4 4 1 4 1 1 4 1 2 4 1 3 MOS CAPACITANCE VOLTAGE CHARACTERISTICS 2 RESULTS BASED ON THE DEPLETION APPROXIMATION 2 Definitions 2 Delta Depletion Approximation 2 Delta Depletion Approximation Deep Depletion 3 4 2 EXACT CHARGE ANALYSIS 3 4 3 MOSC At Intermediate Frequencies 5 4 3 1 4 3 2 4 3 3 4 4 4 4 1 4 4 2 4 4 3 4 4 4 4 4 5 Majority Carrier Response Time 5 Minority Carrier Generation Mechanisms 6 Equivalent Circuit Model 6 MATERIALS AND DEVICE CHARACTERIZATION 8 Oxide And Interface Charge 8 Oxide and Interface Charge Classification 15 Substrate Characterization Pulsed Gate Voltage Technique Zerbst Technique 17 Linear Sweep Technique qualitative discussion 24 Doping Concentration Profiling 25 4 MOS CAPACITANCE VOLTAGE CHARACTERISTICS In this chapter we look in more detail at the MOS CV characteristics including derivations of the capacitance For complete details refer to Tsividis Operation and Modeling of the MOS Transistor McGraw Hill and Nicollian and Brews MOS Physics and Technology Wiley 4 1 RESULTS BASED ON THE DEPLETION APPROXIMATION 4 1 1 Definitions In Chapter 3 we defined the small signal capacitance C dQG dVG where VG VG dc vg i e it comprises ac and dc components For the MOSC C dQG dVG dQG dVox d S or 1 C But in the semiconductor dQG 1 dQG dVox dQS hence 1 C 1 Cox 1 dQG d S 1 CS where Cs is the capacitance due to the semiconductor substrate and Cox is the oxide capacitance We have expressions for the various charge components so we could evaluate the various capacitance components This is tricky since i there is a frequency dependence which is hard to account for easily and ii Q is messy Start with something simpler 4 1 2 Delta Depletion Approximation We make the following simple but inaccurate simplifications to get an idea of the capacitance under the depletion approximation 2 Accumulation C Depletion C Cox C ox C S C ox C S Si where C S w Inversion As discussed qualitatively in the last chapter the inversion layer contributes a capacitance if the frequency is low otherwise the capacitance is determined by the depletion region width which maximizes in inversion Hence for low frequency C Cox 0 C Cox C S Cox C S and for high frequency Si CS wT The resulting capacitance model is crude but simple Considering the depletion region in more detail we obtain an expression for the capacitance in terms of the gate voltage as follows 1 2 2 Si qN B w S Si VG S d ox S ox qN B S w Si Solve these equations for w to get w Si Cox 2VG Cox 2 1 qN B Si Recall the parameter Vo defined earlier 2 1 2 1 qN B Cox2 Vo Si 1 w Now with CS Si w V 1 2 G Vo Si Cox 2 1 we have C ox C 1 ox CS C C ox 1 2 VG Vo Modular Series Vol IV Fig 2 13 4 1 3 Delta Depletion Approximation Deep Depletion Use w derived earlier from Delta Depletion Deep Depletion Section 3 4 2 1 w Si Cox QI Vo Cox 1 2 VG 2 1 Then C ox C QI VO C ox 1 2 VG 4 2 1 2 EXACT CHARGE ANALYSIS C V characteristics based on exact charge analysis are complicated and we present only the results here See Nicollian and Brews Chapters 3 and 4 for details Set up C dQG dVG dQs dVG 3 dQs dU S dU S dVG kT U S U S q VG QS Si Si d ox ox F U s U F LD kT Si U s F U S U F q LD S Result Cox weff C 1 Si weff ox d ox U S L D 2 F U S U F e UF US 1 e e UF e U S 1 1 in accumulation depletion and inversion Also 0 for e U S US 0 e UF 0 US 1 F U S U F U 1 e e U U 1 dU 2 F 3 U U F for Finally at flatbands weff 2 LD e UF e UF 1 2 These results are for p type Si The last equation allows us to calculate the flatband capacitance and to find from there a measured flatband voltage The results just presented are based on the calculation of the charge density we obtained earlier in our Exact Analysis It improves on the exact analysis by allowing a Boltzman distribution of minority carriers based on a constant quasi Fermi level UFn For a p type substrate n ni e U FN eU 1 Assumptions 1 UFn constant i e minority carriers are in thermal equilibrium with themselves 2 Minority carriers can move in response to ac signal but their total number is fixed by dc bias This is the ac inversion layer polarization effect 4 References General Nicollain and Brews Wiley 1981 Low frequency CV A Grove et al Solid State Electronics 8 145 1965 High frequency CV J Brews J Appl Phys 45 1276 1974 Modular Series Vol IV Fig 3 6 4 3 MOSC AT INTERMEDIATE FREQUENCIES We have expressions for the capacitance for 0 and where minority carriers follow the ac signal completely which correspond to situations 0 or not at all w To handle intermediate frequencies we must consider carrier response times We show here how this might be done with reference to Nicollian and Brews MOS Physics and Technology Wiley 1982 Ch 4 The details not presented here are extremely tedious 4 3 1 Majority Carrier Response Time Generally we assume that majority carriers follow the ac signal We estimate the response time as follows Assume a fluctuation in potential of kT q which extends over a distance one extrinsic Debye Length LB recall our earlier discussion of the intrinsic Debye Length LD kT Si q2 NB LB To restore equilibrium majority carriers drift to equalize the disturbed potential The field is kT q LB LB LB LB majority 2 kT q Si majority Si q n 5 D This is the dielectric relaxation time More rigorous development involves Poisson s Eqn see Nicollian and Brews Some numbers Consider depletion which is a worst case i e large D since n is small Take mobility 600 cm2 V s Then at the Si SiO2 interface n ni 1010 cm 3 for which D 10 6 s For the edge of the depletion region n Nd say 1016 cm 3 which gives D 10 12 s Cmeas is dominated by depletion edge capacitance and measurement frequencies of 1MHz majority 1 carriers not a problem since w D 4 3 2 Minority Carrier Generation Mechanisms Minority carrier response time is determined principally by Generation in the space charge region due to bulk and interface traps Supply from an inversion layer beyond the gate edge Generation at the back contact and diffusion through the bulk to the surface We will discuss these mechanisms later when we look at the …