Lecture 22 - The Si surface and the Metal-Oxide-Semiconductor Structure (cont.) April 2, 2007

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1 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 22-1 Lecture 22 - The Si surface and the Metal-Oxide-Semiconductor 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 22-2 Key questions What are the key simplifications that one can make to the Poisson Boltzmann formulation in order to develop simple first-order 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 22-4 Courtesy of S. Mertens, MIT. Used with permission.

5 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 22-5 Courtesy of S. Mertens, MIT. Used with permission.

6 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 22-6 Courtesy of S. Mertens, MIT. Used with permission.

7 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 22-7 Courtesy of S. Mertens, MIT. Used with permission.

8 6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 22-8 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 22-9 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 Poisson-Boltzmann formulation.