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Research Papers

B-Splines and NURBS Based Finite Element Methods for Strained Electronic Structure Calculations

[+] Author and Article Information
Arif Masud

Fellow ASME
3129-E Newmark Civil Engineering Laboratory,
University of Illinois at Urbana-Champaign,
205 N. Mathews Avenue,
Urbana, IL 61801
e-mail: amasud@illinois.edu

Ahmad A. Al-Naseem

3103 Newmark Civil Engineering Laboratory,
University of Illinois at Urbana-Champaign,
205 N. Mathews Avenue,
Urbana, IL 61801
e-mail: alnasee2@illinois.edu

Raguraman Kannan

Mechanical Engineering Department,
Prince Mohammad Bin Fahd University,
e-mail: ragu2.kannan@gmail.com

Harishanker Gajendran

University of Illinois at Urbana-Champaign,
2112 Newmark Civil Engineering Laboratory,
205 N. Mathews Avenue,
Urbana, IL 61801
e-mail: harishankerg@gmail.com

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 23, 2018; final manuscript received May 25, 2018; published online June 18, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(9), 091009 (Jun 18, 2018) (11 pages) Paper No: JAM-18-1167; doi: 10.1115/1.4040454 History: Received March 23, 2018; Revised May 25, 2018

This paper presents B-splines and nonuniform rational B-splines (NURBS)-based finite element method for self-consistent solution of the Schrödinger wave equation (SWE). The new equilibrium position of the atoms is determined as a function of evolving stretching of the underlying primitive lattice vectors and it gets reflected via the evolving effective potential that is employed in the SWE. The nonlinear SWE is solved in a self-consistent fashion (SCF) wherein a Poisson problem that models the Hartree and local potentials is solved as a function of the electron charge density. The complex-valued generalized eigenvalue problem arising from SWE yields evolving band gaps that result in changing electronic properties of the semiconductor materials. The method is applied to indium, silicon, and germanium that are commonly used semiconductor materials. It is then applied to the material system comprised of silicon layer on silicon–germanium buffer to show the range of application of the method.

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References

Viventi, J. , Kim, D. H. , Vigeland, L. , Frechette, E. S. , Blanco, J. A. , Kim, Y. S. , Wulsin, D. F. , Rogers, J. A. , and Litt, B. , 2011, “ Flexible, Foldable, Actively Multiplexed, High-Density Electrode Array for Mapping Brain Activity In Vivo,” Nat. Neurosci., 14(12), p. 1599. [CrossRef] [PubMed]
Lu, N. , Lu, C. , Yang, S. , and Rogers, J. A. , 2012, “ Highly Sensitive Skin-Mountable Strain Gauges Based Entirely on Elastomers,” Adv. Funct. Mater., 22(19), pp. 4044–4050. [CrossRef]
Sun, Y. , and Rogers, J. A. , 2007, “ Inorganic Semiconductors for Flexible Electronics,” Adv. Mater., 19(15), pp. 1897–1916. [CrossRef]
Maiti, A. , 2003, “ Carbon Nanotubes: Bandgap Engineering With Strain,” Nat. Mater., 2(7), p. 440. [CrossRef] [PubMed]
Pitkethly, M. J. , 2004, “ Nanomaterials—The Driving Force,” Mater. Today, 7(12), pp. 20–29. [CrossRef]
Chelikowsky, J. R. , Troullier, N. , and Saad, Y. , 1994, “ Finite Difference-Pseudopotential Method: Electronic Structure Calculations Without a Basis,” Phys. Rev. Lett., 72(8), p. 1240. [CrossRef] [PubMed]
Belytschko, T. , and Xiao, S. P. , 2003, “ Coupling Methods for Continuum Model With Molecular Model,” Int. J. Multiscale Comput. Eng., 1(1), pp. 115–126. [CrossRef]
Liu, W. K. , Karpov, E. G. , Zhang, S. , and Park, H. S. , 2004, “ An Introduction to Computational Nanomechanics and Materials,” Comput. Methods Appl. Mech. Eng., 193(17–20), pp. 1529–1578. [CrossRef]
Kohn, W. , and Sham, L. J. , 1965, “ Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev., 140(4A), pp. 1133–1138. [CrossRef]
Masud, A. , and Kannan, R. , 2012, “ B-Splines and NURBS Based Finite Element Methods for Kohn–Sham Equations,” Comput. Methods Appl. Mech. Eng., 241, pp. 112–127. [CrossRef]
Liu, B. , Jiang, H. , Johnson, H. T. , and Huang, Y. , 2004, “ The Influence of Mechanical Deformation on the Electrical Properties of Single Wall Carbon Nanotubes,” J. Mech. Phys. Solids, 52(1), pp. 1–26. [CrossRef]
Thompson, S. E. , Sun, G. , Choi, Y. S. , and Nishida, T. , 2006, “ Uniaxial-Process-Induced Strained-Si: Extending the CMOS Roadmap,” IEEE Trans. Electron Devices, 53(5), pp. 1010–1020. [CrossRef]
Masud, A. , and Kannan, R. , 2009, “ A Multiscale Framework for Computational Nanomechanics: Application to the Modeling of Carbon Nanotubes,” Int. J. Numer. Methods Eng., 78(7), pp. 863–882. [CrossRef]
Nogueira, F. , Castro, A. , and Marques, M. A. , 2003, “ A Tutorial on Density Functional Theory,” A Primer in Density Functional Theory, Springer, Berlin, pp. 218–256. [CrossRef]
Martin, R. M. , 2004, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge, UK. [CrossRef]
Singh, D. J. , and Nordström, L. , 2006, Planewaves Pseudopotentials and the LAPW Method, Springer, New York.
Gavini, V. , Bhattacharya, K. , and Ortiz, M. , 2007, “ Quasi-Continuum Orbital-Free Density-Functional Theory: A Route to Multi-Million Atom Non-Periodic DFT Calculation,” J. Mech. Phys. Solids, 55(4), pp. 697–718. [CrossRef]
Gavini, V. , Knap, J. , Bhattacharya, K. , and Ortiz, M. , 2007, “ Non-Periodic Finite-Element Formulation of Orbital-Free Density Functional Theory,” J. Mech. Phys. Solids, 55(4), pp. 669–696. [CrossRef]
Suryanarayana, P. , Gavini, V. , Blesgen, T. , Bhattacharya, K. , and Ortiz, M. , 2010, “ Non-Periodic Finite-Element Formulation of Kohn–Sham Density Functional Theory,” J. Mech. Phys. Solids, 58(2), pp. 256–280. [CrossRef]
Hughes, T. J. , Cottrell, J. A. , and Bazilevs, Y. , 2005, “ Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement,” Comput. Methods Appl. Mech. Eng., 194(39–41), pp. 4135–4195. [CrossRef]
Pask, J. E. , Klein, B. M. , Sterne, P. A. , and Fong, C. Y. , 2001, “ Finite-Element Methods in Electronic-Structure Theory,” Comput. Phys. Commun., 135(1), pp. 1–34. [CrossRef]
Pask, J. E. , and Sterne, P. A. , 2005, “ Finite Element Methods in Ab Initio Electronic Structure Calculations,” Modell. Simul. Mater. Sci. Eng., 13(3), p. R71. [CrossRef]
Pask, J. E. , and Sterne, P. A. , 2005, “ Real-Space Formulation of the Electrostatic Potential and Total Energy of Solids,” Phys. Rev. B, 71(11), p. 113101. [CrossRef]
Chelikowsky, J. R. , Troullier, N. , Wu, K. , and Saad, Y. , 1994, “ Higher-Order Finite Difference Pseudopotential Method: An Application to Diatomic Molecules,” Phys. Rev. B, 50(16), p. 11355. [CrossRef]
Pickett, W. E. , 1989, “ Pseudopotential Methods in Condensed Matter Applications,” Comput. Phys. Rep., 9(3), pp. 115–197. [CrossRef]
Piegl, L. , and Tiller, W. , 1997, The NURBS Book, Springer, New York. [CrossRef]
Masud, A. , 2005, “ A 3-D Model of Cold Drawing in Engineering Thermoplastics,” Mech. Adv. Mater. Struct., 12(6), pp. 457–469. [CrossRef]
Masud, A. , 2000, “ A Multiplicative Finite Strain Finite Element Framework for the Modelling of Semicrystalline Polymers and Polycarbonates,” Int. J. Numer. Methods Eng., 47(11), pp. 1887–1908. [CrossRef]
Masud, A. , Truster, T. J. , and Bergman, L. A. , 2011, “ A Variational Multiscale a Posteriori Error Estimation Method for Mixed Form of Nearly Incompressible Elasticity,” Comput. Methods Appl. Mech. Eng., 200(47–48), pp. 3453–3481. [CrossRef]
Masud, A. , and Calderer, R. , 2011, “ A Variational Multiscale Method for Incompressible Turbulent Flows: Bubble Functions and Fine Scale Fields,” Comput. Methods Appl. Mech. Eng., 200(33–36), pp. 2577–2593. [CrossRef]
Masud, A. , and Truster, T. J. , 2013, “ A Framework for Residual-Based Stabilization of Incompressible Finite Elasticity: Stabilized Formulations and F-Bar Methods for Linear Triangles and Tetrahedra,” Comput. Methods Appl. Mech. Eng., 267, pp. 359–399. [CrossRef]
Gullett, P. M. , Horstemeyer, M. F. , Baskes, M. I. , and Fang, H. , 2007, “ A Deformation Gradient Tensor and Strain Tensors for Atomistic Simulations,” Modell. Simul. Mater. Sci. Eng., 16(1), p. 015001. [CrossRef]
Cormier, J. , Rickman, J. M. , and Delph, T. J. , 2001, “ Stress Calculation in Atomistic Simulations of Perfect and Imperfect Solids,” J. Appl. Phys., 89(7), pp. 99–104. [CrossRef]
Kannan, R. , and Masud, A. , 2009, “ Stabilized Finite Element Methods for the Schrödinger Wave Equation,” ASME J. Appl. Mech., 76(2), p. 021203. [CrossRef]
Richard, S. , Aniel, F. , Fishman, G. , and Cavassilas, N. , 2003, “ Energy-Band Structure in Strained Silicon: A 20-Band k⋅p and Bir–Pikus Hamiltonian Model,” J. Appl. Phys., 94(3), pp. 1795–1799. [CrossRef]
Sakata, K. , Magyari-Köpe, B. , Gupta, S. , Nishi, Y. , Blom, A. , and Deák, P. , 2016, “ The Effects of Uniaxial and Biaxial Strain on the Electronic Structure of Germanium,” Comput. Mater. Sci., 112, pp. 263–268. [CrossRef]
Hiemstra, R. R., Calabro, F., Schillinger, D., and Hughes, T. J., 2017, “Optimal and Reduced Quadrature Rules for Tensor Product and Hierarchically Refined Splines in Isogeometric Analysis,” Comput. Methods Appl. Mech. Eng., 316, pp. 966–1004.
Chermette, H. , 1998, “ Density Functional Theory: A Powerful Tool for Theoretical Studies in Coordination Chemistry,” Coord. Chem. Rev., 178, pp. 699–721. [CrossRef]
Monkhorst, H. J. , and Pack, J. D. , 1976, “ Special Points for Brillouin-Zone Integrations,” Phys. Rev. B, 13(12), p. 5188. [CrossRef]
Hartwigsen, C. , Gœdecker, S. , and Hutter, J. , 1998, “ Relativistic Separable Dual-Space Gaussian Pseudopotentials From H to Rn,” Phys. Rev. B, 58(7), p. 3641. [CrossRef]
Perdew, J. P. , and Wang, Y. , 1992, “ Accurate and Simple Analytic Representation of the Electron-Gas Correlation Energy,” Phys. Rev. B, 45(23), p. 13244. [CrossRef]
Smith, C. S. , 1954, “ Piezoresistance Effect in Germanium and Silicon,” Phys. Rev., 94(1), p. 42. [CrossRef]

Figures

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Fig. 1

Schematic plot of (a) unstrained lattice and (b) strained lattice vectors

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Fig. 2

One-way coupled solution procedure

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Fig. 6

Convergence rates attained with five-point Gauss quadrature rule

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Fig. 7

Convergence rates attained using optimal rule for p=2, 4, 5 and using the reduced rule for p=3

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Fig. 3

Plot of the potential along the body diagonal: (a) p=2, (b) p=3, and (c) p=4

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Fig. 4

Error in the potential along the body diagonal: (a) p=2, (b) p=3, and (c) p=4

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Fig. 5

Convergence rates for the Poisson problem with analytical potential in Eq. (31)

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Fig. 12

Convergence plot for total energies of each orbital as a function of radial domain length ξ

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Fig. 13

Conventional unit cell and primitive unit cell

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Fig. 14

First Brillouin zone and irreducible wedge

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Fig. 8

Convergence plot for Et(ρ) as a function of number of elements

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Fig. 9

Convergence plot for energies (Eqs. (43)(47)) as a function of radial domain length ξ

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Fig. 10

Convergence plot for energies (Eqs. (43)(47)) as a function of radial domain length ξ

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Fig. 11

Convergence plot for total energies of each orbital as a function of radial domain length ξ

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Fig. 15

Band diagram for bulk silicon for B-spline order p=2

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Fig. 16

Band diagram for bulk silicon for B-spline order p=3

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Fig. 17

Band diagram for strained silicon on silicon germanium buffer (Si0.9Ge0.1)

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Fig. 18

Band diagram for strained silicon on silicon germanium buffer (Si0.0Ge1.0)

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Fig. 19

Band diagram for unstrained bulk germanium for B-spline order p=2

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Fig. 20

Band diagram for strained bulk germanium for B-spline order p=2 with 3% strain

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