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 - Publisher Copyright:
© 2017
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
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U2 - 10.1016/j.jcp.2017.12.011
DO - 10.1016/j.jcp.2017.12.011
M3 - Article
AN - SCOPUS:85037976591
SN - 0021-9991
VL - 356
SP - 340
EP - 355
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -