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Drift-Diffusion Versus Monte Carlo Simulated ON-Current Variability in Nanowire FETs

Drift-Diffusion Versus Monte Carlo Simulated ON-Current Variability in Nanowire FETs Received December 14, 2018, accepted December 28, 2018, date of publication January 14, 2019, date of current version February 6, 2019. Digital Object Identifier 10.1109/ACCESS.2019.2892592 Drift-Diffusion Versus Monte Carlo Simulated ON-Current Variability in Nanowire FETs 1 1,2 1 DANIEL NAGY , GUILLERMO INDALECIO , ANTONIO J. GARCÍA-LOUREIRO , 1 3,4 GABRIEL ESPIÑEIRA , MUHAMMAD A. ELMESSARY , 3 1 KAROL KALNA , AND NATALIA SEOANE Centro Singular de Investigación en Tecnoloxías da Información, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain Institute for Microelectronics, TU Wien, 1040 Vienna, Austria Nanoelectronic Devices Computational Group, Swansea University, Swansea SA1 8EN, U.K. Department of Engineering Mathematics and Physics, Mansoura University, Mansoura 35516, Egypt Corresponding author: Daniel Nagy ([email protected]) This work was supported in part by the Spanish Government under Project TIN2013-41129-P and Project TIN2016-76373-P, in part by Xunta de Galicia and FEDER funds under Grant GRC 2014/008, in part by the Consellería de Cultura, Educación e Ordenación Universitaria (accreditation 2016-2019), under Grant ED431G/08, and in part by the Centro de Supercomputación de Galicia (CESGA) for the computer resources provided. The work of G. Indalecio was supported by the Programa de Axudas á Etapa Posdoutoral da Xunta de Galicia under Grant 2017/077. The work of N. Seoane was supported by the RyC program of the Spanish Ministerio de Ciencia, Innovación y Universidades under Grant RYC-2017-23312. ABSTRACT Variability of semiconductor devices is seriously limiting their performance at nanoscale. The impact of variability can be accurately and effectively predicted by computer-aided simulations in order to aid future device designs. Quantum corrected (QC) drift-diffusion (DD) simulations are usually employed to estimate the variability of state-of-the-art non-planar devices but require meticulous calibration. More accurate simulation methods, such as QC Monte Carlo (MC), are considered time consuming and elaborate. Therefore, we predict TiN metal gate work-function granularity (MGG) and line edge roughness (LER) induced variability on a 10-nm gate length gate-all-around Si nanowire FET and perform a rigorous comparison of the QC DD and MC results. In case of the MGG, we have found that the QC DD predicted variability can have a difference of up to 20% in comparison with the QC MC predicted one. In case of the LER, we demonstrate that the QC DD can overestimate the QC MC simulation produced variability by a signicant error of up to 56%. This error between the simulation methods will vary with the root mean square (RMS) height and maximum source/drain n-type doping. Our results indicate that the aforementioned QC DD simulation technique yields inaccurate results for the ON-current variability. INDEX TERMS Drift-diffusion, line edge roughness, metal gate granularity, Monte Carlo, quantum corrections, nanowire FET. I. INTRODUCTION metal gate work-function granularity (MGG), and line edge Gate-All-Around (GAA) nanowires (NWs) are showing roughness (LER) [7][13]. Therefore, a rigorous study of arguable promise to be the leading architecture for future all aspects of device performance, including their resistance technological nodes adopted by industry [1][5], due to their against variability sources [3], [4], [14], [15], is critical. This superior electrostatic control of the channel, thus allow- study is often carried out using computer aided design tools ing further scaling of the gate length in comparison with because they are proven to be an economically efcient way the currently used Fin Field-Effect Transistor (FinFET) to do the ground work [15][19]. However, choosing the architecture [6]. However, the devices in the deep nano- right simulation tool without appropriate in-sight can be a regime suffer from various sources of variability which could cumbersome task. greatly affect their performance and yield [7][9]. These Generally, three methods are commonly used for sources of variability are related to either the fabrication nanoscaled device simulations [16], [17]: (i) quantum cor- process or material properties. The most signicant sources rected (QC) drift-diffusion (DD), (ii) QC Monte Carlo (MC) are: random dopants (RD), oxide thickness variation (OTV), and (iii) fully quantum-mechanical Non-Equilibrium Green's 12790 This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/ VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs Functions (NEGF). The later is the most accurate but also the simulator are found in [33]. The MC toolbox accounts for all most computer intensive method that is generally used for relevant electron scattering mechanisms in the silicon transis- ultra small nanoscale transistors in which quantum effects are tor: acoustic and non-polar optical phonons (intra- and inter- expected to be signicant [16], [17], [20]. Therefore, the use valley) [37], [38], ionised impurity scattering using the third of NEGF for statistically signicant variability studies, where body exclusion by Ridley [39], [40], and interface roughness hundreds of simulations are required, is computationally (IR) scattering using Ando's model [41]. The electron screen- prohibitive. The QC MC method is commonly employed ing in the electron-ionised impurity scattering uses a static for the investigations of the device ON-region where carrier screening model [42] with Fermi-Dirac statistics in which the scattering and non-equilibrium transport play an important Fermi energy and electron temperature are calculated self- role [16], [17], [20]. An advantage of the MC over the NEGF consistently in a real space of device simulation domain. is that the implementation of multiple scattering mechanisms We have already argued that quantum connement effects into the MC simulator is less complex in comparison with will play a signicant role in transport at nanoscale dimen- the NEGF method. Finally, the QC DD is the least compu- sions. Therefore, we use 3D density-gradient (DG) QCs in tationally expensive method and often used for variability the DD simulations and 2D Schrödinger based equation cor- studies in the sub-threshold region [3], [15], [16], [21], rections (SCH) in the MC. The former has the disadvantage that involve simulations of thousands of individual devices. that it requires tting against the MC data, as aforesaid, mean- In our case, the QC DD method takes about three times while the later QC approach is calibration free. In case of the less computational time than the QC MC method. However, DG method, we use electron effective masses as calibration the QC DD is disadvantaged by a requirement to calibrate parameters to account for the quantum capacitance (shift of QC parameters against either MC, NEGF or experimental the threshold voltage). The tting parameters used with the data [3], [21], [22]. It was previously shown that QC DD DG method are found in [43]. More details about the QC DD is unable to perform ON-current variability study for planar simulation methodologies can be found in [44] and [45] and MOSFETs without an underestimation because the QC DD about the QC MC in [32], [46], and [47]. cannot capture non-equilibrium effects [21], [23]. A sim- Finally, the in-house simulation toolbox can account for the ilar rigorous study for non-planar multi-gate transistors is following sources of variability: OTV, MGG, LER, gate edge missing from the literature. More importantly, the QC DD roughness (GER), and RD [31]. In this work, we will focus method is still being used in state-of-the-art device variability on the two most inuential ones [7] for GAA NW FET: MGG study [15], [18], [22], [24][28] believing that properly cal- and LER as illustrated in Fig. 2. Note that the same random ibrated QC DD simulations will yield to accurate statistical proles are used in both simulation techniques, the QC DD predictions. and QC MC, for a fair comparison of each variability study. In this paper, we aim to establish how accurate the QC DD Moreover, the QC DD calibration parameters are not adjusted method is when applied to the ON-region variability in com- for each of the proles but use the values calibrated for parison with the more rigorous QC MC simulation technique. the ideal device as this is the standard approach. The 2D We compare the results obtained by applying two of the main Schrödinger equation in the QC MC simulations is solved variability sources affecting the device reliability, the MGG for each random prole of a device as this method does not and LER, on a state-of-the-art 10 nm gate length Si GAA require additional calibration. NW FET that has been scaled down from an experimental In case of the MGG variability, we use the Poisson- device [29], [30]. Voronoi diagrams approach [48] to create the metal grains for the metal gate contact of the simulated device. This II. METHODOLOGY AND DEVICE DESCRIPTION method is believed to mimic more accurately the realistic In this work, we employ a well established in-house sim- metal gates [48] than the square grains approach [49], [50]. ulation toolbox [31][33] that includes 3D DD and MC Furthermore, the MGG prole is characterized by a grain transport models which use the nite element (FE) method size (GS) and by a work function value (WFV) [48]. For for accurate mesh description of a simulation domain. The the current study, we have chosen the titanium nitride (TiN) accurate description of the device nanoscale dimensions is of which is commonly used as a gate material [51]. The metal great importance for accurate simulations in the deep nano- has experimentally observed WFVs of 4:6 eV and 4:4 eV with regime because quantum-mechanical connement in a device a probability of 60% and 40% formation, respectively [52]. channel can signicantly affect transport at nanoscale [34]. In case of the LER variability, we create the uncorre- As mentioned before, the DD approach requires calibra- lated proles using Fourier synthesis with Gaussian auto- tion for the simulations. In this study, we use the readily correlation approach [53]. These are characterized by the available MC simulation toolbox to guide the calibration correlation length (CL) and the root mean square (RMS) of the QC DD. The model used by the DD simulator is values [43], [53]. The current study is limited to a CL of the Caughey-Thomas doping dependent low-eld electron 20 nm and to experimentally observed RMS heights, ranging mobility model [35], together with perpendicular (critical between 0:3 and 1:0 nm [11], [29]. electric eld) and lateral (saturation velocity) electric eld A device used in this study is based on a 10 nm gate models [36]. The calibration parameters used with the DD length GAA NW FET that was scaled down from an exper- VOLUME 7, 2019 12791 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 1. (i) Schematic for the 10 nm gate length GAA NW [30] and (ii) Gaussian doping profiles along the transport direction for three FIGURE 3. Simulated I -V characteristics for the 10 nm gate length GAA D G concentrations of N . (right) Cross-sectional view of the channel for the NW [30] at V D 0:7 V with a channel orientation of h110i. Three different (b) ideal device and two cases when the Fin height was (a) elongated and 19 20 doping concentrations are presented for N : 5 10 , 1 10 and (c) shortened. 20 3 1:5 10 cm . Full lines correspond to 3D QC MC simulations, while dashed lines refer to calibrated (against the QC MC) 3D QC DD simulations. the QC DD is achieved by adjusting the mobility model and QC parameters as described in detail in [43]. To assess the validity of the calibration for the QC DD simulator, two extreme cases of channel height for the NW were chosen as shown in Fig. 1(a) and (c). In each case, the height is increased/decreased symmetrically by 1 nm for an N of 20 3 FIGURE 2. Schematic for the 10 nm gate length GAA NW [30] affected by 1:5  10 cm , without changing any of the calibration LER and MGG variability sources. The LER profile is projected along the parameters. It was found that the QC DD results produce a transport direction (x-axis) and affects the dimension of only the z-axis. The MGG profile with different work function is projected to the gate negligible error, up to 3 %, for both modied devices when area [52]. compared to the results obtained from the QC MC. IV. MGG VARIABILITY imental device [29] following the ITRS [54] guidelines as We have generated 300 random proles with GSs of 3, 5 shown in [30]. The device schematic and dimensions are and 7 nm [52] for a meaningful statistical study of the MGG shown in Fig. 1(i). It has a uniformly p-type doped channel induced variability. These proles were also applied to three 15 3 (110 cm ), a Gaussian n-type doping, with a maximum maximum doping concentrations N to extensively inves- N (see Fig. 1(ii)) and a lateral straggle () of 3:23 nm, and tigate the capabilities of the QC DD and QC MC models. an EOT of 0:8 nm. Finally, it has an elliptical channel cross- Note that the same MGG proles are used in both simulation section with dimensions of 7:17 nm and 5:7 nm as shown techniques, the QC DD and QC MC, for a fair comparison. in Fig. 1(b). III. IDEAL GAA NW FET Even though GAA NWs are considered to be major con- tenders for future technology nodes, they might be unable to deliver a large enough ON-current (I ) [33], [55] in circuits, ON which may be one of the main limiting factors for the adapta- tion of the technology. One way to overcome this issue could be by increasing the maximum N of the S/D region. For this reason we have increased the reversed engineered n-type doping concentration of N from 5  10 that provided a perfect match to the experimental I-V curve [30] to 1 10 FIGURE 4. I due to MGG vs N from the QC DD and the QC MC 20 3 ON D and to 1:5 10 cm . Note that the  was kept constant as simulations using 300 profiles. The difference between the QC DD and QC shown in Fig. 1(ii). We have found that, compared to N D MC simulation results are indicated in percentage. 19 3 5 10 cm , I has increased by 40 % and 60 % for N ON D 20 20 3 of 1 10 and 1:5 10 cm , respectively. Note that I Fig. 4 shows the standard deviation () of the I against ON ON is I at V D V CV , where V is the threshold voltage the maximum N . Both simulation methods show an increas- D G DD T T D and V D 0:7 V. Both the QC MC and the well calibrated ing I with an increasing N . However, the difference DD ON D QC DD simulated I -V characteristics are shown in Fig. 3 between the I (indicated by percentage in the gure) D G ON for the aforementioned cases. Note that the calibration of predicted by both simulation methods is dependent on both 12792 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 5. Scatter plots compare the simulations with 20 3 20 3 19 3 N D 1:5 10 cm and 1 10 cm against N D 5 10 cm D D obtained from (a) QC DD and (b) QC MC. The GS is 7 nm. FIGURE 7. The schematic of the GAA NW FET gate area (a) with a single synthetic profile strip wrapped around the gate. The FSM for the I are ON simulated assuming n-type source/drain concentration (N ) of 20 3 1 10 cm using (a) QC DD and (b) QC MC techniques. 100 synthetic gate profiles with a width of 0:1 nm are simulated. around the gate (see example in Fig. 7(a)), (ii) this prole is then swept along the transport direction and the prole related to I is extracted, and (iii) all the simulated proles and their ON corresponding I are used to create a 2D FSM as shown ON 20 3 FIGURE 6. I due to MGG vs GS for a N D 1 10 cm obtained ON D in Figs. 7(b) and (c) for the QC DD and QC MC simulations, from the QC DD and the QC MC simulations. The difference between QC respectively. DD and QC MC are indicated in percentage. Thanks to the FSM technique, we are able to identify that for a 10 nm gate length GAA NW the most sensitive region of the gate is away from the centre of the gate, close to the gate- the doping value and the grain size. For instance, for a N of 20 3 source junction. However, for the QC MC the maximum value 1 10 cm , the error in the predicted values by QC DD is centered at around 1:8 nm while the QC DD predicts when compared to QC MC ones range from 7 % (7 nm GS) the maximum value at around 1:2 nm. Moreover, the QC to 19 % (3 nm GS). Fig. 5 compares I at N D 5 ON D 19 20 20 3 DD predicts the highest sensitive effective area to be smaller 10 against I at N D 1  10 and 1:5  10 cm ON D than that shown by the QC MC results. Thus, we know that a obtained from the (a) QC DD and (b) QC MC simulations. change in the WFV in the aforementioned region will play a There is a large correlation, as indicated by the correlation signicant role in the I values. ON coefcients (CCs), between the I values produced by both ON simulation methods. This means that the same proles pro- duce a similar variability even when the N is increased. V. LER VARIABILITY Finally, investigation of the effect of the GS is shown in Fig. 6. Section III has shown that the QC DD calibrated to the QC Both simulation methods predict an increasing I with MC simulations can predict the same I for the NW FET. ON ON an increasing GS. However, the QC DD method leads to an This ability has important implications for a LER induced overestimation of the MGG variability of around 20 % for variability study because the LER causes a uctuation in the GSs equal or lower than 5 nm. Furthermore, analysis of the channel dimension along the transport direction. However, mean (1) I showed a negligible difference between the QC what is the accuracy of the QC DD produced variability when ON DD and QC MC methods. the channel cross-section dimension uctuates? To answer A Fluctuation Sensitivity Map (FSM) [56] that analyzes this question, we generate 300 random LER proles assum- the spatial effect of the MGG variability in key gure of ing a correlation length (CL) of 20 nm for four experimentally merits (FoMs) (e.g. I ) is employed in order to reveal the observed RMS heights [11], [29], [30] and three maximum ON most sensitive regions of the studied device to the MGG. doping concentrations N . The same LER proles are used The procedure is as follows: (i) a single synthetic prole is for both simulation techniques, the QC DD and QC MC, for created, which has a WFV localized in a small strip wrapped a fair comparison. VOLUME 7, 2019 12793 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 8. I due to LER vs N from the QC DD and the QC MC ON D simulations using 300 profiles. The LER characteristic values are: CL D 20 nm and RMS heights of 1:0 and 0:5 nm. The difference between the QC DD and QC MC simulation results are indicated in percentage. FIGURE 9. Scatter plots compare the simulations with 20 3 20 3 19 3 N D 1:5 10 cm and 1 10 cm against N D 5 10 cm D D obtained from (a) QC DD and (b) QC MC simulations, respectively. The Fig. 8 shows the standard deviation () of the I against ON RMS height is 1 nm. the maximum N . The predicted I by the QC DD and D ON QC MC simulation techniques has very similar values, with a difference of up to 7 %, for the devices with a N of 19 3 5 10 cm . Yet, the error in the estimation given by the QC DD simulations increases with N reaching a staggering 56% difference when compared to the results from QC MC 20 3 simulations for a N of 1:5 10 cm . Finally, note that I is practically constant with dependence on N when ON D obtained from the QC MC simulations, whereas the QC DD results predicts an increasing I with N . Note that the ON D difference in the predicted behaviour lays in the implemen- tation of quantum correction methods as well as the different models, classical DD vs. semi-classical MC. The Schrödinger 20 3 FIGURE 10. I due to LER vs RMS height for a N D 1 10 cm ON D based quantum corrections in the QC MC simulations are obtained from the QC DD and the QC MC simulations. The difference between QC DD and QC MC are indicated in percentage. able to accurately capture the physics when some modica- tion in the device architecture occurs, for example, doping, LER, MGG, etc. However, the simulation approach using density gradient quantum corrections would require adjusting the calibration parameters for each of the aforementioned modications against a more complex simulation model. Furthermore, the MC method accounts for non-equilibrium electron transport as well as the inclusion of the important scattering models, which the DD model is not capable of. Further analysis of this behaviour is shown in Fig. 9 that 19 20 compares I at N D 510 against I at N D 110 ON D ON D 20 3 and 1:510 cm . The correlation between the I values ON produced by the LER proles from the QC DD simulations (Fig. 9(a)) is lower than for the QC MC ones (Fig. 9(b)) as indicated by the correlation coefcients (CCs). Finally, observe that the regression lines (red lines in Fig. 9) are FIGURE 11. The GAA NW FET schematic (a) is scaled to the I FSM (b). ON shifted by a constant value for the QC MC obtained results 100 synthetic profiles with a width deformation are simulated for N of 20 3 and yet, for the QC DD ones, they also change the slope. The 1:5 10 cm using QC DD and QC MC techniques as indicated. investigation of the effect of RMS height is shown in Fig. 10. The QC DD results give up to 22 % overestimation of a predicted I from the QC MC simulations. Additional I . The procedure is similar to the one used for the MGG ON ON analysis of the 1I showed a negligible difference between variability: (i) a single synthetic prole is created, which has ON the QC DD and QC MC methods. a Gaussian vertical deformation localized in a small region FSM [43] introduced in Section IV is also used to ana- of the device (see Fig. 11(a)), (ii) the prole is then swept lyze the spatial effect of the LER variability induced by along the transport direction and a prole related to I is ON 12794 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs extracted, and (iii) each prole and the corresponding I are ity studies that involve the variation of the channel cross- ON used to create a 1D FSM as shown in Fig. 11(b). Note that section in the ON-region regardless their calibration against a synthetic deformation for the LER can lead to an increase reliable data. This is because the QC DD method has (negative sensitivity) or decrease (positive sensitivity) of the xed calibration parameters which are ``device dimension I . Therefore, the normalized scale from 1 to 1 is used. specic'' while the QC MC uses the calibration free 2D ON Fig. 11 shows that the QC MC technique predicts the most Schrödinger equation to account for the actual quantum- sensitive regions to the LER variability closer to the source- mechanical connement effect. gate junction than the locations predicted by the QC DD technique. Notice that there is not only a shift between the QC ACKNOWLEDGMENT DD and QC MC largest absolute sensitive areas, but also the The authors would like to thank Centro de Supercomputación magnitude of the sensitivity is different. Finally, we can say de Galicia (CESGA) for the computer resources provided. that if a change in the diameter of a NW FET occurs near the middle of the gate or around the source-gate junction, it will REFERENCES heavily impact the I , as shown by the FSM. However, ON [1] O. Badami, F. Driussi, P. Palestri, L. Selmi, and D. Esseni, ``Performance changes in other parts of the NW FET dimensions will only comparison for FinFETs, nanowire and stacked nanowires FETs: Focus on the inuence of surface roughness and thermal effects,'' in IEDM Tech. have a negligible inuence in the I . ON Dig., Dec. 2017, pp. 13.2.113.2.4. [2] M. Li et al., ``Sub-10 nm gate-all-around CMOS nanowire transistors on bulk Si substrate,'' in Proc. Symp. VLSI Technol., Jun. 2009, pp. 9495. VI. CONCLUSION [3] K. Nayak, S. Agarwal, M. Bajaj, K. V. R. M. Murali, and V. R. 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Nagy, G. Indalecio, A. J. García-Loureiro, M. A. Elmessary, K. Kalna, device behaviour under metal grain work-function variability effects,'' and N. Seoane, ``FinFET versus gate-all-around nanowire FET: Perfor- IEEE Trans. Electron Devices, vol. 64, no. 4, pp. 16951701, Apr. 2017. mance, scaling, and variability,'' IEEE J. Electron Devices Soc., vol. 6, pp. 332340, Feb. 2018. [34] M. Stadele et al., ``A comprehensive study of corner effects in tri-gate transistors,'' in Proc. Eur. Solid-State Device Res. Conf. (ESSDERC), Sep. 2004, pp. 165168. [35] R. E. Thomas, ``Carrier mobilities in silicon empirically related to doping DANIEL NAGY received the M.Res. degree in and eld,'' Proc. IEEE, vol. 55, no. 12, pp. 21922193, Dec. 1967. nanoscience to nanotechnology and the Ph.D. [36] K. Yamaguchi, ``Field-dependent mobility model for two-dimensional degree in electronic and electrical engineering numerical analysis of MOSFET's,'' IEEE Trans. Electron Devices, from Swansea University, Swansea, U.K., in 2013 vol. ED-26, no. 7, pp. 10681074, Jul. 1979. and 2016, respectively. [37] K. Tomizawa, Numerical Simulation of Submicron Semiconductor Devices He currently holds a Postdoctoral position with (Artech House Materials Science Library). Norwood, MA, USA: the Centro Singular de Investigación en Tec- Artech House, 1993. [38] C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor noloxías da Información, University of Santiago Device Simulation (Computational Microelectronics). Vienna, Austria: de Compostela, Santiago de Compostela, Spain Springer, 2012. 12796 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs GUILLERMO INDALECIO received the B.S. MUHAMMAD A. ELMESSARY received the degree in physics and the Ph.D. degree in semicon- B.Sc. degree (Hons.) in computer and systems ductor device simulation from the University of engineering and the M.Sc. degree in engineer- Santiago de Compostela, Santiago de Compostela, ing physics from Mansoura University, Mansoura, Spain, in 2010 and 2016, respectively. Egypt, in 2004 and 2010, respectively, and the He was a Visiting Researcher with the Uni- Ph.D. degree in semiconductor device simulation versity of Swansea, Swansea, U.K., in 2015. from Swansea University, Swansea, U.K., in 2017. His current research interests include electronic He is currently a Research Assistant with devices simulation with a focus on computational Swansea University. techniques and novel techniques to understand variability sources. KAROL KALNA received the M.Sc. (Hons.) ANTONIO J. GARCÍA-LOUREIRO received the and Ph.D. degrees from Comenius University, Ph.D. degree from the University of Santiago Bratislava, Czechoslovakia, in 1990 and 1998, de Compostela, Santiago de Compostela, Spain, respectively. in 1999. He is currently an Associate Professor leading He is currently an Associate Professor with the Nanoelectronic Devices Computational Group, the Department of Electronics and Computer Sci- Swansea University, Swansea, U.K. He has held ence, University of Santiago de Compostela. His the EPSRC Advanced Research Fellowship and current research interests include the multidimen- has pioneered IIIV MOSFETs, since 2002. He sional simulations of nanoscale transistors and has published 93 peer-review papers and has given solar cells. over 20 invited talks. NATALIA SEOANE received the Ph.D. degree from the University of Santiago de Compostela, GABRIEL ESPIÑEIRA received the B.S. degree in Santiago de Compostela, Spain, in 2007. physics from the University of Santiago de Com- She was a Visiting Postdoctoral Researcher postela, Santiago de Compostela, Spain, in 2018, with the University of Glasgow, Glasgow, U.K., where he is currently pursuing the M.Res. degree from 2007 to 2009, The University of Edinburgh, in HPC and also with the Centro Singular de Inves- Edinburgh, U.K., in 2011, and Swansea Univer- tigación en Tecnoloxías da Información. sity, Swansea, U.K., from 2013 to 2015. She is currently with the University of Santiago de Compostela. VOLUME 7, 2019 12797 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IEEE Access Unpaywall

Drift-Diffusion Versus Monte Carlo Simulated ON-Current Variability in Nanowire FETs

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Received December 14, 2018, accepted December 28, 2018, date of publication January 14, 2019, date of current version February 6, 2019. Digital Object Identifier 10.1109/ACCESS.2019.2892592 Drift-Diffusion Versus Monte Carlo Simulated ON-Current Variability in Nanowire FETs 1 1,2 1 DANIEL NAGY , GUILLERMO INDALECIO , ANTONIO J. GARCÍA-LOUREIRO , 1 3,4 GABRIEL ESPIÑEIRA , MUHAMMAD A. ELMESSARY , 3 1 KAROL KALNA , AND NATALIA SEOANE Centro Singular de Investigación en Tecnoloxías da Información, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain Institute for Microelectronics, TU Wien, 1040 Vienna, Austria Nanoelectronic Devices Computational Group, Swansea University, Swansea SA1 8EN, U.K. Department of Engineering Mathematics and Physics, Mansoura University, Mansoura 35516, Egypt Corresponding author: Daniel Nagy ([email protected]) This work was supported in part by the Spanish Government under Project TIN2013-41129-P and Project TIN2016-76373-P, in part by Xunta de Galicia and FEDER funds under Grant GRC 2014/008, in part by the Consellería de Cultura, Educación e Ordenación Universitaria (accreditation 2016-2019), under Grant ED431G/08, and in part by the Centro de Supercomputación de Galicia (CESGA) for the computer resources provided. The work of G. Indalecio was supported by the Programa de Axudas á Etapa Posdoutoral da Xunta de Galicia under Grant 2017/077. The work of N. Seoane was supported by the RyC program of the Spanish Ministerio de Ciencia, Innovación y Universidades under Grant RYC-2017-23312. ABSTRACT Variability of semiconductor devices is seriously limiting their performance at nanoscale. The impact of variability can be accurately and effectively predicted by computer-aided simulations in order to aid future device designs. Quantum corrected (QC) drift-diffusion (DD) simulations are usually employed to estimate the variability of state-of-the-art non-planar devices but require meticulous calibration. More accurate simulation methods, such as QC Monte Carlo (MC), are considered time consuming and elaborate. Therefore, we predict TiN metal gate work-function granularity (MGG) and line edge roughness (LER) induced variability on a 10-nm gate length gate-all-around Si nanowire FET and perform a rigorous comparison of the QC DD and MC results. In case of the MGG, we have found that the QC DD predicted variability can have a difference of up to 20% in comparison with the QC MC predicted one. In case of the LER, we demonstrate that the QC DD can overestimate the QC MC simulation produced variability by a signicant error of up to 56%. This error between the simulation methods will vary with the root mean square (RMS) height and maximum source/drain n-type doping. Our results indicate that the aforementioned QC DD simulation technique yields inaccurate results for the ON-current variability. INDEX TERMS Drift-diffusion, line edge roughness, metal gate granularity, Monte Carlo, quantum corrections, nanowire FET. I. INTRODUCTION metal gate work-function granularity (MGG), and line edge Gate-All-Around (GAA) nanowires (NWs) are showing roughness (LER) [7][13]. Therefore, a rigorous study of arguable promise to be the leading architecture for future all aspects of device performance, including their resistance technological nodes adopted by industry [1][5], due to their against variability sources [3], [4], [14], [15], is critical. This superior electrostatic control of the channel, thus allow- study is often carried out using computer aided design tools ing further scaling of the gate length in comparison with because they are proven to be an economically efcient way the currently used Fin Field-Effect Transistor (FinFET) to do the ground work [15][19]. However, choosing the architecture [6]. However, the devices in the deep nano- right simulation tool without appropriate in-sight can be a regime suffer from various sources of variability which could cumbersome task. greatly affect their performance and yield [7][9]. These Generally, three methods are commonly used for sources of variability are related to either the fabrication nanoscaled device simulations [16], [17]: (i) quantum cor- process or material properties. The most signicant sources rected (QC) drift-diffusion (DD), (ii) QC Monte Carlo (MC) are: random dopants (RD), oxide thickness variation (OTV), and (iii) fully quantum-mechanical Non-Equilibrium Green's 12790 This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/ VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs Functions (NEGF). The later is the most accurate but also the simulator are found in [33]. The MC toolbox accounts for all most computer intensive method that is generally used for relevant electron scattering mechanisms in the silicon transis- ultra small nanoscale transistors in which quantum effects are tor: acoustic and non-polar optical phonons (intra- and inter- expected to be signicant [16], [17], [20]. Therefore, the use valley) [37], [38], ionised impurity scattering using the third of NEGF for statistically signicant variability studies, where body exclusion by Ridley [39], [40], and interface roughness hundreds of simulations are required, is computationally (IR) scattering using Ando's model [41]. The electron screen- prohibitive. The QC MC method is commonly employed ing in the electron-ionised impurity scattering uses a static for the investigations of the device ON-region where carrier screening model [42] with Fermi-Dirac statistics in which the scattering and non-equilibrium transport play an important Fermi energy and electron temperature are calculated self- role [16], [17], [20]. An advantage of the MC over the NEGF consistently in a real space of device simulation domain. is that the implementation of multiple scattering mechanisms We have already argued that quantum connement effects into the MC simulator is less complex in comparison with will play a signicant role in transport at nanoscale dimen- the NEGF method. Finally, the QC DD is the least compu- sions. Therefore, we use 3D density-gradient (DG) QCs in tationally expensive method and often used for variability the DD simulations and 2D Schrödinger based equation cor- studies in the sub-threshold region [3], [15], [16], [21], rections (SCH) in the MC. The former has the disadvantage that involve simulations of thousands of individual devices. that it requires tting against the MC data, as aforesaid, mean- In our case, the QC DD method takes about three times while the later QC approach is calibration free. In case of the less computational time than the QC MC method. However, DG method, we use electron effective masses as calibration the QC DD is disadvantaged by a requirement to calibrate parameters to account for the quantum capacitance (shift of QC parameters against either MC, NEGF or experimental the threshold voltage). The tting parameters used with the data [3], [21], [22]. It was previously shown that QC DD DG method are found in [43]. More details about the QC DD is unable to perform ON-current variability study for planar simulation methodologies can be found in [44] and [45] and MOSFETs without an underestimation because the QC DD about the QC MC in [32], [46], and [47]. cannot capture non-equilibrium effects [21], [23]. A sim- Finally, the in-house simulation toolbox can account for the ilar rigorous study for non-planar multi-gate transistors is following sources of variability: OTV, MGG, LER, gate edge missing from the literature. More importantly, the QC DD roughness (GER), and RD [31]. In this work, we will focus method is still being used in state-of-the-art device variability on the two most inuential ones [7] for GAA NW FET: MGG study [15], [18], [22], [24][28] believing that properly cal- and LER as illustrated in Fig. 2. Note that the same random ibrated QC DD simulations will yield to accurate statistical proles are used in both simulation techniques, the QC DD predictions. and QC MC, for a fair comparison of each variability study. In this paper, we aim to establish how accurate the QC DD Moreover, the QC DD calibration parameters are not adjusted method is when applied to the ON-region variability in com- for each of the proles but use the values calibrated for parison with the more rigorous QC MC simulation technique. the ideal device as this is the standard approach. The 2D We compare the results obtained by applying two of the main Schrödinger equation in the QC MC simulations is solved variability sources affecting the device reliability, the MGG for each random prole of a device as this method does not and LER, on a state-of-the-art 10 nm gate length Si GAA require additional calibration. NW FET that has been scaled down from an experimental In case of the MGG variability, we use the Poisson- device [29], [30]. Voronoi diagrams approach [48] to create the metal grains for the metal gate contact of the simulated device. This II. METHODOLOGY AND DEVICE DESCRIPTION method is believed to mimic more accurately the realistic In this work, we employ a well established in-house sim- metal gates [48] than the square grains approach [49], [50]. ulation toolbox [31][33] that includes 3D DD and MC Furthermore, the MGG prole is characterized by a grain transport models which use the nite element (FE) method size (GS) and by a work function value (WFV) [48]. For for accurate mesh description of a simulation domain. The the current study, we have chosen the titanium nitride (TiN) accurate description of the device nanoscale dimensions is of which is commonly used as a gate material [51]. The metal great importance for accurate simulations in the deep nano- has experimentally observed WFVs of 4:6 eV and 4:4 eV with regime because quantum-mechanical connement in a device a probability of 60% and 40% formation, respectively [52]. channel can signicantly affect transport at nanoscale [34]. In case of the LER variability, we create the uncorre- As mentioned before, the DD approach requires calibra- lated proles using Fourier synthesis with Gaussian auto- tion for the simulations. In this study, we use the readily correlation approach [53]. These are characterized by the available MC simulation toolbox to guide the calibration correlation length (CL) and the root mean square (RMS) of the QC DD. The model used by the DD simulator is values [43], [53]. The current study is limited to a CL of the Caughey-Thomas doping dependent low-eld electron 20 nm and to experimentally observed RMS heights, ranging mobility model [35], together with perpendicular (critical between 0:3 and 1:0 nm [11], [29]. electric eld) and lateral (saturation velocity) electric eld A device used in this study is based on a 10 nm gate models [36]. The calibration parameters used with the DD length GAA NW FET that was scaled down from an exper- VOLUME 7, 2019 12791 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 1. (i) Schematic for the 10 nm gate length GAA NW [30] and (ii) Gaussian doping profiles along the transport direction for three FIGURE 3. Simulated I -V characteristics for the 10 nm gate length GAA D G concentrations of N . (right) Cross-sectional view of the channel for the NW [30] at V D 0:7 V with a channel orientation of h110i. Three different (b) ideal device and two cases when the Fin height was (a) elongated and 19 20 doping concentrations are presented for N : 5 10 , 1 10 and (c) shortened. 20 3 1:5 10 cm . Full lines correspond to 3D QC MC simulations, while dashed lines refer to calibrated (against the QC MC) 3D QC DD simulations. the QC DD is achieved by adjusting the mobility model and QC parameters as described in detail in [43]. To assess the validity of the calibration for the QC DD simulator, two extreme cases of channel height for the NW were chosen as shown in Fig. 1(a) and (c). In each case, the height is increased/decreased symmetrically by 1 nm for an N of 20 3 FIGURE 2. Schematic for the 10 nm gate length GAA NW [30] affected by 1:5  10 cm , without changing any of the calibration LER and MGG variability sources. The LER profile is projected along the parameters. It was found that the QC DD results produce a transport direction (x-axis) and affects the dimension of only the z-axis. The MGG profile with different work function is projected to the gate negligible error, up to 3 %, for both modied devices when area [52]. compared to the results obtained from the QC MC. IV. MGG VARIABILITY imental device [29] following the ITRS [54] guidelines as We have generated 300 random proles with GSs of 3, 5 shown in [30]. The device schematic and dimensions are and 7 nm [52] for a meaningful statistical study of the MGG shown in Fig. 1(i). It has a uniformly p-type doped channel induced variability. These proles were also applied to three 15 3 (110 cm ), a Gaussian n-type doping, with a maximum maximum doping concentrations N to extensively inves- N (see Fig. 1(ii)) and a lateral straggle () of 3:23 nm, and tigate the capabilities of the QC DD and QC MC models. an EOT of 0:8 nm. Finally, it has an elliptical channel cross- Note that the same MGG proles are used in both simulation section with dimensions of 7:17 nm and 5:7 nm as shown techniques, the QC DD and QC MC, for a fair comparison. in Fig. 1(b). III. IDEAL GAA NW FET Even though GAA NWs are considered to be major con- tenders for future technology nodes, they might be unable to deliver a large enough ON-current (I ) [33], [55] in circuits, ON which may be one of the main limiting factors for the adapta- tion of the technology. One way to overcome this issue could be by increasing the maximum N of the S/D region. For this reason we have increased the reversed engineered n-type doping concentration of N from 5  10 that provided a perfect match to the experimental I-V curve [30] to 1 10 FIGURE 4. I due to MGG vs N from the QC DD and the QC MC 20 3 ON D and to 1:5 10 cm . Note that the  was kept constant as simulations using 300 profiles. The difference between the QC DD and QC shown in Fig. 1(ii). We have found that, compared to N D MC simulation results are indicated in percentage. 19 3 5 10 cm , I has increased by 40 % and 60 % for N ON D 20 20 3 of 1 10 and 1:5 10 cm , respectively. Note that I Fig. 4 shows the standard deviation () of the I against ON ON is I at V D V CV , where V is the threshold voltage the maximum N . Both simulation methods show an increas- D G DD T T D and V D 0:7 V. Both the QC MC and the well calibrated ing I with an increasing N . However, the difference DD ON D QC DD simulated I -V characteristics are shown in Fig. 3 between the I (indicated by percentage in the gure) D G ON for the aforementioned cases. Note that the calibration of predicted by both simulation methods is dependent on both 12792 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 5. Scatter plots compare the simulations with 20 3 20 3 19 3 N D 1:5 10 cm and 1 10 cm against N D 5 10 cm D D obtained from (a) QC DD and (b) QC MC. The GS is 7 nm. FIGURE 7. The schematic of the GAA NW FET gate area (a) with a single synthetic profile strip wrapped around the gate. The FSM for the I are ON simulated assuming n-type source/drain concentration (N ) of 20 3 1 10 cm using (a) QC DD and (b) QC MC techniques. 100 synthetic gate profiles with a width of 0:1 nm are simulated. around the gate (see example in Fig. 7(a)), (ii) this prole is then swept along the transport direction and the prole related to I is extracted, and (iii) all the simulated proles and their ON corresponding I are used to create a 2D FSM as shown ON 20 3 FIGURE 6. I due to MGG vs GS for a N D 1 10 cm obtained ON D in Figs. 7(b) and (c) for the QC DD and QC MC simulations, from the QC DD and the QC MC simulations. The difference between QC respectively. DD and QC MC are indicated in percentage. Thanks to the FSM technique, we are able to identify that for a 10 nm gate length GAA NW the most sensitive region of the gate is away from the centre of the gate, close to the gate- the doping value and the grain size. For instance, for a N of 20 3 source junction. However, for the QC MC the maximum value 1 10 cm , the error in the predicted values by QC DD is centered at around 1:8 nm while the QC DD predicts when compared to QC MC ones range from 7 % (7 nm GS) the maximum value at around 1:2 nm. Moreover, the QC to 19 % (3 nm GS). Fig. 5 compares I at N D 5 ON D 19 20 20 3 DD predicts the highest sensitive effective area to be smaller 10 against I at N D 1  10 and 1:5  10 cm ON D than that shown by the QC MC results. Thus, we know that a obtained from the (a) QC DD and (b) QC MC simulations. change in the WFV in the aforementioned region will play a There is a large correlation, as indicated by the correlation signicant role in the I values. ON coefcients (CCs), between the I values produced by both ON simulation methods. This means that the same proles pro- duce a similar variability even when the N is increased. V. LER VARIABILITY Finally, investigation of the effect of the GS is shown in Fig. 6. Section III has shown that the QC DD calibrated to the QC Both simulation methods predict an increasing I with MC simulations can predict the same I for the NW FET. ON ON an increasing GS. However, the QC DD method leads to an This ability has important implications for a LER induced overestimation of the MGG variability of around 20 % for variability study because the LER causes a uctuation in the GSs equal or lower than 5 nm. Furthermore, analysis of the channel dimension along the transport direction. However, mean (1) I showed a negligible difference between the QC what is the accuracy of the QC DD produced variability when ON DD and QC MC methods. the channel cross-section dimension uctuates? To answer A Fluctuation Sensitivity Map (FSM) [56] that analyzes this question, we generate 300 random LER proles assum- the spatial effect of the MGG variability in key gure of ing a correlation length (CL) of 20 nm for four experimentally merits (FoMs) (e.g. I ) is employed in order to reveal the observed RMS heights [11], [29], [30] and three maximum ON most sensitive regions of the studied device to the MGG. doping concentrations N . The same LER proles are used The procedure is as follows: (i) a single synthetic prole is for both simulation techniques, the QC DD and QC MC, for created, which has a WFV localized in a small strip wrapped a fair comparison. VOLUME 7, 2019 12793 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs FIGURE 8. I due to LER vs N from the QC DD and the QC MC ON D simulations using 300 profiles. The LER characteristic values are: CL D 20 nm and RMS heights of 1:0 and 0:5 nm. The difference between the QC DD and QC MC simulation results are indicated in percentage. FIGURE 9. Scatter plots compare the simulations with 20 3 20 3 19 3 N D 1:5 10 cm and 1 10 cm against N D 5 10 cm D D obtained from (a) QC DD and (b) QC MC simulations, respectively. The Fig. 8 shows the standard deviation () of the I against ON RMS height is 1 nm. the maximum N . The predicted I by the QC DD and D ON QC MC simulation techniques has very similar values, with a difference of up to 7 %, for the devices with a N of 19 3 5 10 cm . Yet, the error in the estimation given by the QC DD simulations increases with N reaching a staggering 56% difference when compared to the results from QC MC 20 3 simulations for a N of 1:5 10 cm . Finally, note that I is practically constant with dependence on N when ON D obtained from the QC MC simulations, whereas the QC DD results predicts an increasing I with N . Note that the ON D difference in the predicted behaviour lays in the implemen- tation of quantum correction methods as well as the different models, classical DD vs. semi-classical MC. The Schrödinger 20 3 FIGURE 10. I due to LER vs RMS height for a N D 1 10 cm ON D based quantum corrections in the QC MC simulations are obtained from the QC DD and the QC MC simulations. The difference between QC DD and QC MC are indicated in percentage. able to accurately capture the physics when some modica- tion in the device architecture occurs, for example, doping, LER, MGG, etc. However, the simulation approach using density gradient quantum corrections would require adjusting the calibration parameters for each of the aforementioned modications against a more complex simulation model. Furthermore, the MC method accounts for non-equilibrium electron transport as well as the inclusion of the important scattering models, which the DD model is not capable of. Further analysis of this behaviour is shown in Fig. 9 that 19 20 compares I at N D 510 against I at N D 110 ON D ON D 20 3 and 1:510 cm . The correlation between the I values ON produced by the LER proles from the QC DD simulations (Fig. 9(a)) is lower than for the QC MC ones (Fig. 9(b)) as indicated by the correlation coefcients (CCs). Finally, observe that the regression lines (red lines in Fig. 9) are FIGURE 11. The GAA NW FET schematic (a) is scaled to the I FSM (b). ON shifted by a constant value for the QC MC obtained results 100 synthetic profiles with a width deformation are simulated for N of 20 3 and yet, for the QC DD ones, they also change the slope. The 1:5 10 cm using QC DD and QC MC techniques as indicated. investigation of the effect of RMS height is shown in Fig. 10. The QC DD results give up to 22 % overestimation of a predicted I from the QC MC simulations. Additional I . The procedure is similar to the one used for the MGG ON ON analysis of the 1I showed a negligible difference between variability: (i) a single synthetic prole is created, which has ON the QC DD and QC MC methods. a Gaussian vertical deformation localized in a small region FSM [43] introduced in Section IV is also used to ana- of the device (see Fig. 11(a)), (ii) the prole is then swept lyze the spatial effect of the LER variability induced by along the transport direction and a prole related to I is ON 12794 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs extracted, and (iii) each prole and the corresponding I are ity studies that involve the variation of the channel cross- ON used to create a 1D FSM as shown in Fig. 11(b). Note that section in the ON-region regardless their calibration against a synthetic deformation for the LER can lead to an increase reliable data. This is because the QC DD method has (negative sensitivity) or decrease (positive sensitivity) of the xed calibration parameters which are ``device dimension I . Therefore, the normalized scale from 1 to 1 is used. specic'' while the QC MC uses the calibration free 2D ON Fig. 11 shows that the QC MC technique predicts the most Schrödinger equation to account for the actual quantum- sensitive regions to the LER variability closer to the source- mechanical connement effect. gate junction than the locations predicted by the QC DD technique. Notice that there is not only a shift between the QC ACKNOWLEDGMENT DD and QC MC largest absolute sensitive areas, but also the The authors would like to thank Centro de Supercomputación magnitude of the sensitivity is different. Finally, we can say de Galicia (CESGA) for the computer resources provided. that if a change in the diameter of a NW FET occurs near the middle of the gate or around the source-gate junction, it will REFERENCES heavily impact the I , as shown by the FSM. However, ON [1] O. Badami, F. Driussi, P. Palestri, L. Selmi, and D. Esseni, ``Performance changes in other parts of the NW FET dimensions will only comparison for FinFETs, nanowire and stacked nanowires FETs: Focus on the inuence of surface roughness and thermal effects,'' in IEDM Tech. have a negligible inuence in the I . ON Dig., Dec. 2017, pp. 13.2.113.2.4. [2] M. Li et al., ``Sub-10 nm gate-all-around CMOS nanowire transistors on bulk Si substrate,'' in Proc. Symp. VLSI Technol., Jun. 2009, pp. 9495. VI. CONCLUSION [3] K. Nayak, S. Agarwal, M. Bajaj, K. V. R. M. Murali, and V. R. 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Nagy, G. Indalecio, A. J. García-Loureiro, M. A. Elmessary, K. Kalna, device behaviour under metal grain work-function variability effects,'' and N. Seoane, ``FinFET versus gate-all-around nanowire FET: Perfor- IEEE Trans. Electron Devices, vol. 64, no. 4, pp. 16951701, Apr. 2017. mance, scaling, and variability,'' IEEE J. Electron Devices Soc., vol. 6, pp. 332340, Feb. 2018. [34] M. Stadele et al., ``A comprehensive study of corner effects in tri-gate transistors,'' in Proc. Eur. Solid-State Device Res. Conf. (ESSDERC), Sep. 2004, pp. 165168. [35] R. E. Thomas, ``Carrier mobilities in silicon empirically related to doping DANIEL NAGY received the M.Res. degree in and eld,'' Proc. IEEE, vol. 55, no. 12, pp. 21922193, Dec. 1967. nanoscience to nanotechnology and the Ph.D. [36] K. Yamaguchi, ``Field-dependent mobility model for two-dimensional degree in electronic and electrical engineering numerical analysis of MOSFET's,'' IEEE Trans. Electron Devices, from Swansea University, Swansea, U.K., in 2013 vol. ED-26, no. 7, pp. 10681074, Jul. 1979. and 2016, respectively. [37] K. Tomizawa, Numerical Simulation of Submicron Semiconductor Devices He currently holds a Postdoctoral position with (Artech House Materials Science Library). Norwood, MA, USA: the Centro Singular de Investigación en Tec- Artech House, 1993. [38] C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor noloxías da Información, University of Santiago Device Simulation (Computational Microelectronics). Vienna, Austria: de Compostela, Santiago de Compostela, Spain Springer, 2012. 12796 VOLUME 7, 2019 D. Nagy et al.: DD Versus MC Simulated ON-Current Variability in NW FETs GUILLERMO INDALECIO received the B.S. MUHAMMAD A. ELMESSARY received the degree in physics and the Ph.D. degree in semicon- B.Sc. degree (Hons.) in computer and systems ductor device simulation from the University of engineering and the M.Sc. degree in engineer- Santiago de Compostela, Santiago de Compostela, ing physics from Mansoura University, Mansoura, Spain, in 2010 and 2016, respectively. Egypt, in 2004 and 2010, respectively, and the He was a Visiting Researcher with the Uni- Ph.D. degree in semiconductor device simulation versity of Swansea, Swansea, U.K., in 2015. from Swansea University, Swansea, U.K., in 2017. His current research interests include electronic He is currently a Research Assistant with devices simulation with a focus on computational Swansea University. techniques and novel techniques to understand variability sources. KAROL KALNA received the M.Sc. (Hons.) ANTONIO J. GARCÍA-LOUREIRO received the and Ph.D. degrees from Comenius University, Ph.D. degree from the University of Santiago Bratislava, Czechoslovakia, in 1990 and 1998, de Compostela, Santiago de Compostela, Spain, respectively. in 1999. He is currently an Associate Professor leading He is currently an Associate Professor with the Nanoelectronic Devices Computational Group, the Department of Electronics and Computer Sci- Swansea University, Swansea, U.K. He has held ence, University of Santiago de Compostela. His the EPSRC Advanced Research Fellowship and current research interests include the multidimen- has pioneered IIIV MOSFETs, since 2002. He sional simulations of nanoscale transistors and has published 93 peer-review papers and has given solar cells. over 20 invited talks. NATALIA SEOANE received the Ph.D. degree from the University of Santiago de Compostela, GABRIEL ESPIÑEIRA received the B.S. degree in Santiago de Compostela, Spain, in 2007. physics from the University of Santiago de Com- She was a Visiting Postdoctoral Researcher postela, Santiago de Compostela, Spain, in 2018, with the University of Glasgow, Glasgow, U.K., where he is currently pursuing the M.Res. degree from 2007 to 2009, The University of Edinburgh, in HPC and also with the Centro Singular de Inves- Edinburgh, U.K., in 2011, and Swansea Univer- tigación en Tecnoloxías da Información. sity, Swansea, U.K., from 2013 to 2015. She is currently with the University of Santiago de Compostela. VOLUME 7, 2019 12797

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