SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

382 CHAPTER 8. FIELD EFFECT TRANSISTORS

2. It effectively acts as a voltage divider between the gate and source as the Fermi level in
   the channelEF,chis raised relative to the Fermi level in the sourceEF,s, and hence only
   part of the applied gate-source voltageVGSis used for charge control. Thus the intrinsic
   source-gate biasVGS,intis related to the applied source gate voltageVGSby

VGS,int=VGS−(EF,ch−EF,s) (8.5.3)

This voltage division (or reduced charge control) can also be represented by a displacement
of the centroid of the 2DEGΔdaway from the heterointerface (see figure 8.15c), effectively
increasing the gate to channel distance tod+Δdand reducing the gate capacitanceCGto

CG=

A

d+Δd

This is sometimes referred to as gate capacitance reduction due to a quantum capacitance asso-
ciated with motion of the fermi level. We now calculate an analytic expression forΔd. We first
assume that the 2DEG forms a triangular potential well, as shown in figure 8.16. The sub-band
energies are well known to be

Ei

(

^2

2 m∗

) 1 / 3 [

3

2

eE 2 π

(

i+

3

4

)] 2 / 3

(8.5.4)

We assume that the electric fieldE 2 is generated by only the 2DEG chargens, yielding

Ei

(

^2

2 m∗

) 1 / 3 [

3

2

eπ

(

i+

3

4

)] 2 / (^3) (
ens


) 2 / 3

= γin^2 s/^3 (8.5.5)

wherei=0, 1 , 2 , ..., n. The coefficientsγiare material dependent, explicitly related to the
density of states effective massm∗. Typical values for GaAs are

γ 0 =2. 5 × 10 −^12 eV·cm^4 /^3

γ 1 =3. 2 × 10 −^12 eV·cm^4 /^3

The 2DEG concentration is related to the position of the Fermi level via the Fermi-Dirac
distribution

ns=Ds

kBT

e

∑n

i=0

ln

[

1+exp

(

e(EF−Ei)

kBT

)]

(8.5.6)

whereDsis the 2D density of states

Ds=

em∗

π^2

(8.5.7)

Assuming only the first sub-band is dominant, we can write

ns=Ds

kBT

e

ln

[

1+exp

(

e(EF−E 0 )

kBT

)]

(8.5.8)