An adjoint-based adaptive error approximation of functionals by the hybridizable discontinuous Galerkin method for second-order elliptic equations

Bernardo Cockburn, Shiqiang Xia

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This paper presents a novel, fully computable error approximation and mesh adaptation approach for functionals defined by second-order elliptic equations. The functionals are approximated by the hybridizable discontinuous Galerkin (HDG) method and the error approximation is obtained by the adjoint-based method and a local post-processing technique of the HDG method. Unlike most adjoint-based error estimations in the literature, the novelty of our method is that the error approximation is obtained without requiring an auxiliary finer mesh or higher order approximation spaces for solving the adjoint problem. This reduces the computational cost and eases the implementation of the problem. What's more, the local post-processing technique can be carried out in parallel, which speeds up the method even more. We illustrate the method with a second-order elliptic problem and we present examples of three types of functionals: volume integrals; boundary integrals and eigenvalue problems. Numerical tests with adaptive mesh refinements for non-smooth solutions are presented to show that our method is efficient and robust.

Original languageEnglish (US)
Article number111078
JournalJournal of Computational Physics
StatePublished - May 15 2022

Bibliographical note

Funding Information:
a dose of 5E15 cm.*. The dopant was activated by rapid thermal annealing. Next, the contact holes and A1 metallization was performed. Finally, each device was passivated by NH3 plasma treatment for 1 hr at 300 OC. Results and discussion Figure 3a presents SEM picture of active pattern of proposed TFTs. Figure 3b presents SEM picture of ten nano-wire channels, each of which is 67 nm wide. Figure 4 to 7 plots a typical transfer curves and output curves of all poly-Si TFT devices. For investigation of devices process variation, five TFTs in different area on the 6-inch wafer were taken into account. Table I1 lists all poly-Si TFT parameters. Figure 8 to 12 shows TFTs' pFEO, N/OFE Vth, SS and DIBL versus multi-channel with different'widths TFTs. Fig. 8 reveals that the SI, M2, M5 and MI0 poly-Si TkTs yield almost the same carrier mobility, indicating that the canier mobility and the turn-on current are not degraded in TFTs with various numbers of channels. Moreover, in Fig. 9 to 12 MI0 TFT performs best in switching applications, with the highest ON/OFF ratio, most stable V,h, the smallest SS, and an absence of DIBL effect. These results are summarized by stating that the active channels around the tri-gate of the MI0 TFT exhibits the best gate control, resulting in its nano-wire structure (Fig. IC), hence the short-channel effect is highly reduced. Simulation results of Fig. 2 present the tri-gate TFT has superior gate control, which can confine the electrical field penetrated from drain to single-gate TFT. Additionally, experimental results also indicate that the gate control increases with the number of channels from SI, M2, and M5 to MIO, as similar as the structure changes from a single-gate to tri-gate sequentially. Moreover, NH3 plasma passivation more efficiently affects MI0 TFT than it does other TFTs. The MI0 TFT has a split nano-wire structure (Fig. 3h), most of which is exposed to NH3 plasma, further reducing the number of grain boundary defects as shown in Fig. 13, which is responding for achieving low leakage current and steep SS. In output characteristics, the kink-effect is reduced form S1, M2, M5 to MI0 TIT. The MI0 TFI even exhibits almost kink-free characteristics, indicating that the thin 50 nm-thick, and 67 nm-wide channels of MI0 TlT are fully-depleted by the tr-gate electrode. Therefore, at high drain bias (Vb),th e impact ionization is lower than that on other dimension TFk Figure I4 depicts typical single-channel (Sl) poly-Si TFT Id-Vg curves after hot carrier stress at 40 and 160 min, respectively. The hot carrier stress condition is determined at kink-effect occurrence. Fig. 15 depicts typical MI0 poly-Si TFT ZcVg curves after hot carrier stress at 40 and 160 mi.n, respectively. The mobility, ON/OFE V, and SS, after hot carrier stress duration for S1, M2, M5 and MI0 are displayed in Fig. 16 to Fig. 19, respectively. ON/OFF ratio and mobility of all TFT has the identical degradation behavior. However, M10 TFT's V,h and SS remain constant after stress duration. According to its tri-gate nano-wires structure can effectively reduce the lateral electric field penetration from drain side and has highly effective NH3 plasma passivation, leading to low deep-state generation in the poly-Si grain boundaries. Conclusion Experiment results show that the gate control is increasing with channel number from SI, M2, M5 to MI0 as the structure changes from a single-gate to a tri-gate sequentially. The M10 TFT exhibits superior and stable characteristics, such as low leakage current in the off-state, a high ON/OFF current ratio, a-steep SS, an absence of DIBL, and a favorable output characteristics. In reliability study, ON/OFF ratio and mobility of all TlT has the identical degradation behavior. The Vr,, and SS of MI0 TFI remain constant after stress duration. The fabrication of multi-channel TFTs is quite easy and involves no additional processes. Such TFTs are thus highly promising for use in future high-performance poly-Si TFI applications. Acknowledge The authors would like to acknowledge the National Science Council of the Republic of China, Taiwan for supporting this research under Contract No. NSC 92-2215-E-009-013. The National Nan0 Device Laboratory is appreciated for its technical support. Ref e re ne e s

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© 2022 Elsevier Inc.


  • Adaptivity
  • Adjoint-based method
  • Goal-oriented error approximation
  • Hybridizable discontinuous Galerkin method (HDG)
  • Output functionals


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