Lecture 22  The Si surface and the MetalOxideSemiconductor Structure (cont.) April 2, 2007


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1 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 221 Lecture 22  The Si surface and the MetalOxideSemiconductor Structure (cont.) April 2, 2007 Contents: 1. Ideal MOS structure under bias (cont.) Reading assignment: del Alamo, Ch. 8, 8.3 ( )
2 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 222 Key questions What are the key simplifications that one can make to the Poisson Boltzmann formulation in order to develop simple firstorder models for the various regimes of operation of the MOS structure? How thick are the inversion and accumulations layers?
3 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Ideal MOS structure outside equilibrium (cont.) Next few viewgraphs: results of calculations for: W M = 4.04 ev (n + polysi gate) x ox = 4.5 nm N A = cm 3 T = 300 K Courtesy of S. Mertens, MIT. Used with permission.
4 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 224 Courtesy of S. Mertens, MIT. Used with permission.
5 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 225 Courtesy of S. Mertens, MIT. Used with permission.
6 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 226 Courtesy of S. Mertens, MIT. Used with permission.
7 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 227 Courtesy of S. Mertens, MIT. Used with permission.
8 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 228 Properties of F(φ) function: F(φ) = φ qφ n 2 i qφ qφ [(exp qφ + 1) + φ N 2 (exp 1)] 1/2 A For φ = 0, F(0) = 0. Courtesy of S. Mertens, MIT. Used with permission. d 2 F For φ = 0, dφ 2 0 = 0. For φ < 0, F(φ) < 0. For φ > 0, F(φ) > 0. For all φ, df > 0. dφ
9 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture 229 Last three terms multiplied by small prefactor: 2 n i NA 2 For φ < 0 and φ more than a few : q F(φ) exp qφ 2 For φ > 0 and φ more than a few q, but not too large: qφ F(φ) For φ > 0 and sufficiently large: F(φ) n i N A exp qφ 2
10 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Approximations for depletion qφ F(φ) Yields: 2ɛ s φ s x d = qn A Relationship between φ and V : γ 2 V V FB φ s = [ ] 2 4 γ 2 Depletion charge: 1 γ 2 V V FB Q s = C ox [ ] 2 γ 2
11 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Approximations for Accumulation F(φ) exp qφ 2 Results in accumulation region thickness: x acc 2L D Relationship between V and φ s : 2 q 1 φ s ln[ q γ (V FB V )] Accumulation charge: In terms of V : Q h = Q s 2ɛ s N A exp qφ s 2 Q h C ox (V FB V )
12 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Approximations for Inversion Piecewise description of F(φ): qφ F(φ) for φ < 2φ f q2φ f q(φ 2φ f ) F(φ) + exp 1 for φ > 2φ f Results in inversion region thickness: x inv 2L D Depletion region thickness: 2ɛ s 2φ f x d = x inv + qn A x dmax Relationship between φ s and V : φ s 2φ f + q 1 q2φ f ln[ q γ 2(V V FB 2φ f ) 2 + 1]
13 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Electron charge in inversion layer: q2φ f q(φ s 2φ f ) q2φ f Q e 2ɛ s N A [ + exp 1 ] In terms of V : Q e C ox (V V th ) with: V th = V FB + 2φ f + γ 2φ f Assumed threshold surface potential φ sth = 2φ f. Empirically, often: with m around 5. φ sth 2φ f + m q
14 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Key conclusions Charge control relationship for inversion charge: Q e C ox (V V th ) Charge control relationship for accumulation charge: Q h C ox (V FB V ) Inversion and accumulation layer thickness of order of Debye length. Better choice for surface potential at threshold: with m around 5. φ sth 2φ f + m q
15 6.720J/3.43J  Integrated Microelectronic Devices  Spring 2007 Lecture Self study Derivation of analytical approximations for electrostatics in depletion, accumulation and inversion from PoissonBoltzmann formulation.
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