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

N1 - Funding Information:
Huang was partially supported by the Ministry of Science and Technology, Taiwan (MOST) 105-2115-M-003-009-MY3, National Center for Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST 106-2628-M-009-004-, NCTS and ST Yau Center in Taiwan. Tian and Chen would like to acknowledge the support from University of Hong Kong Grant Council (AoE/P-04/08). Tian is also grateful for the chance of short-term visit to the ST Yau Center in NCTU, which breeds this joint project. Mr. Jun Li at University of Hong Kong (HKU) provided the original data of matrices H0,S0,H1,S1 and Dr. Shu-Guang Chen at HKU gave us valuable feedback and helped us to prepare Fig. 1 in this article. Both of them deserve gratitude, too.
Funding Information:
Huang was partially supported by the Ministry of Science and Technology, Taiwan (MOST) 105-2115-M-003-009-MY3 , National Center for Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST 106-2628-M-009-004- , NCTS and ST Yau Center in Taiwan . Tian and Chen would like to acknowledge the support from University of Hong Kong Grant Council ( AoE/P-04/08 ). Tian is also grateful for the chance of short-term visit to the ST Yau Center in NCTU, which breeds this joint project. Mr. Jun Li at University of Hong Kong (HKU) provided the original data of matrices H 0 , S 0 , H 1 , S 1 and Dr. Shu-Guang Chen at HKU gave us valuable feedback and helped us to prepare Fig. 1 in this article. Both of them deserve gratitude, too.

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 -