ECSE-6230 Semiconductor Devices and Models I Lecture 9

1. ECSE-6230Semiconductor Devices and Models ILecture 9 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 March 12, 2014 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html 1

2. Lecture Outline Homework concepts PN Junction Forward Characteristics Quasi-Fermi Levels Ideal Diode Theory Non-ideal introduction

3. Hmwk 1 Brillouin Zone problem Show how the first Brillouin Zone of the BCC lattice is obtained From primitive BCC lattice vectors, determine reciprocal lattice vectors using equations (FCC lattice) Draw FCC primitive cell as Draw vectors joining center atom to edge atom Create planes intercepting these lines Draw perpendicular bisector planes bounded by intercept lines

4. Hmwk 1 Conduction band electron in silicon is in the (100) valley and has a k-vector of 2π/a (1.0, 0.1, 0.1) Calculate the energy of the electron measured from the conduction band edge. The valley for Si in the x direction or the bottom of the conduction band is given in the handout

5. Hmwk 1 Find Δk, as the energy measured from the bottom:

6. PN Junction Forward Characteristics PN junction ideal forward characteristics are based on: Depletion Approximation Boltzmann Approximation in depletion layer Low-Level Injection No generation within the depletion layer I = I0 exp ( q V / k T) Boltzmann Relation: thermal equilibrium relation sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

7. Quasi-Fermi Levels At thermal equilibrium, np = ni2 EF is constant With forward bias, np > ni2 EF  constant With reverse bias, np < ni2 EF  constant Define the quasi-Fermi levels for electrons and holes, EFn and EFp (solve Boltzmann eq. for EF) The product pn becomes At forward bias At reverse bias sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

8. Quasi-Fermi Levels under Forward Bias • The electron and hole current densities are proportional to he gradients of the electron and hole quasi Fermi levels • The minority carrier concentration varies the most, on either side of the junction • Majority carrier quasi-Fermi level is not affected much (close to original EF) • Gradients within the depletion region are very small, why? Within the dep. region: sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

9. Quasi-Fermi Levels Electron density at the boundary of the depletion layer on the p side Electron density at the boundary of the depeltion layer on the n side np0 and pn0 are equilibrium concentrations sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

10. Carrier Concentration Forward Bias Reverse Bias sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

11. Diffusion Equation One-dimensional steady-state continuity equations are for n-side Assumption 1: Under low-level injection (pn<<nn~nno), Assumption 2: In the neutral region where  = 0, This equation is the Diffusion Equation. Majority carrier electrons Minority carrier holes sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

12. Long Diode Long Diode where pn0 ( xn   ) = pn0 At x = xn,the hole diffusion current is Similarly, on the p-side, at x = - xp, the electron diffusion current is Shockley Equation xn sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

13. Long Diode Currentsunder Forward Bias • Total current must be constant throughout the device • The majority carrier component of current at any point is just the difference between I and the minority component • Since Jp(xn) is proportional to the excess hole concentration at each position in the n material it decreases exponentially in xn with the decreasing δp(xn) • Thus the electron component must increase appropriately with xn to maintain the total current J Current density plot sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

14. Long Diode In the forward direction (positve bias on the p-side) for V>2kT/q the rate of current rise is constant At 300 K, for every decade change in current, the voltage changes by 60mV (= 2.3 kT/q ) In reverse the current density saturates at –J0 sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

15. Deviations from Ideal Diode Theory The Shockley Equation only adequately predicts the I-V characteristics of Ge pn junctions at low current densities. For Si and GaAs pn junctions, deviations are mainly due to: Surface effects, Generation and recombination in the depletion layer, Tunneling of carriers between states in the bandgap, High level injection, Series resistance effects. For modern Si pn junctions, surface leakage currents are much less than the generation current in the depletion layer. Generation-Recombination follows SRH model. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

16. Generation-Recombination Process Under reverse bias, np << ni2 within the depletion layer so the dominant generation-recombination processes are those of carrier emission. Electron-hole pair generate rate is If the lifetime has a slow temperature dependence, Jgen will then have the same temperature dependence as ni. Generation current in depl. Reg. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

17. Generation Current At a given temperature, Jge is proportional to the depletion layer width, which is dependent on the applied reverse bias For abrupt junctions For linearly graded junctions Total reverse current = Jdiff + Jge For Ge junctions, Jdiffusion > Jgen at room temperature For Si and other semiconductors with wider bandgaps, Jdiffusion < Jgen at room temperature. α α sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

18. Recombination Current Under forward bias, the major generation-recombination recombination processes are the carrier capture processes. We have a recombination current, Jrec, in addition to the diffusion current. When np > ni2, Assuming ET=Eiσn=σp=σ sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

19. Recombination Current The maximum value of recombination in the depletion region is where Ei is halfway between EFn and Efp where V = EFn - EFp For V > k T / q, Recombination current Total Forward current = Jdiff +Jre JF αexp(qV/ηkT) (empirical form) sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

20. Forward Current Experimentally, JF ~ exp ( q V / n k T ) n is called the ideality factor. Generation recombination current region Diffusion current region High injection region Series resistance effect Reverse leakage current due to g-r and surface effects When n = 2, the recombination current dominates. When n = 1, the diffusion current dominates. When both currents are comparable, n is between 1 and 2. sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

21. High Level Injection Gradual sloping due applied voltage in no longer totally dropped across the depletion region At high current densities minority carrier concentration is comparable to the majority carrier concentration (both drift and diffusion must be considered) sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html

22. Example A silicon p+n junction has the following parameters at 300K: τp= τg=10-6 s, ND=1015 cm-3. Find the generation current density in the depletion region and the total reverse current density at a bias of 5V. Dp=11.6 cm2/s, Vbi=0.86V • Total reverse current = Jdiff + Jge • Find WD sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html