### Abstract

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener–Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000×4000 at the density functional tight binding level, corresponding to a 8×8nm^{2} cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Original language | English |
---|---|

Pages (from-to) | 340-355 |

Number of pages | 16 |

Journal | Journal of Computational Physics |

Volume | 356 |

DOIs | |

Publication status | Published - 2018 Mar 1 |

### Fingerprint

### Keywords

- G⊤SHIRA
- Non-equivalence deflation
- Palindromic quadratic eigenvalue problem
- Quantum transport
- Surface Green's function

### ASJC Scopus subject areas

- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational Physics*,

*356*, 340-355. https://doi.org/10.1016/j.jcp.2017.12.011

**Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations.** / Huang, Tsung Ming; Lin, Wen Wei; Tian, Heng; Chen, Guan Hua.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 356, pp. 340-355. https://doi.org/10.1016/j.jcp.2017.12.011

}

TY - JOUR

T1 - Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

AU - Huang, Tsung Ming

AU - Lin, Wen Wei

AU - Tian, Heng

AU - Chen, Guan Hua

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener–Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000×4000 at the density functional tight binding level, corresponding to a 8×8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

AB - Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener–Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000×4000 at the density functional tight binding level, corresponding to a 8×8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

KW - G⊤SHIRA

KW - Non-equivalence deflation

KW - Palindromic quadratic eigenvalue problem

KW - Quantum transport

KW - Surface Green's function

UR - http://www.scopus.com/inward/record.url?scp=85037976591&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037976591&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2017.12.011

DO - 10.1016/j.jcp.2017.12.011

M3 - Article

AN - SCOPUS:85037976591

VL - 356

SP - 340

EP - 355

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -