ECSE-6230 Semiconductor Devices and Models I Lecture 8

1. ECSE-6230Semiconductor Devices and Models ILecture 8 Prof. Shayla Sawyer Bldg. CII, Rooms 8225 Rensselaer Polytechnic Institute Troy, NY 12180-3590 Tel. (518)276-2164 Fax. (518)276-2990 e-mail: sawyes@rpi.edu April 1, 2014 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html 1

2. Lecture Outline PN Junctions Built in Potential Depletion Approximation Depletion Width Depletion Capacitance Linearly Graded Junction

3. PN Junction Background The pn junction theory serves as the foundation of the physics of semiconductor physics The basic equations are used to develop the ideal static and dynamic characteristics of pn junctions Departures from ideal Generation and recombination in the depletion layer High injection Series resistance Junction breakdown sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

4. Built-in Potential: Abrupt pn junction Joining the two regions of p and n type doping causes diffusion Large carrier concentration gradient at the junction Space charge region: diffusion carriers leave behind uncompensated donor ions in the n region (Nd+) and uncompensated acceptor ions in the p region (Na-) Drift current opposes diffusion current n p n p sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

5. Built-In Potential: Abrupt pn junction • Constant Fermi level throughout the sample • Band bending and built in potential occurs • Electric field is proportional to the slope of the bands, charge density proportional to the curvature sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

6. Built-In Potential: Derivation Finding potential Solving for the electric field from JN equation and with Einstein’s relationship, integrate the electric field to find Vbi Vbi for a non degenerate semiconductor is sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

7. Built-In Potential: Derivation Built-in potential is the internal potential difference between the p-side and the n-side of the junction sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

8. Depletion Approximation Electrostatic Solution of an abrupt, uniformly doped pn junction at thermal equilibrium ( J = 0 ) Poisson’s Equation Depletion approximation allows Poisson’s equation to be solved easily It assumes that the mobile carriers (n and p) are small in number compared to the donor and acceptor ion concentrations in the depletion region The device is charge neutral elsewhere sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

9. Depletion Width, Vbi, Carrier Conc. Poisson’s Equation Integrate above: For -xp < x < 0 ( p-side), For 0 < x < xn ( n-side), Maximum electric field is given by sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

10. Depletion Width, Vbi, Carrier Conc. Integrate electric field equation to get potential distribution Potential across different regions =Ψbi -Ψn Ψp sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

11. Depletion Width, Vbi, Carrier Conc. Depletion widths are calculated to be Penetration of the transition region into the n and p materials sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

12. Depletion Capacitance Variation of charge within a pn junction with an applied voltage variation yields a capacitance Capacitance in non-linear; derive small signal capacitance associated with the depletion layer of a pn junction. Define change of charge DC signal VA has a small signal superimposed onto it va, W increases or decreseas by an increment of ΔW, for small signals va<<VA, ΔW<<W Charge is only added and removed at the edge of the depletion region When va>0, W decreases (neutralizing ions) When va < 0, W increases (more depletion) sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

13. Depletion Capacitance Two observations: Charges are majority carriers which respond to the voltage change at roughly the dielectric relaxation time of the material At normal doping levels the majority carrier response time is from 10-10 to 10-12 sec. Therefore, the phenomena will be independent of the frequency of vaup to very high frequencies The incremental charge diagrams are similar to charge fluctuations in parallel plates of a capacitor with area A and width separation W sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

14. Depletion Capacitance Depletion layer capacitance per unit area pn junction Depletion layer capacitance per unit area for a one sided abrupt junction p+ n where Na>>>Nd and xno~W and xpo is negligible Parameter extraction Rearrange and solve for 1/C2 plot vs Voltage sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

15. Depletion Layer Widths in PN Junctions sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

16. Debye Length Capacitance voltage data are insensitive to changes in doping profiles that occur in a distance less than a Debye Length Limit of a potential change in response to an abrupt change in the doping profile If the depletion width is smaller than the Debye Length, the analysis using the Poisson’s equation is no longer valid Also defined as the width of the transition region in which the carrier depletion goes from 100% to 0% and can written as Typically, W = 8LD in Si = 6 LD in Ge = 10 LD in GaAs sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

17. Extrinsic Debye Length sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

18. Linearly Graded PN Junction • In practical devices the doping profiles are not abrupt • Near the metallurgical junction the two types compensate each other • When depletion widths terminate within this transition region, the doping profile can be approximated as a linear function Poisson’s equation a is the doping gradient in cm-4 Electric field (integrate above) sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

19. Linearly Graded PN Junction • Built in potential related to the depletion width or • Depletion layer capacitance sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

20. Example • An abrupt Si p-n junction has Na=1018 cm-3 on one side and Nd = 5 x 1015 cm-3 on the other. It also has a circular cross section with a diameter of 10 μm. a) Calculate the built in voltage b) Calculate the depletion width c) The penetration of the depletion width into the n and p region (xpo and xno d) Calculate the maximum for the electric field ξm e) Sketch ξ(x) and the charge density to scale sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html