SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

442 CHAPTER 9. FIELD EFFECT TRANSISTORS: MOSFET

negative bias is needed to cause inversion. From Chapter 2, using Boltzmann statistics,

φF=

kBT

e

ln

p

ni

∼

kBT

e

ln

Na

ni

(9.3.2)

whereNais the acceptor density andniis the intrinsic carrier concentration. The strong inver-
sion criterion then becomes

ψs(inv)=2

kBT

e

ln

Na

ni

(9.3.3)

Later we will develop a model for sub-threshold current based on a more gradual transition
in the electron density. At the onset of strong inversion there is an electron charge density of
∼ 1011 cm−^2 at the surface so that the interface region’s conductivity is high. Let us now
evaluate the charge in the semiconductor channel. The electron concentration is approximately
given by the Boltzmann distribution. In the bulk region, this concentration is

np 0 =niexp (EF−EFi)/kBT=niexp

(

eφF

kBT

)

(9.3.4)

We are interested in calculating the carrier concentration in the semiconductor near the Si-SiO 2
interface.
A detailed overview of the charge, electric field, and potential in the inversion regime is shown
in figure 9.10. The areal charge density on the metalQmis balanced by the channel depletion
chargeQdand the inversion chargeQn. We are interested in calculating the gate voltage needed
to cause inversion in the channel. This voltage is called the threshold voltage.
The total surface charge density is related to the surface field by Gauss’ law and is

|Qs|=s|Es| (9.3.5)

This chargeQsis the total surface charge density at the semiconductor-oxide interface region
and includes the induced free charge (in inversion) and the background ionic charge. The charge
Qsgoes to zero when the bands are flat.
We can relate the gate voltage to the surface potentialψsby using the continuity of the electric
displacement across the oxide-semiconductor interface (EsandEoxare the electric fields in the
semiconductor and the oxide at the interface):

sEs=oxEox (9.3.6)

The voltage between the gate and the semiconductor is best understood by starting from the flat
band condition such that
VGS−Vfb=ΔVox+ψs

or the applied voltage difference from flat-band is the sum of the change the oxide voltage,ΔVox
andψs. (Note: In the absence of additional fixed charges and traps in the system,Voxat flat-band
is zero andVfb=φms.) In general

VGS−Vfb=ΔVox+ψs (9.3.7)