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Title TCAD ready density gradient calculation of channel charge for Strained Si/Strained Si1−xGex dual channel pMOSFETs on (001) Relaxed Si1−y Gey C D Nguyen, A T Pham, C Jungemann, and B Meinerzhagen Institut fă ur Netzwerktheorie und Schaltungstechnik Technische Universită at Braunschweig C Jungemann IWCE 2004 Outline • Motivation ã Schră odinger/Poisson Solver for Strained Si and SiGe ã Density Gradient Model • Extraction of the heterojunction valence band offsets needed for TCAD simulators • Conclusion and Outlook C Jungemann IWCE 2004 Motivation C Jungemann IWCE 2004 Motivation multi stacked strained structure Vg SiO2 (4.4nm) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ Strained Si0.4 Ge0.6 (5nm) ✁ Strained Si (3.3nm) III III/II ∆EV II II/I ∆EV I Relaxed Si0.7 Ge0.3 EV EF EC Changes in the band structure and small thickness of the strained layers =⇒ Size Quantization Solution: Schră odinger equation (SE) with a full band description using the k · p-method For TCAD use, directly solving the SE is too CPU intensive =⇒ Density Gradient Method (DGM) Problem: unknown model parameters e g effective band offsets z Vb C Jungemann IWCE 2004 Motivation Effective band offsets can be determined by: • Measurement: The effective band offsets can be extracted by inverse modeling of CV measurements based on the DGM [1] =⇒ Uncertainty due to incomplete knowledge of the investigated devices • Simulation: Based on the self-consistent solution of the SE and Poisson equations, the effective band offsets can be extracted and the errors of the DGM approximation can be investigated [1] C Ni Chleirigh et al., “Extraction of band offsets in Strained Si/Strained SiGe on relaxed SiGe dual-channel enhanced mobility structures” to be presented at SiGe Materials, Processing and Devices Symposium, Hawai, 2004 C Jungemann IWCE 2004 Schră odinger/Poisson Solver for Strained Si and SiGe C Jungemann IWCE 2004 Schră odinger/Poisson Solver for Strained Si and SiGe × k · p SE for holes: ∂ n + ˆI · eV (z) Fn k (z) = En (k)Fk (z) ∂z ˆ =H ˆ kp + H ˆ so + H ˆ str and with k = (kx, ky ), H ˆ k, kz = −i H V (z) = Ψ(z) + ∆Evav/e, ∆Evav [2]: “natural” valance band offset step of the Si/ SiGe heterostructure The quantum-mechanical charge density: pqm(z) = n (2π) |Fkn|2f (En(k) + EF ) d2k , (1) In contrast to nextnano3, a modified discretization scheme for the twodimensional k space is used in order to reduce the computation time and to calculate (1) with high accuracy Moreover, the CV characteristics for mobility and band-offset extraction are determined by 1st order perturbation theory =⇒ About 30 times less CPU intensive than nextnano [2] C G van de Wall Phys Rev B, vol 35, no 15, pp 81548165, 1987 C Jungemann IWCE 2004 Schră odinger/Poisson Solver for Strained Si and SiGe New interpolation method and grid 0.378 0.40 0.376 0.35 k||=0.2 [π/a0,Si] Energy [eV] Energy [eV] 0.374 0.372 0.370 Nφ=45 0.368 0.366 φ=00 0.30 0.25 0.20 0.15 Nk||=45 Nφ=8, linear inter 0.10 Nk||=8, linear inter Nφ=8, harmonic inter 0.05 Nk||=8, cubic spline inter 0.364 0 10 15 20 25 30 35 40 45 0.1 0.2 o φ[ ] =⇒ C Jungemann 0.3 0.4 Si k|| [π/a0 ] 0.5 0.6 CPU-time gain = 25-30 IWCE 2004 Schră odinger/Poisson Solver for Strained Si and SiGe Band structure of first three subbands (ND = × 1017 cm−3, VG = −2.5V, φ = 0o) and the wave function of the first energy level 1.0 subband subband subband 0.9 0.8 0.4 1,1 F0 (z) strained Si0.4Ge0.6 strained Si 0.7 0.6 Energy [eV] Energy [eV] 0.3 relaxed Si0.7Ge0.3 0.5 0.4 0.2 0.3 0.1 0.2 a) 0.1 0.0 0.1 0.2 C Jungemann 0.3 0.4 0.5 0.6 Si k|| [ π /a0 ] 0.7 0.8 0.9 10 1.0 IWCE 2004 Schră odinger/Poisson Solver for Strained Si and SiGe Hole density at room temperature for two gate biases evaluated by SE 50 FBSC (VG=-4V) FBSC (VG=-2V) Hole density [x10 18 -3 cm ] 40 30 20 10 C Jungemann z [nm] IWCE 2004 10 10 Density Gradient Model C Jungemann IWCE 2004 11 Density Gradient Model Approximate quantum correction by the density gradient model (DGM): p dg (z) = Nv exp Ev + Φm + Λ − EF kB T Here, Φm = (3/2)kB T log(m∗) and Λ is obtained by solving a differential equation: 2γ ¯ − EF ¯ − EF Φ Φ ∇·∇ + ∇ Λ= 12m kB T kB T C Jungemann 2 ¯ = Ev + Φm + Λ , with Φ IWCE 2004 13 Density Gradient Model What is new in strained material compared to relaxed material? ✁ ✁ ✁ ✁ Electrons: ∆Ec(y) known from literature ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ∆Ec(y) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ∆Ev (y, k||) ✁ ✁ tSSi C Jungemann Relaxed Si1−y Gey Holes: ∆Ev (y, k||) depends on k|| ¯v (y) independent from k|| but ∆E required for TCAD (effective valence band offsets) IWCE 2004 14 Extraction of the band offsets for TCAD C Jungemann IWCE 2004 14 Extraction of the band offsets for TCAD 20 Cgd[pF] 15 10 I/II ∆Ev I/II ∆Ev + 40meV - 40meV I/II ana ∆Ev Ii/III ∆Ev II/III ∆Ev -4 C Jungemann -3 II/III and ∆Ev - 40 meV + 40 meV -2 VG[V] -1 • Based on the CV data calculated by SE, the valance band offsets have been extracted by matching the CV data calculated by DGM • The conduction band offsets are fixed during the fitting procedure • Note that in this version the effective mass of Si for DGM was used because no values are available for strained Si and strained SiGe IWCE 2004 15 Extraction of the band offsets for TCAD Gate capacitance with different thickness of strained Si region (T = 300 K) 20 20 DGM FBSC DGM FBSC 15 Cgd[pF] Cgd[pF] 15 10 -4 10 -3 -2 VG[V] -1 0 -4 tSSi = 3.3 [nm] C Jungemann -3 -2 VG[V] -1 tSSi = 4.0 [nm] IWCE 2004 16 Conclusion and Outlook C Jungemann IWCE 2004 17 Conclusion and Outlook • Conclusions – Efficient evaluation of low frequency CV characteristic for multi stacked strained Si structures with a complete description of the valance band structure is now possible – Accurate calculation of CV-characteristics for strained Si/SiGe dual channel pMOSFETs based on Density Gradient Method with the corresponding extracted valance band offsets • Outlook – Improvement of the state of art Density Gradient Model for holes in strained Si and strained Si1−xGex based on our SE/PE solver – Extraction of the heterojunction valence band offsets and other parameters for wide range of Ge contents – Verification of the extracted results by comparison with measured CVdata C Jungemann IWCE 2004 18 ... Motivation ã Schră odinger/Poisson Solver for Strained Si and SiGe • Density Gradient Model • Extraction of the heterojunction valence band offsets needed for TCAD simulators • Conclusion and... – Improvement of the state of art Density Gradient Model for holes in strained Si and strained Si1 −xGex based on our SE/PE solver – Extraction of the heterojunction valence band offsets and other... is now possible – Accurate calculation of CV-characteristics for strained Si/ SiGe dual channel pMOSFETs based on Density Gradient Method with the corresponding extracted valance band offsets •
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