EE105 Fall 2006 Microelectronic Devices and Circuits Prof Jan M Rabaey jan eecs Lecture 4 Capacitors PN Junctions Overview Last lecture Diffusion currents Overview of IC fabrication process Review of electrostatics This lecture Capacitances pn Junctions 2 1 Administrativia Another Make up Lecture Monday at 4pm streamed NO LECTURE ON TUESDAY 3 IC MIM Capacitor Bottom Plate Top Plate Bottom Plate Contacts Thin Oxide Q CV By forming a thin oxide and metal or polysilicon plates a capacitor is formed Contacts are made to top and bottom plate Parasitic capacitance exists between bottom plate and substrate 4 2 Review of Capacitors Q E dS Vs Q E dS E dl E t 0 ox Vs E dS E0 A Q Vs tox Vs Q A tox Q CVs E0 C A tox For an ideal metal all charge must be at surface Gauss law Surface integral of electric field over closed surface equals charge inside volume 5 Capacitor Q V Relation Q y Vs Q y y Q CVs Total charge is linearly related to voltage Charge density is a delta function at surface for perfect metals 6 3 A Non Linear Capacitor y Q Q y Vs y Q f Vs We ll soon meet capacitors that have a non linear Q V relationship If plates are not ideal metal the charge density can penetrate into surface 7 What s the Capacitance For a non linear capacitor we have Q f Vs CVs We can t identify a capacitance Imagine we apply a small signal on top of a bias voltage Q f Vs vs f Vs df V vs dV V Vs Constant charge The incremental charge is therefore Q Q0 q f Vs df V vs dV V Vs 8 4 Small Signal Capacitance Break the equation for total charge into two terms Incremental Charge Q Q0 q f Vs df V vs dV V Vs Constant Charge q df V vs C vs dV V Vs C df V dV V Vs 9 Example of Non Linear Capacitor Next lecture we ll see that for a PN junction the charge is a function of the reverse bias Q j V qN a x p 1 Charge At N Side of Junction V Voltage Across NP Junction b Constants Small signal capacitance C j V dQ j dV qN a x p 2 b 1 1 V b C j0 1 V b 10 5 PN Junctions Diodes 11 Carrier Concentration and Potential In thermal equilibrium there are no external fields and we thus expect the electron and hole current densities to be zero J n 0 qn0 n E0 qDn dno dx dno q d n no E0 no 0 dx kT dx Dn kT dn dn d 0 o Vth 0 n0 q n0 12 6 Carrier Concentration and Potential 2 We have an equation relating the potential to the carrier concentration kT dn dn d 0 o Vth 0 n0 q n0 If we integrate the above equation we have 0 x 0 x0 Vth ln n0 x n0 x0 We define the potential reference to be intrinsic Si 0 x0 0 n0 x0 ni 13 Carrier Concentration Versus Potential The carrier concentration is thus a function of potential n0 x ni e 0 x Vth Check that for zero potential we have intrinsic carrier concentration reference If we do a similar calculation for holes we arrive at a similar equation p0 x ni e 0 x Vth Note that the law of mass action is upheld n0 x p0 x ni2e 0 x Vth e 0 x Vth ni2 14 7 The Doping Changes Potential Due to the log nature of the potential the potential changes linearly for exponential increase in doping n x n x n x 0 x Vth ln 0 26mV ln 0 26mV ln 10 log 0 10 ni x0 ni x0 10 0 x 60mV log n0 x 1010 0 x 60mV log p0 x 1010 Quick calculation aid For a p type concentration of 1016 cm 3 the potential is 360 mV N type materials have a positive potential with respect to intrinsic Si 15 PN Junction Overview 16 8 PN Junction Overview 17 PN Junction Overview Present in most IC structures 18 9 PN Junctions Overview The most important device is a junction between a p type region and an n type region When the junction is first formed due to the concentration gradient mobile charges transfer near junction Electrons leave n type region and holes leave p type region These mobile carriers become minority carriers in new region can t penetrate far due to recombination p type NA V Due to charge transfer a voltage difference occurs between regions This creates a field at the junction that causes drift currents to oppose the diffusion current ND n type In thermal equilibrium drift current and diffusion must balance 19 PN Junction Currents Consider the PN junction in thermal equilibrium Again the currents have to be zero so we have J n 0 qn0 n E0 qDn qn0 n E0 qDn E0 dno dx dno dx dno dx kT 1 dn0 n0 n q n0 dx Dn dpo dx kT 1 dp0 E0 n0 p q p0 dx Dp 20 10 PN Junction Fields p type n type NA ND p0 N a p0 x J diff E0 x p0 n0 xn 0 p0 ni2 Nd n0 N d J diff ni2 Na E0 21 Transition Region Total Charge in Transition Region To solve for the electric fields we need to write down the charge density in the transition region 0 x q p0 n0 N d N a In the p side of the junction there are very few electrons and only acceptors 0 x q p0 N a x p0 x 0 Since the hole concentration is decreasing on the pside the net charge is negative N a p0 0 x 0 22 11 Charge on N Side Analogous to the p side the charge on the n side is given by 0 x q n0 N d 0 x xn 0 The net charge here is positive since 0 x 0 N d n0 n0 N d n0 ni2 Na J diff E0 23 Transition Region Exact Solution for Fields Given the above approximations we now have an expression for the charge density q ni e 0 x Vth N a 0 x 0 x Vth q N d ni e x po x 0 0 x xn 0 We also have the following result from electrostatics dE0 d 2 0 x 2 dx dx s Notice that the potential appears on both sides of the equation difficult problem to solve A much simpler way to solve the problem 24 12 Depletion Approximation Let s assume that the transition region is completely depleted of free carriers only immobile dopants exist Then the charge density is given by qN a x po x 0 qN d 0 x xn 0 0 x The solution for electric field is now easy dE0 0 x Field zero outside dx s transition region x 0 x E0 x …
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