Simudo: a device model for intermediate band materials

  • Eduard C. Dumitrescu
  • Matthew M. Wilkins
  • Jacob J. KrichEmail author


We describe Simudo, a free Poisson/drift-diffusion steady state device model for semiconductor and intermediate band materials, including self-consistent optical absorption and generation. Simudo is the first freely available device model that can treat intermediate band materials. Simudo uses the finite element method (FEM) to solve the coupled nonlinear partial differential equations in two dimensions, which is different from the standard choice of the finite volume method in essentially all commercial semiconductor device models. We present the continuous equations that Simudo solves, show the FEM formulations we have developed, and demonstrate how they allow robust convergence with double-precision floating point arithmetic. With a benchmark semiconductor pn junction device, we show that Simudo has a higher rate of convergence than Synopsys Sentaurus, converging to high accuracy with a considerably smaller mesh. Simudo includes many semiconductor phenomena and parameters and is designed for extensibility by the user to include many physical processes.


Device model Intermediate band materials Optoelectronics Finite element method Photovoltaics 



We acknowledge funding from US Army Research Laboratory (W911NF-16-2-0167), the Natural Sciences and Engineering Research Council of Canada TOP-SET training program, and computing resources from Compute Canada. We thank Emily Zinnia Zhang for alpha testing Simudo, contributing the first code implementing trapping processes, and valuable conversations.

Supplementary material

10825_2019_1414_MOESM1_ESM.pdf (78 kb)
Supplementary material 1 (PDF 78 kb)


  1. 1.
    Bank, R.E., Rose, D.J., Fichtner, W.: Numerical methods for semiconductor device simulation. IEEE Trans. Electron Devices 30(9), 1031 (1983)zbMATHCrossRefGoogle Scholar
  2. 2.
    Fichtner, W., Rose, D., Bank, R.: Semiconductor device simulation. IEEE Trans. Electron Devices 30(9), 1018 (1983)zbMATHCrossRefGoogle Scholar
  3. 3.
    Markowich, P.A.: The Stationary Semiconductor Device Equations. Computational Microelectronics. Springer, Vienna (1986)CrossRefGoogle Scholar
  4. 4.
    Piprek, J. (ed.): Handbook of Optoelectronic Device Modeling & Simulation. CRC Press, Boca Raton (2018)Google Scholar
  5. 5.
    Schenk, A.: Advanced Physical Models for Silicon Device Simulation. Springer, Wien (1998)zbMATHCrossRefGoogle Scholar
  6. 6.
    Luque, A., Martí, A.: Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels. Phys. Rev. Lett. 78(26), 5014 (1997)CrossRefGoogle Scholar
  7. 7.
    Okada, Y., Ekins-Daukes, N.J., Kita, T., Tamaki, R., Yoshida, M., Pusch, A., Hess, O., Phillips, C.C., Farrell, D.J., Yoshida, K., Ahsan, N., Shoji, Y., Sogabe, T., Guillemoles, J.F.: Intermediate band solar cells: recent progress and future directions. Appl. Phys. Rev. 2(2), 021302 (2015)CrossRefGoogle Scholar
  8. 8.
    Mailoa, J.P., Akey, A.J., Simmons, C.B., Hutchinson, D., Mathews, J., Sullivan, J.T., Recht, D., Winkler, M.T., Williams, J.S., Warrender, J.M., Persans, P.D., Aziz, M.J., Buonassisi, T.: Room-temperature sub-band gap optoelectronic response of hyperdoped silicon. Nat. Commun. 5, 3011 (2014)CrossRefGoogle Scholar
  9. 9.
    Berencén, Y., Prucnal, S., Liu, F., Skorupa, I., Hübner, R., Rebohle, L., Zhou, S., Schneider, H., Helm, M., Skorupa, W.: Room-temperature short-wavelength infrared Si photodetector. Sci. Rep. 7, 43688 (2017)CrossRefGoogle Scholar
  10. 10.
    Wang, M., Berencén, Y., García-Hemme, E., Prucnal, S., Hübner, R., Yuan, Y., Xu, C., Rebohle, L., Böttger, R., Heller, R., Schneider, H., Skorupa, W., Helm, M., Zhou, S.: Extended infrared photoresponse in Te-hyperdoped Si at room temperature. Phys. Rev. Appl. 10(2), 024054 (2018)CrossRefGoogle Scholar
  11. 11.
    Brown, A.S., Green, M.A.: Impurity photovoltaic effect: fundamental energy conversion efficiency limits. J. Appl. Phys. 92(3), 1329 (2002)CrossRefGoogle Scholar
  12. 12.
    Martí, A., Antolín, E., Stanley, C.R., Farmer, C.D., López, N., Díaz, P., Cánovas, E., Linares, P.G., Luque, A.: Production of photocurrent due to intermediate-to-conduction-band transitions: a demonstration of a key operating principle of the intermediate-band solar cell. Phys. Rev. Lett. 97(24), 247701 (2006)CrossRefGoogle Scholar
  13. 13.
    Wang, W., Lin, A.S., Phillips, J.D.: Intermediate-band photovoltaic solar cell based on ZnTe:O. Appl. Phys. Lett. 95(1), 011103 (2009)CrossRefGoogle Scholar
  14. 14.
    López, N., Reichertz, L.A., Yu, K.M., Campman, K., Walukiewicz, W.: Engineering the electronic band structure for multiband solar cells. Phys. Rev. Lett. 106(2), 028701 (2011)CrossRefGoogle Scholar
  15. 15.
    Sullivan, J.T., Simmons, C.B., Buonassisi, T., Krich, J.J.: Targeted search for effective intermediate band solar cell materials. IEEE J. Photovolt. 5(1), 212 (2015)CrossRefGoogle Scholar
  16. 16.
    der Maur, M.A.: A multiscale simulation environment for electronic and optoelectronic devices. Ph.D. thesis, Universita’ degli Studi di Roma Tor Vergata (2008)Google Scholar
  17. 17.
    Birner, S., Zibold, T., Andlauer, T., Kubis, T., Sabathil, M., Trellakis, A., Vogl, P.: Nextnano: General purpose 3-D simulations. IEEE Trans. Electron Devices 54(9), 2137 (2007)CrossRefGoogle Scholar
  18. 18.
    Clugston, D.A., Basore, P.A.: PC1D version 5: 32-bit solar cell modeling on personal computers. In: Twenty Sixth IEEE Photovoltaic Specialists Conference, pp. 207–210 (1997)Google Scholar
  19. 19.
    Haug, H., Greulich, J.: PC1Dmod 6.2-improved simulation of c-Si devices with updates on device physics and user interface. Energy Procedia 92(1876), 60 (2016)CrossRefGoogle Scholar
  20. 20.
    Varache, R., Leendertz, C., Gueunier-Farret, M., Haschke, J., Muñoz, D., Korte, L.: Investigation of selective junctions using a newly developed tunnel current model for solar cell applications. Sol. Energy Mater. Sol. Cells 141, 14 (2015)CrossRefGoogle Scholar
  21. 21.
    Burgelman, M., Nollet, P., Degrave, S.: Modelling polycrystalline semiconductor solar cells. Thin Solid Films 361–362, 527 (2000)CrossRefGoogle Scholar
  22. 22.
    Alonso-Álvarez, D., Wilson, T., Pearce, P., Führer, M., Farrell, D., Ekins-Daukes, N.: Solcore: a multi-scale, Python-based library for modelling solar cells and semiconductor materials. J. Comput. Electron. 17(3), 1099 (2018)CrossRefGoogle Scholar
  23. 23.
    Shockley, W., Read, W.T.: Statistics of the recombinations of holes and electrons. Phys. Rev. 87(5), 835 (1952)zbMATHCrossRefGoogle Scholar
  24. 24.
    Marti, A., Cuadra, L., Luque, A.: Quasi-drift diffusion model for the quantum dot intermediate band solar cell. IEEE Trans. Electron Devices 49(9), 1632 (2002)CrossRefGoogle Scholar
  25. 25.
    Strandberg, R., Reenaas, T.W.: Drift-diffusion model for intermediate band solar cells including photofilling effects. Prog. Photovolt.: Res. Appl. 19(1), 21 (2011)CrossRefGoogle Scholar
  26. 26.
    Tobías, I., Luque, A., Martí, A.: Numerical modeling of intermediate band solar cells. Semicond. Sci. Technol. 26(1), 014031 (2011)CrossRefGoogle Scholar
  27. 27.
    Yoshida, K., Okada, Y., Sano, N.: Device simulation of intermediate band solar cells: effects of doping and concentration. J. Appl. Phys. 112(8), 084510 (2012)CrossRefGoogle Scholar
  28. 28.
    Cuadra, L., Martí, A., Luque, A.: Influence of the overlap between the absorption coefficients on the efficiency of the intermediate band solar cell. IEEE Trans. Electron Devices 51(6), 1002 (2004)CrossRefGoogle Scholar
  29. 29.
    Levy, M.Y., Honsberg, C.: Intraband absorption in solar cells with an intermediate band. J. Appl. Phys. 104(11), 113103 (2008)CrossRefGoogle Scholar
  30. 30.
    Hu, W.G., Inoue, T., Kojima, O., Kita, T.: Effects of absorption coefficients and intermediate-band filling in InAs/GaAs quantum dot solar cells. Appl. Phys. Lett. 97(19), 193106 (2010)CrossRefGoogle Scholar
  31. 31.
    Strandberg, R.: Analytic \({JV}\)-characteristics of ideal intermediate band solar cells and solar cells with up and downconverters. IEEE Trans. Electron Devices 64(5), 2275 (2017)CrossRefGoogle Scholar
  32. 32.
    Strandberg, R.: The \({JV}\)-characteristic of intermediate band solar cells with overlapping absorption coefficients. IEEE Trans. Electron Devices 64(12), 5027 (2017)CrossRefGoogle Scholar
  33. 33.
    Lin, A.S., Wang, W., Phillips, J.D.: Model for intermediate band solar cells incorporating carrier transport and recombination. J. Appl. Phys. 105(6), 064512 (2009)CrossRefGoogle Scholar
  34. 34.
    Navruz, T., Saritas, M.: Determination of the optimum material parameters for intermediate band solar cells using diffusion model. Prog. Photovolt: Res. Appl. 22(5), 593 (2014) CrossRefGoogle Scholar
  35. 35.
    Krich, J.J., Trojnar, A.H., Feng, L., Hinzer, K., Walker, A.W.: Modeling intermediate band solar cells: a roadmap to high efficiency. In: Proceedings SPIE 8981, Physics, Simulation, and Photonic Engineering of Photovoltaic Devices III, p. 89810O (2014)Google Scholar
  36. 36.
    Yoshida, K., Okada, Y., Sano, N.: Self-consistent simulation of intermediate band solar cells: effect of occupation rates on device characteristics. Appl. Phys. Lett. 97(13), 133503 (2010)CrossRefGoogle Scholar
  37. 37.
    Alnæs, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Arch. Numer Softw. 3(100), 9–23 (2015)Google Scholar
  38. 38.
    Eymard, R., Gallouët, T., Herbin, R.: Handbook of Numerical Analysis, vol. 7, pp. 713–1018. Elsevier, Amsterdam (2000)Google Scholar
  39. 39.
    He, Y., Cao, G.: A generalized Scharfetter–Gummel method to eliminate crosswind effects (semiconduction device modeling). IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 10(12), 1579 (1991)CrossRefGoogle Scholar
  40. 40.
    Nachaoui, A.: Iterative solution of the drift-diffusion equations. Numer. Algorithms 21, 323 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Bochev, P., Peterson, K., Perego, M.: A multiscale control volume finite element method for advection-diffusion equations. Int. J. Numer. Methods Fluids 77(11), 641 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)Google Scholar
  43. 43.
    Kumar, G., Singh, M., Bulusu, A., Trivedi, G.: A framework to simulate semiconductor devices using parallel computer architecture. In: Journal of Physics: Conference Series, vol. 759, p. 012098. IOP Publishing, Bristol (2016)Google Scholar
  44. 44.
    Kumar, G., Singh, M., Ray, A., Trivedi, G.: An FEM based framework to simulate semiconductor devices using streamline upwind Petrov–Galerkin stabilization technique. In: 2017 27th International Conference Radioelektronika, pp. 1–5 (2017)Google Scholar
  45. 45.
    Poupaud, F., Schmeiser, C.: Charge transport in semiconductors with degeneracy effects. Math. Methods Appl. Sci. 14(5), 301 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Marshak, A.H., Van Vliet, C.: Electrical current and carrier density in degenerate materials with nonuniform band structure. Proc. IEEE 72(2), 148 (1984)CrossRefGoogle Scholar
  47. 47.
    Nelson, J.: The Physics of Solar Cells. Imperial College Press, London (2003)CrossRefGoogle Scholar
  48. 48.
    McIntosh, K.R., Black, L.E.: On effective surface recombination parameters. J. Appl. Phys. 116(1), (2014)CrossRefGoogle Scholar
  49. 49.
    Zhao, J., Tan, J., Liu, L.: A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media. J. Comput. Phys. 232(1), 431 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  51. 51.
    Gockenbach, M.S.: Understanding and Implementing the Finite Element Method. SIAM, New Delhi (2006)zbMATHCrossRefGoogle Scholar
  52. 52.
    Jean Donea, A.H.: Finite Element Methods for Flow Problems. Wiley, Hoboken (2003)zbMATHCrossRefGoogle Scholar
  53. 53.
    Gaury, B., Sun, Y., Bermel, P., Haney, P.M.: Sesame: a 2-dimensional solar cell modeling tool. Sol. Energy Mater. Sol. Cells 198, 53 (2019)CrossRefGoogle Scholar
  54. 54.
    Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Roberts, J., Thomas, J.M.: In Finite Element Methods (Part 1), Handbook of Numerical Analysis, vol. 2, pp. 523–639. Elsevier, Hoboken (1991)CrossRefGoogle Scholar
  56. 56.
    Cummings, D.J., Law, M.E., Cea, S., Linton, T.: Comparison of discretization methods for device simulation. Int. Conf. Simul. Semicond. Process. Devices 2009, 1–4 (2009)Google Scholar
  57. 57.
    Synopsys Inc.: Sentaurus Device User Guide, vK-2015. Synopsys Inc., Mountain View (2015)Google Scholar
  58. 58.
    Logg, A., Mardal, K.A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)zbMATHCrossRefGoogle Scholar
  59. 59.
    Marti, A., Cuadra, L., Luque, A.: Quasi-drift diffusion model for the quantum dot intermediate band solar cell. IEEE Trans. Electron Devices 49, 1632 (2002)CrossRefGoogle Scholar
  60. 60.
    Strandberg, R., Reenaas, T.W.: Photofilling of intermediate bands. J. Appl. Phys. 105(12), 124512 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of OttawaOttawaCanada
  2. 2.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada

Personalised recommendations