EECS 105 Spring 2005 Lecture 27 Lecture 27 PN Junctions Prof Niknejad Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Diffusion z z z z Diffusion occurs when there exists a concentration gradient In the figure below imagine that we fill the left chamber with a gas at temperate T If we suddenly remove the divider what happens The gas will fill the entire volume of the new chamber How does this occur Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Diffusion cont z z z z The net motion of gas molecules to the right chamber was due to the concentration gradient If each particle moves on average left or right then eventually half will be in the right chamber If the molecules were charged or electrons then there would be a net current flow The diffusion current flows from high concentration to low concentration Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Diffusion Equations z z Assume that the mean free path is Find flux of carriers crossing x 0 plane n 0 n n 1 dn dn F vth n 0 n 0 2 dx dx 1 dn n vth 2 F vth dx 1 n vth 2 Department of EECS 1 F vth n n 2 0 J qF qvth dn dx University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Einstein Relation z The thermal velocity is given by kT 1 2 mn vth2 12 kT vth c Mean Free Time kT q c vth v kT mn q mn 2 th c c kT dn dn J qvth q n dx q dx kT Dn n q Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Total Current and Boundary Conditions z When both drift and diffusion are present the total current is given by the sum J J drift J diff z z dn q n nE qDn dx In resistors the carrier is approximately uniform and the second term is nearly zero For currents flowing uniformly through an interface no charge accumulation the field is discontinous J1 1 J 2 2 Department of EECS J1 J 2 1 E1 2 E2 E1 2 E2 1 University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Carrier Concentration and Potential z In thermal equilibrium there are no external fields and we thus expect the electron and hole current densities to be zero dno J n 0 qn0 n E0 qDn dx n dno d q no E0 no 0 dx kT dx Dn kT dno dn0 Vth d 0 n0 q n0 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Carrier Concentration and Potential 2 z We have an equation relating the potential to the carrier concentration kT dno dn d 0 Vth 0 n0 q n0 z If we integrate the above equation we have n0 x 0 x 0 x0 Vth ln n0 x0 z We define the potential reference to be intrinsic Si 0 x0 0 Department of EECS n0 x0 ni University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Carrier Concentration Versus Potential z The carrier concentration is thus a function of potential n0 x ni e 0 x Vth z z 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 z Note that the law of mass action is upheld n0 x p0 x ni2 e 0 x Vth e 0 x Vth ni2 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad The Doping Changes Potential z Due to the log nature of the potential the potential changes linearly for exponential increase in doping n0 x n0 x n0 x 0 x Vth ln 26mV ln 26mV ln 10 log 10 ni x0 ni x0 10 n0 x 0 x 60mV log 10 10 p0 x 0 x 60mV log 10 10 z z 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 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad PN Junctions Overview z z z z z z z 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 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 In thermal equilibrium drift current and diffusion must balance Department of EECS p type NA V ND n type University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad PN Junction Currents z z Consider the PN junction in thermal equilibrium Again the currents have to be zero so we have J n 0 qn0 n E0 qDn dno dx dno qn0 n E0 qDn dx E0 dno dx kT 1 dn0 n0 n q n0 dx Dn dpo Dp kT 1 dp0 dx E0 n0 p q p0 dx Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad PN Junction Fields p type n type NA ND p0 N a p0 x E0 x p0 ni2 n0 Na E0 J diff xn 0 ni2 p0 Nd n0 N d J diff Transition Region Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Total Charge in Transition Region z 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 z In the p side of the junction there are very few electrons and only acceptors 0 x q p0 N a z x p0 x 0 Since the hole concentration is decreasing on the pside the net charge is negative N a p0 Department of EECS 0 x 0 University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Charge on N Side z Analogous to the p side the charge on the n side is given by 0 x q n0 N d z 0 x xn 0 The net charge here is positive since 0 x 0 N d n0 n0 N d ni2 n0 Na E0 J diff Department of EECS Transition Region University of California Berkeley EECS 105 Fall 2003 Lecture 27 Prof A Niknejad Exact Solution for Fields z Given the above approximations we now have an expression for …
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