1. Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 8 Lecture 8: Capacitors and PN Junctions Prof. Niknejad

2. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Lecture Outline Review of Electrostatics IC MIM Capacitors Non-Linear Capacitors PN Junctions Thermal Equilibrium

3. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Electrostatics Review (1) Electric field go from positive charge to negative charge (by convention) Electric field lines diverge on charge In words, if the electric field changes magnitude, there has to be charge involved! Result: In a charge free region, the electric field must be constant! +++++++++++++++++++++ − − − − − − − − − − − − − − −

4. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Electrostatics Review (2) Gauss’ Law equivalently says that if there is a net electric field leaving a region, there has to be positive charge in that region: +++++++++++++++++++++ − − − − − − − − − − − − − − − Electric Fields are Leaving This Box! Recall:

5. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Electrostatics in 1D Everything simplifies in 1-D Consider a uniform charge distribution Zero field boundary condition

6. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Electrostatic Potential The electric field (force) is related to the potential (energy): Negative sign says that field lines go from high potential points to lower potential points (negative slope) Note: An electron should “float” to a high potential point:

7. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley More Potential Integrating this basic relation, we have that the potential is the integral of the field: In 1D, this is a simple integral: Going the other way, we have Poisson’s equation in 1D:

8. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Boundary Conditions 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: Field discontiuity implies charge density at surface!

9. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley IC MIM Capacitor 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 Bottom Plate Contacts Bottom Plate Top Plate Thin Oxide

10. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Review of Capacitors For an ideal metal, all charge must be at surface Gauss’ law: Surface integral of electric field over closed surface equals charge inside volume +++++++++++++++++++++ − − − − − − − − − − − − − − − +−+− VsVs

11. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Capacitor Q-V Relation Total charge is linearly related to voltage Charge density is a delta function at surface (for perfect metals) +++++++++++++++++++++ − − − − − − − − − − − − − − −

12. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley A Non-Linear Capacitor 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 +++++++++++++++++++++ − − − − − − − − − − − − − − −

13. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley What’s the Capacitance? For a non-linear capacitor, we have We can’t identify a capacitance Imagine we apply a small signal on top of a bias voltage: The incremental charge is therefore: Constant charge

14. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Small Signal Capacitance Break the equation for total charge into two terms: Constant Charge Incremental Charge

15. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Example of Non-Linear Capacitor Next lecture we’ll see that for a PN junction, the charge is a function of the reverse bias: Small signal capacitance: Constants Charge At N Side of Junction Voltage Across NP Junction

16. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley 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:

17. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Carrier Concentration and Potential (2) We have an equation relating the potential to the carrier concentration If we integrate the above equation we have We define the potential reference to be intrinsic Si:

18. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Carrier Concentration Versus Potential The carrier concentration is thus a function of potential 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 Note that the law of mass action is upheld

19. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley The Doping Changes Potential Due to the log nature of the potential, the potential changes linearly for exponential increase in doping: Quick calculation aid: For a p-type concentration of 10 16 cm -3, the potential is -360 mV N-type materials have a positive potential with respect to intrinsic Si

20. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley n-type p-type NDND NANA 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) 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 − − − + + + + + − − − −V+−V+

21. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley PN Junction Currents Consider the PN junction in thermal equilibrium Again, the currents have to be zero, so we have

22. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley PN Junction Fields n-type p-type NDND NANA Transition Region – – + +

23. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Total Charge in Transition Region To solve for the electric fields, we need to write down the charge density in the transition region: In the p-side of the junction, there are very few electrons and only acceptors: Since the hole concentration is decreasing on the p- side, the net charge is negative:

24. EECS 105 Fall 2003, Lecture 8Prof. A. Niknejad Department of EECS University of California, Berkeley Charge on N-Side Analogous to the p-side, the charge on the n-side is given by: The net charge here is positive since: Transition Region – – + +