Chapter 1 · Visualization 1 of 6 · Sections 1.2.1 to 1.2.3

Carriers and the Fermi Level

Dope silicon with acceptors or donors, change the temperature, and watch the Fermi level slide through the band gap while the product of electron and hole densities stays fixed.

Energy bands with the Fermi level (dots: electrons near Ec, open circles: holes near Ev; the number of dots grows with log density)

Equilibrium carrier densities (log scale) compared with ni and the doping

Fermi potential ΦF versus doping at the chosen temperature

What to look for

n0p0 = ni², ΦF = φt ln(p0/ni) = (Ei − EF)/q

  • The product n0p0 stays equal to ni² no matter how you dope. Adding majority carriers removes minority carriers by the same factor.
  • The Fermi level sits below Ei for p-type and above it for n-type, by q|ΦF| = kT ln(N/ni): one decade of doping moves it by 60 mV at 300 K.
  • Heat the sample: ni grows roughly exponentially with T. When ni approaches N the silicon turns intrinsic and EF moves back to midgap. Above about 10¹⁹ cm⁻³ Boltzmann statistics fails and EF reaches the band edge (the book then uses ΦF ≈ ±0.56 V).

Try this

  1. Set p-type, N = 10¹⁷ cm⁻³, T = 300 K. Read ΦF (about 0.416 V; the book rounds to 0.417 V using ni = 10¹⁰) and n0 (about 10³ cm⁻³). How many decades separate majority and minority carriers?
  2. Lower N to 10¹² cm⁻³ and raise T to 450 K. Which carrier now dominates, and where is EF?
  3. At 300 K compare ΦF for N = 10¹⁵ and 10¹⁶. Is the difference what kT ln10 predicts?
  4. Switch to n-type. Which quantities change sign and which do not?