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應力於結晶與奈米級矽固體之作用 Strain Effect on Crystalline and Nano-scale Silicon Solids

應力於結晶與奈米級矽固體之作用 Strain Effect on Crystalline and Nano-scale Silicon Solids. 指導教授 : 劉致為 博士 學生 : 黃筱鈞 國立臺灣大學電子工程學研究所. Outline. Thesis organization Chapter 2 ： Strain-induced Raman Shift Chapter 3 ： Carrier Mobility in Orthorhombically Strained Silicon

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應力於結晶與奈米級矽固體之作用 Strain Effect on Crystalline and Nano-scale Silicon Solids

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  1. 應力於結晶與奈米級矽固體之作用Strain Effect on Crystalline and Nano-scale Silicon Solids 指導教授: 劉致為 博士 學生: 黃筱鈞 國立臺灣大學電子工程學研究所

  2. Outline • Thesis organization • Chapter 2：Strain-induced Raman Shift • Chapter 3：Carrier Mobility in Orthorhombically Strained Silicon • Chapter 4：2-D Electrons in Strained Silicon Inversion Layers • Chapter 5：Surface Effect on Strained Silicon Clusters • Chapter 6：Strain Effect on Silicon Atomic Wires • Summary and Future Work

  3. Thesis organization

  4. Thesis organization：Strain Type

  5. Strain-induced Raman Shift • Raman spectra of a typical thin Si epilayer grown above a thick Si1-xGex buffer layer on Si (001) substrate

  6. Strain-induced Raman Shift • Qualitative and quantitative prediction of Raman shift • Simplified unit cell in Si epi-layer (instead of diamond structure) • Backscattering geometry (only singlet is observed) [D. J. Lockwood, PRB, 1992] • Forced to vibrate at a different force constant when strain is applied

  7. Strain-induced Raman Shift • Spring equation form of Hooke’s Law • Frequency is related to the square root of U’s second derivative

  8. Strain-induced Raman Shift • U from Harrison’s total/cohesive energy (1972, 1981) • U’s second derivative

  9. Strain-induced Raman Shift • Sqrt(k) vs. bond length • Region of interest: 2.35 ~ 2.4 A (Si1-xGex, 0<x<0.5) • Compare with Raman data from published empirical equation • a~200, a good prediction

  10. Carrier Mobility in Orthorhombically Strained Silicon • Vertical MOSFET • Unstrained Si substrate • Compressively strained SiGe pillar • Orthorhombically strained Si sidewall layer

  11. Carrier Mobility in Orthorhombically Strained Silicon

  12. Carrier Mobility in Orthorhombically Strained Silicon • Band splitting of orthorhombically strained silicon

  13. Carrier Mobility in Orthorhombically Strained Silicon • Electron and hole mobility of orthorhombically strained silicon • Two-fold electron mobility enhancement at 20% Ge • Two-fold hole mobility enhancement at 30% Ge

  14. 2-D Electrons in Strained Silicon Inversion Layers • Planar MOSFET • Channel mobility modeled as 2DEG

  15. 2-D Electrons in Strained Silicon Inversion Layers • Constant-energy ellipses (6 equivalent valleys) of Si conduction band • Energy lineups of Si conduction band w. and w/o tensile strain

  16. 2-D Electrons in Strained Silicon Inversion Layers

  17. 2-D Electrons in Strained Silicon Inversion Layers • Airy function vs. SC wavefunctions for delta 2 and delta 4 valleys

  18. 2-D Electrons in Strained Silicon Inversion Layers • Airy function vs. SC subband levels for delta 2 and delta 4 valleys

  19. 2-D Electrons in Strained Silicon Inversion Layers • Phonon-limited mobility vs. effective field • Mobility enhancement factor vs. substrate Ge content

  20. Surface Effect on Strained Silicon Clusters • Generalized Hooke’s Law -0.77

  21. Surface Effect on Strained Silicon Clusters • Horizontal fixed (5.65A); vertical tuned various α • Searching for min E(α) • Simulation building block: single silicon unit cell (diamond structure) 1x1y1z • 2x1y1z, 1x2y1z, 1x1y2z represent two unit cells stacking up in x, y, z direction, respectively • From 1x1y1z (18 atoms) to 3x3y1z (110 atoms)

  22. Surface Effect on Strained Silicon Clusters • Gaussian 03 and GaussView • Model Chemistry [theoretical method/basis set]: BLY3P/6-31G(d) • No min E(α) on the plot of total E versus α candidates for 1x1y1z, 2x2y1z, etc • Squeezed (more negative α), total energy goes down

  23. Surface Effect on Strained Silicon Clusters • Clusters w. bare silicon (w. dangling bonds) • Clusters w. silicon and valence hydrogen atoms (instead of dangling bonds) • Min E(α) on the plot of total E versus α candidates for 1x1y1z • Min E(α) by (1) squeezed (more negative α), total energy goes up (2) a energy step (4.8 eV) for all α> -0.77

  24. Surface Effect on Strained Silicon Clusters • Valence hydrogen pair with angle of 54.7 degree (instead of 109.8) • 1x1y1z (1), 2x2y1z (5), 3x3y1z (13): yes

  25. Surface Effect on Strained Silicon Clusters Antenna check for 2x2y1z

  26. Surface Effect on Strained Silicon Clusters • Only 1x1y2z has antenna (same with 1x1y3z, 1x1y4z, etc) • 3x2y1z no, 2x2y2z yes • Square symmetry on x-y plane required?

  27. Surface Effect on Strained Silicon Clusters • 2x2y1z minus one (3), 3x3y1z minus one (11): yes • 3x3y1z minus two, 2x1y2z, 3x1y3z: no

  28. Surface Effect on Strained Silicon Clusters • Simulation of up to 9 unit cells • Bare silicon clusters: unstable with dangling bonds • With surface hydrogen: obey the same rule with bulk silicon- deformation of shorten heights with α = -0.77 by (1) squeezed, total energy goes up (2) energy step starting at α = -0.77 • Bond angle effect under deformation • Near-square symmetry on one of the surface of the x-y plane

  29. Strain Effect on Silicon Atomic Wires • Molecule systems (equilibrium) coupled to electrodes and bias voltage is applied (non-equilibrium) • TranSIESTA-C: Density Functional Theory (DFT) and Non-equilibrium Greens Function (NEGF) solving self-consistently • Several approximation is adopted

  30. Strain Effect on Silicon Atomic Wires • Molecular system: Si3 cluster (Si3 atomic wire in zigzag fashion) • Electrode: Li [He]2s1 closely resembles Au [Xe]4f145d106s1 • Two-probe system: Si3 cluster coupled to lithium electrode

  31. Strain Effect on Silicon Atomic Wires • Isolated Si3 cluster (van der Waals radii; HOMO; LUMO) • Relaxed Si3 atomic wire (new MPSH LUMO as a channel) • MPSH: Molecular Projected Self-Consistent Hamiltonian

  32. Strain Effect on Silicon Atomic Wires • I-V characteristic of relaxed Si3 atomic wire

  33. Strain Effect on Silicon Atomic Wires • Transmission spectrum vs. MPSH eigenstates (red dot) • T(E, Vb) at Vb= 0, 1, 2V; LUMO closely associate with the peak

  34. Strain Effect on Silicon Atomic Wires • Three strain type CASE I n=1~4 (a1, a2, a3, a4) CASE II n=1~4 (m1, m2, m3, m4) CASE III n=1~2 (d1, d2)

  35. Strain Effect on Silicon Atomic Wires • I-V characteristic of strained Si3 atomic wire (CASE I) • 0V ~ 1.2V: a4 < a3 < a2 < a1 < relax • 1.2V ~ 2V: relax < a1 < a2 < a3 < a4

  36. Strain Effect on Silicon Atomic Wires • I-V characteristic of strained Si3 atomic wire (CASE II) • 0V and 2V: m4 ~ m3 ~ m2 ~ m1 ~ relax • Between 0V and 2V (esp. 1V): relax < m1 < m2 < m4 < m3

  37. Strain Effect on Silicon Atomic Wires • I-V characteristic of strained Si3 atomic wire (CASE III) • 0V ~ 2V: d2 < d1 < relax

  38. Strain Effect on Silicon Atomic Wires • T(E, Vb) for CASE I, n=1~4 • T(E, Vb) for CASE II, n=1~4 • T(E, Vb) for CASE III, n=1~2 Relax

  39. Strain Effect on Silicon Atomic Wires • Current is obtained by Landauer-Buttiker formula • Bias window: the energy region which contributes to the current integral (only positive part is shown)

  40. Strain Effect on Silicon Atomic Wires • Transmission spectrum within bias window at Vb= 1 and 2 V • LUMO peak (1) bottom (2) move to center (3) bottom

  41. Strain Effect on Silicon Atomic Wires • MPSH eigenstates at Vb= 1 and 2 V • LUMO with (1) HOMO/HOMO+1 (2) LUMO+1; HOMO/HOMO-1 (3) LUMO+2; HOMO

  42. Summary and Future Work：Summary • A simple spring model is developed to make qualitative and quantitative predictions of Raman peak red-shift in tensile strain silicon epi-layer. • Phonon-limited bulk mobility under orthorhombic strain is calculated. Strong electron and hole mobility enhancement is observed. • Phonon-limited electron channel mobility under tensile strain is calculated. Airy function is a fair approximation. Enhancement factor saturates at 20% Ge content. • Surface hydrogen atoms is necessary to stabilize silicon clusters up to 9 unit cells in a morphology of shorten heights (α = -0.77) under tensile strain. Near square symmetry is required for above observation. • I-V characteristic of relaxed and strained Si3 atomic wire is investigated. Bias window and MPSH eigenstates are helpful in understanding the changes in I-V characteristic in three strain conditions.

  43. Summary and Future Work：Future Work • Experimental confirmation • More sophisticated molecular electronics with realistic metal electrodes

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