# ECE 440 Lecture 20 : PN Junction Electrostatics II Class Outline:

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1 ECE 440 Lecture 20 : PN Junction Electrostatics II Class Outline: Depletion Approximation Step Junction

2 Things you should know when you leave Key Questions What is the space charge region? What are the important quantities? How are the important quantities related to one another? How would bias change my analysis?

3 Depletion Approximation But we already know what will happen when we join them together P-type N-type Dashed Arrows = Particle Flow Solid Arrows = Resulting Currents

4 Depletion Approximation Let s begin to develop a quantitative relationship for the contact potential Let s start by developing a relationship between the contact potential (V 0 ) and the doping concentrations on either side of the junction Begin with the current densities Electrons Holes Drift Diffusion Focus for the moment on the hole current. In equilibrium, the different components should cancel.

5 Depletion Approximation Now begin to simplify the equation Relate the electric field in the above equation to the gradient in the potential, and apply the Einstein relations. After we arrive at: Einstein Relations This is a simple differential equation we can solve by integration. But what are the limits?

6 Depletion Approximation Now we assume that we are dealing with a step-potential In doing so we can further assume that the electron and hole concentrations outside of the transition region are at their equilibrium values. So let s integrate Finish representing the contact potential by relating it to the hole (or electron) concentrations on either side of the junction But this may not be the most useful as we are often dealing with extrinsic semiconductors, so let s rewrite it

7 Depletion Approximation By assuming the majority carrier concentration to be the doping, we can find another form of the equation Using the equilibrium carrier concentration equations, we can extend the above equation to include the electron concentrations on either side of the junction Will be useful later when we apply bias. Or again assuming equilibrium condition of Fermi level continuity across the junction and using the charge density relations, we can derive a final relationship ( E f Ec ) kbt n = Nce ( Ev E f ) kbt p = N e v

8 Depletion Approximation To gain a qualitative understanding of the solution for the electrostatic variables we need Poisson s equation: Most times a simple closed form solution will not be possible, so we need an approximation from which we can derive other relations. Consider the following Use the depletion approximation

9 Depletion Approximation What does the depletion approximation tell us Poisson equation becomes

10 Depletion Approximation What does the depletion approximation look like

11 Depletion Approximation Let s solve a few problems: A silicon step junction is maintained at room temperature with doping concentrations such that E F = E V 2kT on the p-side and E F = E C E G /4 on the n- side. (a) Draw the band diagram (b) Determine the contact potential Consider the p1-p2 isotope junction shown here: (a) Draw the band diagram for the junction taking the doping to be non-degenerate and N A1 > N A2. (b) Derive an expression for the contact potential. (c) Make rough sketches of the electric field, potential and charge density.

12 Step Junction We are already well aware of the formation of the space charge region The space charge region is characterized by: N a < N d Electrons and holes moving across the junction. Only a few carriers at a time being in the space charge region (depletion approximation). Space charge is primarily composed of uncompensated donors and acceptors. We are forming a series of dipoles at the junction.

13 Step Junction What about the electric field in the space charge region? We again begin with Poisson s equation Apply the depletion approximation So how do we get the electric field out of the charge distribution? We integrate

14 Step Junction Let s find the electric field What characteristics does the electric field possess It should have two slopes, positive on the n-side and negative on the p-side. There should be a maximum field at x = 0. P side: These characteristics come from Gauss Law but we could have obtained these qualitatively. E(x) should be negative as it points in the x direction. N side: E(x) goes to zero at the edges of the space charge region.

15 Step Junction p-doped region I N a (-) ionized acceptors W I N d (+) ionized donors n-doped region N d on n-type side of junction > N a on p-type side x po > x no -x po x no # of E field flux lines

16 Step Junction How do we find the maximum field Integrate the Poisson solution. Thus we now have the maximum value of the electric field What about the potential?

17 Step Junction Let s find the potential It is easy to find the contact potential once we have the field in the space charge region The negative of the contact potential is the area under the electric field curve. We can also relate this to the width of the space charge region But we can go farther

18 Step Junction Now we have the width of the space charge region as a function of contact potential, doping concentrations, and other constants However, there are other variations Simplify using the contact potential relation to obtain an equation which depends on doping only N-type P-type

19 Step Junction Closing thoughts about the space charge region

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