4. MOS CAPACITANCE VOLTAGE CHARACTERISTICS 2 4.1 RESULTS BASED ON THE DEPLETION APPROXIMATION 2 4.1.1 Definitions 2 4.1.2 Delta-Depletion Approximation 2 4.1.3 Delta-Depletion Approximation: Deep Depletion 3 4.2 EXACT CHARGE ANALYSIS 3 4.3 MOSC At Intermediate Frequencies 5 4.3.1 Majority Carrier Response Time 5 4.3.2 Minority Carrier Generation Mechanisms 6 4.3.3 Equivalent Circuit Model 6 4.4 MATERIALS AND DEVICE CHARACTERIZATION 8 4.4.1 Oxide And Interface Charge 8 4.4.2 Oxide and Interface Charge Classification 15 4.4.3 Substrate Characterization: Pulsed Gate Voltage Technique (Zerbst Technique) 17 4.4.4 Linear Sweep Technique (qualitative discussion) 24 4.4.5 Doping Concentration Profiling 252 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 CdQdVGG where VG‟ = VG‟,dc + vg‟, i.e., it comprises ac and dc components. For the MOSC, CdQdVdQdV dGGGox S or 1 1 1CdQdVdQdGoxGS But in the semiconductor, SGdQdQ, hence, 1 1 1C C Cox S 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 Cox Depletion CC CC Cox Sox Swhere CwSSi 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 and for high frequency, CC CC CCwox Sox SSSiT;( ) 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. wqNSiBS21 2/ V dG SSioxox S SBSiqNw Solve these equations for w to get wCVqNCSioxGBoxSi12121 2/ Recall the parameter Vo (defined earlier):3 VqNCwCVVoB SioxSioxGo21 2 112 Now withCwSSi, we have oGoxSoxoxVVCCCCC211 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): wCVQCVSioxGIoxo1 2 112/ Then 2/1/21OoxIGoxVCQVCC 4.2 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 CdQdVdQdVdQdUdUdVGGsGsSSG4 VkTqU U dF U ULG S SSioxoxs FD( , ) Q UkTq LF U US Si S sSiDS F( , ) Result CCwdw U LF U Ue e e eoxeff oxSi oxeff S DS FU U U UF S F S121 1 1( , )( ) ( ) / ( ) in accumulation, depletion, and inversion. Also 0 011 1230fore U F U Ue e e U dUF U UforUS S FU U UFUSFS( ) / ( , )( )( )( , ) Finally, at flatbands, wLe eeffDU UF F21 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 n e eiU UFN( )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.5 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 which correspond to situations where minority carriers follow the ac signal completely ( )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). LkTq NBSiB2 To restore equilibrium, majority carriers drift to equalize the disturbed potential. The field is kT qLB/ qkTLLLBBBmajority2 DSiSimajoritynq6 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 carriers not a problem since wD1. 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 Zerbst characterization technique. For now we simply note that at room temperature, generation in the space charge region dominates the generation of minority carriers, and because of the low defect density in both the bulk and at the