EECS 105 Spring 2004 Lecture 9 EECS 105 Spring 2004 Lecture 9 Prof J S Smith Electrostatics Review Lecture 9 Diffusion Electrostatics review and Capacitors z z z Prof J S Smith Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 9 Prof J S Smith The force between any two charges is given by Coulomb s law r qq F e 1 2 2 4 r Where e is unit vector in the direction away from the other charge Since Maxwell s equations are linear we can add up all the forces from other charges and define the electric field at a point r r q2 E 1 F1n e 1n 4 r12n q1 q1 n Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 9 Prof J S Smith Context z z In the last lecture we looked at the carriers in a neutral semiconductor and drift currents In this lecture we will continue to study transport the motion of carriers due to diffusion and the influence of charge distributions on the electric fields z Department of EECS Electrostatic fields z Since we are going to be dealing with large numbers of charges we can use a continuum model The charges are given as va smooth density The electric field is a smooth vector field which diverges from positive charge and converges on negative charges r t Coulombs cm3 University of California Berkeley Review of Electrostatics Diffusion IC MIM Capacitors In the next lecture we will study P N diodes v r r r t E r t Which is one of Maxwell s equations Department of EECS University of California Berkeley 1 EECS 105 Spring 2004 Lecture 9 Prof J S Smith EECS 105 Spring 2004 Lecture 9 Prof J S Smith Gauss s Law z Electrostatic Potential Gauss s Law state that the total amount of E field flux leaving a volume is equal to the net charge enclosed z The electric field force is related to the potential energy E d dx The E field is the slope of the potential E dS Recall z Q Negative sign says that field lines go from high potential points to lower potential points negative slope Note Electrons float to a high potential point Fe qE e Q E dV dV Q V z E dV E dS V V EECS 105 Spring 2004 Lecture 9 Prof J S Smith Department of EECS x E x E x0 x0 z x Department of EECS r r dl C z In 1D this is a simple integral x x x x E x dx 0 x 0 Integrating this basic relation we have that the x potential is the integral of the field x x0 E dl x dx 0 x1 z dx Consider a uniform charge distribution E x Zero field boundary condition Prof J S Smith More Potential In EE105 we are almost always going to use 1 D models so it simplifies dE University of California Berkeley EECS 105 Spring 2004 Lecture 9 Electrostatics in 1D dE dx e d dx 2 University of California Berkeley E Fe e S Department of EECS z 1 d dx 0 x1 x x0 E x dx E x0 z Since the derivative of the E field is the charge we can integrate again to get Poisson s equation in 1D d 2 x x dx 2 x1 University of California Berkeley x0 Department of EECS University of California Berkeley 2 EECS 105 Spring 2004 Lecture 9 Prof J S Smith EECS 105 Spring 2004 Lecture 9 Boundary Conditions z z Note Band edge diagrams Potential must be a continuous function If not the fields forces would be infinite Electric fields need not be continuous We have already seen that the electric fields diverge on charges In fact across an interface we have x E dS E S 1 E1 1 1 2 z z E2 S Qinside Qinside x 0 0 E1 2 E2 1 Field discontinuity implies charge density at surface E field Force on electrons Energy z S We will often draw a diagram of the valence and the conduction band edges as a function of position The energy at the band edge corresponds to the potential energy that an electron has which is the negative of the electrostatic potential Thus the slope of the band edge with distance is the electric field Silicon 1 E1S 2 E2 S 0 E2 2 Prof J S Smith P type N type Distance Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 9 Prof J S Smith Department of EECS EECS 105 Spring 2004 Lecture 9 Boundary conditions Metals z z z z z The presence of metals greatly simplifies boundary conditions Inside a metal the E fields are very small otherwise the current would be very large The inside of a metal has no free charges if it had free charges they are free to move and would very rapidly scatter to the edges of the metal So all of the net charge on a metal occurs on its surface and The surface of a metal is therefore all nearly at the same potential exception long wires conducting a current Department of EECS University of California Berkeley University of California Berkeley Prof J S Smith 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 perfume at temperate T If we suddenly remove the divider what happens The perfume will fill the entire volume of the new chamber How does this occur Department of EECS University of California Berkeley 3 EECS 105 Spring 2004 Lecture 9 Prof J S Smith EECS 105 Spring 2004 Lecture 9 Prof J S Smith Diffusion z z z Diffusion Even though there is no force acting on the perfume molecules because there are more on the left than on the right their random motions take more from the left to the right than are going from right to left Electrons and holes do the same thing but since they are charged they carry current with them Diffusion moves particles in addition to the motion from electric forces z z z If the electrons move long distances without collisions they would quickly spread out diffusion would be large diffusion is proportional to the mean free path If the larger the particles thermal velocities are then the faster diffusion will be diffusion is proportional to vth the mean unidirectional thermal velocity Diffusion is also proportional to the rate of change of the number of particles with distance diffusion is dn dx Flux vth Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 9 Prof J S Smith Department of EECS z z Prof J S Smith Diffusion Equations The net motion of gas molecules to the right chamber was due to the concentration gradient Diffusion will lead to a net flow of particles as long as the distribution of particles is not uniform Diffusion causes a flow of particles from places of high concentration to …
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