential barrier for which an empirical re- lationship is used to estimate the density of the trapping states 23 where some modifications have been introduced to check the effects of the grain size and doping concen- tration in polycrystalline materials. 2 METHODOLOGY The used model is based on the principle developed by Seto and other researchers who succeeded him. 19,22,24–26 The formation of trap states (N t ) is due to the existence of a large number of defects at grain boundaries, able to trap charge carriers and immobilize them. The trapped charges are compensated by charged depletion regions surrounding the grain boundaries, which cause a curvature of the energy bands and the re- Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 3 UDK 544.228 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 57(1)3(2023) *Corresponding author's e-mail: smerabet@univ-jijel.dz (Souad Merabet) sulting potential barriers (see Figure 1), 22 impeding the movement of carriers from one crystallite to another. This model, which was established by the researchers as shown in Figure 2, predicts a strong resistivity below the critical doping concentration N * (cm –3 ), which drops rapidly for N > N * . The principle can be summarized as follows: For N N * , the concentration of free carriers is low and the resistivity is high (see Equation (1)): LNQVq LN G t b G 2 Si ×<⇒=× × ×× 8 0 (1) where V b is the potential barrier (V), L G is the average grain size (nm), N is the dopant concentration (cm –3 ), Si is the silicon dielectric permittivity, q is the elementary charge (C), and Q t is the charge on the traps (cm –2 ). ForN N*, the concentration of free carriers in- creases and the height of the potential barrier decreases, as expressed by Equation (4). The resistivity decreases rapidly as the doping concentration increases until it reaches that of the monocrystalline silicon. LNQVq Q N G t b t 2 Si ×>⇒=× ×× 8 0 (4) Where 0 =8.85×10 –12 F/m and Si = 11.68. In accordance with these concepts, the effect of the variation in the grain size and doping concentration on the assessment of the potential barrier height, thermionic current and effective mobility is studied below. Several models have been proposed to describe the density of trap states N t (cm –2 ). In this work, an empirical equation established during previous work 24 is used with some modifications: NL t G =+×× − (.) 35 255 10 10 (5) These changes refer to the value of parameter ,ac - cording to which the calculations are strictly dependent on the doping rate; the chosen values have been verified and validated during the comparison with the previous experimental results, 19,24 and the details of the calcula- tions are reported in Table 1. The obtained values show that the critical doping concentration N* (cm –3 ) and pa- rameter evolve in proportion to the increase in the dop- ant concentration of the sample. Two case studies were considered. In the first step, a choice about the doping rates used during the previous S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD 4 Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 Figure 1: Energy-band diagram of the n-type silicon 22 Figure 2: Potential barrier height as a function of grain-boundary dop- ing Table 1: Parameters needed for the simulation L G (× 10 –7 cm) Doping concentration (cm –3 ) Trapping state density Nt Experimental values (× 10 12 cm –2 ) (Ref. 19 ) Computed values (× 10 12 cm –2 ) 20 5×10 18 0.3533 2.98 2.9802 7×10 18 0.3630 (Ref. 20 ) / 3.3372 1×10 19 0.3649 3.41 3.4126 5×10 19 0.3704 3.64 3.6418 works was made 18,23 to validate the proposed calculation method with a fixed grain size of about 20 nm. In the second step with a fixed doping concentration, the grain size was varied. 3 RESULTS AND DISCUSSION 3.1 Potential barrier height The computed variations in the potential barrier height for different doping concentrations are shown in Figure 3a, calculated with Equations (1), (3) and (4) above. The potential barrier increases linearly as the dop- ing level reaches a maximum value, due to the addition of more impurity atoms to the traps. The grain-boundary traps increase the potential barrier and decrease the car- rier density of the grain. 27 After this, the potential barrier decreases rapidly with a high doping concentration be- cause the traps are filled and the free-carrier concentra- tion increases. This is very consistent with the previous findings. 28–30 Following the line along the doping axis, which cor- responds to the layer doping along the curve, allows us to determine the value corresponding to the potential barrier represented by the intersection point (see Fig- ure 3b). Table 2 shows the obtained results, which cor- respond well with the experimental values found in the literature, 19 and the values are mostly identical. They show that the trap density and the potential barrier evolve proportionally with the increase in the dopant concentra- tion of the polysilicon layer at a constant average grain size. Likewise, as before, the doping is set to1×10 19 (cm –3 ). The first value of the grain size is 20 nm so that we can compare and then validate the results obtained for the new, randomly chosen grain sizes. The computed values are represented in Figure 4a. The critical concentration N* is found to decrease with the increasing grain size. Corresponding to the different grain sizes selected, the values of the potential barrier are determined, as be- fore, as shown in Figure 4b, by the point of intersection of the line acrossing the various curves drawn and the corresponding doping axis to1×1 0 19 (cm –3 ). In sum- mary, the obtained results are reported in Table 3.Itis found that the density of trapping states and the maxi- mum value of the potential barrier decrease with the in- creasing grain size at grain boundary levels, thus facili- tating the passage of carriers from one grain to another; S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 5 Figure 4: a) Potential barrier height as a function of the grains size, and b) potential barrier values Figure 3: a) Potential barrier height as a function of the doping level, and b) potential barrier values this is explained with the decrease in the disorder in larger grains. 22,26–31 3.2 Thermionic current density The current flow in polycrystalline materials is lim- ited by the carrier movement across the grain boundary barrier rather than the current flow in the grains them- selves. 32 The maximum flux of carriers, considered as free particles by several models, 33–36 is provided by the thermionic emission above the potential barrier depend- ing on its height rather than its shape. The thermionic current density across the height of the barrier is given by the number of carriers that have sufficient energy un- der an applied voltage to move towards the grain bound- aries that can overcome this barrier. 19,32 Figure 5a shows the variation in the thermionic cur- rent density according to the applied potential and the change in the doping level. The choice of the applied po- tential values (U a ) must satisfy the ones imposed by the models. 19,32 Therefore, U a is the polarization across the sample divided by the number of grains, assuming all the grain boundaries are identical. 32 In general, the applied bias voltage is non-uniformly divided between the two sides of a grain boundary. However, for low voltages, ap- proximately half of this voltage appears on each side. 32 The thermionic current density is initially high at low doping rates due to a complete depletion of the grains and the potential barriers are not yet formed as a result of the partial filling of the traps due the low doping levels. At increasing doping levels, the thermionic current den- S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD 6 Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 Figure 6: Thermionic-emission current density as a function of ap- plied voltage and: a) doping concentration, b) grain size Figure 5: Thermionic-emission current density as a function of: a) doping concentration, b) grain size Table 2: Obtained results as a function of the doping concentration Doping concen- tration (cm –3 ) N* (× 10 18 cm –3 ) Maximum height of poten- tial barrier qV bmax (eV) Potential barrier qV b (eV) (Ref. 19 ) Potential barrier qV b (eV) Variation q(V b –V b(i-1) ) (eV) 5×10 18 1.49 0.1122 0.0335 0.0334 0.0034 7×10 18 1.67 0.1257 / 0.030 1×10 19 1.7063 0.1285 0.022 0.022 0.008 5×10 19 1.82 0.1370 0.005 0.005 0.017 sity decreases, and a minimum value dip occurs at a criti- cal doping concentration when the barrier height is maxi- mum, and, at higher doping levels, the barrier height is reduced, allowing an increase in the thermionic current density resulting from the rapid conduction. A strong shift was observed between the four curves due to the increase in the doping concentration, which led to a decrease in the potential barrier height, verified by the results obtained from Table 2. The observed devi- ation between the curves was found to be in reasonable agreement with the theoretical predictions as a function of the doping concentration. As the grain size increases at a fixed doping concen- tration (see Figure 5b) and at low doping levels, a ther- mionic current density value decrease dip occurs at a critical doping concentration. This decrease highly de- pends on the height of the corresponding potential bar- rier. It is worth noting that the corresponding curves at low grain sizes, 20 nm and 100 nm, are higher than those at higher grains sizes. These results are consistent with those shown in Figure 4. For any grain size, the position of the minimum value depends on the maximum value of the potential barrier and the density of the trapping states; after this value, the barrier decreases for high dop- ing levels and the current density increases proportion- ally with the grain size. The thermionic current density versus the applied po- tential change is shown in Figure 6. The current density grows weakly at low voltages. This is attributed to the fact that the carriers moving through the grain bound- aries have more opportunities to be trapped at the trap- ping sites. As the applied voltage rises, the effect of the barrier becomes insignificant. 37,38 The carriers have enough energy to bypass the barrier, so the current den- sity increases rapidly. 3.3 Effective mobility carrier Effective mobility describes the ease of the carrier movement from one grain to another and also the restric- tion of the current flow by the potential barrier at grain boundaries. 15 Therefore, it depends considerably on the doping concentration. In this last part of our work, the mobility is plotted in Figure 7 as a function of the dop- ing-concentration variation in the first stage and as a function of the grain-size variation in the second stage. The mobility plot is remarkably similar to the one for the thermionic current density. It can be clearly seen that the mobility presents the minimum value for the critical doping concentration N*, which decreases at low doping (high V b ) and then increases at doping concentrations above N* (low V b ). This result is in good agreement with the other research findings published in the literature. 24 The evolution of the obtained curves agrees well with the curve results from Figures 3 and 4. The observed discrepancy between the curves ob- tained for the effective mobility and thermionic current density at high doping levels (see Figures 6 and 7)isat - tributed to the difference in the obtained potential-barrier values at the grain boundaries. When the doping concen- tration increases at a fixed grain size, the discrepancy be- comes important, increasing from 0.0034 eV to 0.008 eV, and from 0.008 eV to 0.017 eV (see Table 2). It decreases from 0.014 eV to 0.002 eV, and from S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 7 Figure 7: Effective carrier mobility as a function of: a) doping level, b) grain size Table 3: Obtained results as a function of the grain size Grain size (× 10 –7 cm) Trapping state density N t (× 10 12 cm –2 ) N * (cm –3 ) Maximum height of poten- tial barrier qV bmax (eV) Potential barrier qV b (eV) Variation q(V bi –V b(i-1) ) (eV) 20 3.4126 1.7063 × 10 18 0.12853 0.022 0.014 100 2.0523 2.0523 × 10 17 0.3865 0.008 160 1.7840 1.115 × 10 17 0.5376 0.006 0.002 240 1.5868 6.6117 × 10 16 0.7172 0.0048 0.0012 0.002 eV to 0.0012 eV when the grain size increases with a constant doping concentration (see Table 3). 4 CONCLUSIONS In this study, the method used to determine the poten- tial barrier as a function of trapping density states in polycrystalline layers was verified by feeding calculated parameters into the plotting of equations and comparing them with the experimental data reported in the previous research works. Almost all the results were in agreement. The obtained results showed that when the layers have the same grain size, the states of trapping density and critical concentration are proportional to the increas- ing doping concentration (see Table 2) and the corre- sponding potential barrier is inversely proportional. An increasing discrepancy is observed between the thermi- onic current density curves due to the growing difference between the calculated potential barrier values (from 0.0034 eV to 0.017 eV). This is explained by the fact that the trapping levels become pretty full and the rate of free carriers increases, inducing a barrier reduction. Also, when the layers have the same doping concen- tration, the density states of traps, the critical concentra- tion and the corresponding potential barrier are inversely proportional to the increasing grain size (see Table 3). The spacing between the thermionic current density curves becomes small with the grain growth, which is also related to the decreasing difference between the cal- culated potential barrier values. This may be due to the reduced number and width of the grain boundaries in the layers. Hence, increased grain size and doping concentration help to reduce the number of grain boundaries, resulting in a reduced potential barrier and improved transport properties. These results are in good agreement with the other research findings published in the literature. Acknowledgement This work was supported by the Algerian Ministry of Higher Education and Scientific Research. We are in- debted to all the members of the LER Laboratory of the Jijel University for their services and assistance, as well as to our research teams in Jijel and Constantine. 5 REFERENCES 1 B. Olyaeefar, S. Ahmadi-Kandjani, A. Asgari, Classical modelling of grain size and boundary effects in polycrystalline perovskite solar cells, J. Solar Energy Materials and Solar Cells, 180 (2018), 76–82, doi:10.1016/j.solmat.2018.02.026 2 S. Park, M. A. Shehzad, M. F. Khan, G. Nazir, J. Eom, H. Noh, Y. Seo, Effect of grain boundaries on electrical properties of poly- crystalline graphene, J. Carbon, 112 (2017), 142–148, doi:10.1016/ j.carbon.2016.11.010 3 M. Hogyoku, T. Izumida, H. Tanimoto, N. Aoki, S. Onoue, Grain-boundary-limited carrier mobility in polycrystalline silicon with negative temperature dependence: modeling carrier conduction through grain-boundary traps based on trap-assisted tunneling, Jpn. J. Appl. Phys., 58 (2019), SBBA01, doi:10.7567/1347-4065/aaf7fa 4 M. V. Frischbier, H. F. Wardenga, M. Weidner, O. Bierwagen, J. Jia, Y. Shigesato, A. Klein, Influence of dopant species and concentration on grain boundary scattering in degenerately doped In2O3 thin films, J. Thin Solid Films, Part B, 614 (2016), 62–68, doi:10.1016/j.tsf. 2016.03.022 5 L. Carnel, I. Gordon, K. Van Nieuwenhuysen, D. Van Gestel, G. Beaucarne, J. Poortmans, Defect passivation in chemical vapour de- posited fine-grained polycrystalline silicon by plasma hydrogenation, J. Thin Solid Films, 487 (2005), 147–151, doi:10.1016/j.tsf.2005. 01.081 6 S. Merabet, M. Boukezzata, Analysis of boron profiles as function of nitrogen content in silicon, J. Thin Solid Films, 690 (2019), 137537, doi:10.1016/j.tsf.2019.137537 7 Y. Zhao, Q. Song, H. Ji, W. Cai, Z. Liu, Y. Cai, Multi-scale modeling method for polycrystalline materials considering grain boundary misorientation angle, Materials & Design, 221 (2022), 110998, doi:10.1016/j.matdes.2022.110998 8 K. Sharma, Study of photovoltaic properties of silicon solar cell and their dependence on grain boundaries, J. Advanced Research in En- gineering and Technology, 11 (2020) 9, 1112–1119, doi:10.34218/ ijaret.11.9.2020.111 9 A. Abass, B. Maes, D. Van Gestel, K. Van Wichelen, M. Burgelman, Effects of inhomogeneous grain size distribution in polycrystalline silicon solar cells, Energy Proc., 10 (2011), 55–60, doi:10.1016/ j.egypro.2011.10.152 10 A. Abass, D. Van Gestel, K. Van Wichelen, B. Maes, M. Burgelman, On the diffusion length and grain size homogeneity requirements for efficient thin-film polycrystalline silicon solar cells, J. Phys. D: Appl. Phys., 46 (2012), 045105, doi:10.1088/0022-3727/46/4/045105 11 M. Burgelman, P. Nollet, S. Degrave, Modelling polycrystalline semiconductor solar cells, Thin Solid Films, 361 (2000), 527–532, doi:10.1016/S0040-6090(99)00825-1 12 Y. Yanfa, Y. Wan-Jian, W. Yelong, S. Tingting, P. R. Naba, L. Chen, J. Poplawsky, Z. Wang, J. Moseley, H. Guthrey, H. Moutinho, S. J. Pennycook, M. M. Al-Jassim, Physics of grain boundaries in polycrystalline photovoltaic semiconductors, J. Appl. Phys., 117 (2015) 112807, doi:10.1063/1.4913833 13 B. Gaury, P. M. Haney, Charged grain boundaries and carrier recom- bination in polycrystalline thin film solar cells, Physical Review Ap- plied, 8 (2017), 054026, doi:10.1103/PhysRevApplied.8.054026 14 R. Gegevicius, M. Franckevicius, V. Gulbinas, The role of grain boundaries in charge carrier dynamics in polycrystalline metal halide perovskites, Eur. J. Inor. Chem., (2021) 35, 3519–3527, doi:10.1002/ ejic.202100360 15 C. Persson, A. Zunger, Anomalous grain boundary physics in polycrystalline CuInSe2: the existence of a hole barrier, Physical Re- view Letters, 91 (2003) 26, 266401, doi:10.1103/PhysRevLett.91. 266401 16 R. Herberholz, U. Rau, H. W. Schock, T. Haalboom, T. Gödecke, F. Ernst, C. Beilharz, K. W. Benz, D. Cahen, Phase segregation, Cu mi- gration and junction formation in Cu(In,Ga)Se2, Eur. Phys. J. Appl. Phys., 6 (1999) 2, 131–139, doi:10.1051/epjap:1999162 17 A. Niemegeers, M. Burgelman, R. Herberholz, U. Rau, D. Hariskos, Model for electronic transport in Cu(In,Ga)Se2 solar cells, Prog. Photovolt. Res. Appl., 6 (1998) 6, 407–421, doi:10.1002/ (SICI)1099-159X(199811/12)6:6<407::AID PIP230>3.0.CO;2-U 18 M. J. Romero, K. Ramanathan, M. A. Contreras, M. M. Al-Jassim, R. Noufi, P. Sheldon, Cathodoluminescence of Cu(In,Ga)Se2 thin films used in high-efficiency solar cells, Appl. Phys. Lett., 83 (2003), 4770, doi:10.1063/1.1631083 19 J. Y. W. Seto, The electrical properties of polycrystalline silicon films, Journal of Applied Physics, 46 (1975) 12, 5247–5254, doi:10.1063/1.321593 20 T. Kamins, Polycrystalline silicon for integrated circuits applications, Stanford University, 1988, Kluwer Academic Publishers S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD 8 Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 21 E. Canessa, V. L. Nguyen, Non-linear I–V characteristics of double Schottky barriers and polycrystalline semiconductors, Physica B: Condensed Matter, 179 (1992) 4, 335–341, doi:10.1016/0921- 4526(92)90634-5 22 F. Greuter, G. Blatter, Electrical properties of grain boundaries in polycrystalline compound semiconductors, Semicond. Sci. Technol,. 5( 1990) 2, 111, doi:10.1088/0268-1242/5/2/001 23 S. Tsurekawa, K. Kido, T. Watanabe, Measurements of potential bar- rier height of grain boundaries in polycrystalline silicon by Kelvin probe force microscopy, J. Philosophical Magazine Letters, 85 (2005) 1, 41–49, doi:10.1080/09500830500153859 24 N. Gupta, B. P. Tyagi, An analytical model of the influence of grain size on the mobility and transfer characteristics of polysilicon thin-film transistors (TFTs), J. Physica Scripta, 71 (2006)2 , 225–228, doi:10.1238/Physica.Regular.071a00225 25 J. P. Colinge, E. Demoulin, F. Delannay, M. Lobet, J. M. Temerson, Grain size and resistivity of LPCVD polycrystalline silicon films, J. Electrochem. Soc., 128 (1981), 2009–2014, doi:10.1149/1.2127785 26 B-H. Yan, B. Li, R-H. Yao, W-J. Wu, A physics-based effective mo- bility model for polycrystalline silicon thin film transistor consider- ing discontinuous energy band at grain boundaries, Jpn. J. Appl. Phys., 50 (2011), 094302, doi:10.1143/jjap.50.094302 27 J. G. Lee, T. W. Kim, Effects of the grain boundary and interface traps on the electrical characteristics of 3D NAND flash memory de- vices, J. Nanosci. Nanotechnol., 18 (2018) 3, 1944–1947, doi:10.1166/jnn.2018.15000 28 G. Baccarani, B. Riccò, G. Spadini, Transport properties of polycrystalline silicon films, J. Appl. Phys., 49 (1978), 5565–5570, doi:10.1063/1.324477 29 T. Takagi, F. Koyama, K. Iga, Potential barrier height analysis of AlGaInP multi-quantum barrier (MQB), Jpn. J. Appl. Phys., 29 (1990), L1977, doi:10.1143/JJAP.29.L1977 30 H. Dong, J. Sun, S. Ma, J. Liang, T. Lu, Z. Jia, X. Liu, B. Xu, Effect of potential barrier height on the carrier transport in InGaAs/GaAsP multi-quantum wells and photoelectric properties of laser diode, J. Phys. Chem. Chem. Phys., 18 (2016) 9, 6901–6912, doi:10.1039/ C5CP07805A 31 A. Shamir, I. Amit, D. Englander, D. Horvitz, Y. Rosenwaks, Poten- tial barrier height at the grain boundaries of a poly-silicon nanowire, Nanotechnology, 26 (2015) 35, 355201, doi:10.1088/0957-4484/26/ 35/355201 32 S. D. Brotherton, Introduction to thin film transistors, Chapter 8, Poly-Si TFT Performance, Springer International Publishing, 2013, Switzerland 33 A. T. Hatzopoulos, D. H. Tassis, N. A. Hastas, C. A. Dimitriadis, G. Kamarinos, On-state drain current modeling of large-grain poly-Si TFTs based on carrier transport through latitudinal and longitudinal grain boundaries, IEEE Transactions on Electron Devices, 52 (2005) 8, 1727–1733, doi:10.1109/ted.2005.852732 34 H. Ikeda, Analysis of grain boundary induced nonlinear output char- acteristics in polycrystalline-silicon thin-film transistors, Jpn. J. Appl. Phys.,45( 2006) 3R, 1540, doi:10.1143/JJAP.45.1540 35 M. Wang, M. Wong, An effective channel mobility-based analytical on-current model for polycrystalline silicon thin-film transistors, IEEE Transactions on Electron Devices, 54 (2007) 4, 869–874, doi:10.1109/ted.2007.891248 36 M. Wong, T. Chow, C. C. Wong, D. Zhang, A quasi two-dimensional conduction model for polycrystalline silicon thin-film transistor based on discrete grains, IEEE Transactions on Electron Devices, 55 (2008) 8, 2148–2156, doi:10.1109/ted.2008.926277 37 A. Valletta, P. Gaucci, L. Mariucci, A. Pecora, M. Cuscunà, L. Maiolo, G. Fortunato, Threshold voltage in short channel poly- crystalline silicon thin film transistors: Influence of drain induced barrier lowering and floating body effects, J. Appl. Phys, 107 (2010) 7, 074505, doi:10.1063/1.3359649 38 B. Du, C. Han, Z. Li, C. Han, J. Li, M. Xiao, Z. Yang, Effect of po- larity-reversal voltage on charge accumulation and carrier mobility in silicone rubber/silicon carbide composites, IET Sci. Meas. Technol., 15 (2021), 184–192, doi:10.1049/smt2.12020 S. MERABET, B. DJELLIL: POTENTIAL BARRIER ESTIMATION WITH A GRAPHICAL METHOD Materiali in tehnologije / Materials and technology 57 (2023) 1, 3–9 9