Chapter 1 · Visualization 4 of 6 · Section 1.4

Work Functions and Contact Potentials

Put two conductors in contact and watch their Fermi levels line up. The mismatch of work functions appears as a potential step between them, and an ideal voltmeter never sees it.

Before contact: each material has its own Fermi level, a common vacuum level is the reference

After contact (equilibrium): Fermi levels are aligned, vacuum levels differ by the contact-potential step ψ

The ideal voltmeter

What to look for

ψM1,M2 = φw,M2 − φw,M1, ψM1,Mx + ψMx,M2 = ψM1,M2

  • Metals have their Fermi level inside the band; for silicon the work function depends on doping: φw,S = χ + Eg/2q + ΦF, so it moves by about 60 mV per decade of doping.
  • When M1 and M2 touch, electrons flow until EF is the same everywhere. The difference of the work functions is then a potential step ψM1,M2 = φw,M2 − φw,M1 across the contact (Al to p-Si at 10¹⁷ cm⁻³: 5.03 − 4.10 = 0.93 V).
  • In chain mode the two steps add to φw,M2 − φw,M1 for any middle material. This difference between the gate and body work functions is the origin of ΦMS, and so of the flatband voltage VFB in Chapter 2.

Try this

  1. Press “Load book example” and check 5.03 − 4.10 = 0.93 V. Then change the p-Si doping from 10¹⁴ to 10¹⁸: how much does the step move?
  2. Choose n⁺ polysilicon for M1 and p-type silicon for M2. Compare ψ with the same gate on n-type silicon: which has the larger step?
  3. Turn on chain mode, keep M1 = Al and M2 = p-Si, and cycle through the five middle materials. What stays the same and what changes in the diagram?
  4. Replace the gate by p⁺ polysilicon in a polysilicon-to-p-Si pair. By how much does the step differ from the n⁺ gate (compare with Chapter 2's VFB shift of about 1.1 V)?