SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

(Greg DeLong) #1
2.7. MOBILE CARRIERS 57

electron density in the conduction band is


n =

∫∞

Ec

Ne(E)f(E)dE

n =

1

2 π^2

(

2 m∗e
^2

) 3 / 2 ∫∞

Ec

(E−Ec)^1 /^2 dE
exp (Ek−BETF)+1

(2.7.7)

For small values ofn(non-degenerate statistics where we can ignore the unity in the Fermi
function) we get
n=Ncexp [(EF−Ec)/kBT] (2.7.8)


where the effective density of statesNcis given by


Nc=2

(

m∗ekBT
2 π^2

) 3 / 2

A similar derivation for hole density gives


p=Nvexp [(Ev−EF)/kBT] (2.7.9)

where the effective density of statesNvis given by


Nv=2

(

m∗hkBT
2 π^2

) 3 / 2

We also obtain
np=4

(

kBT
2 π^2

) 3

(m∗em∗h)^3 /^2 exp (−Eg/kBT) (2.7.10)

Notice that within our low carrier density approximation, the productnpis independent of the
position of the Fermi level and is dependent only on the temperature and intrinsic properties of
the semiconductor. This is the law of mass action. Ifnincreases,pmust decrease, and vice
versa. For the intrinsic casen=ni=p=pi, we have from the square root of the equation
above


ni=pi=2

(

kBT
2 π^2

) 3 / 2

(m∗em∗h)^3 /^4 exp (−Eg/ 2 kBT)

EFi=

Ec+Ev
2

+

3

4

kBTln (m∗h/m∗e) (2.7.11)

Thus the Fermi level of an intrinsic material lies close to the midgap.
In Table 2.3 we show the effective densities and intrinsic carrier concentrations in Si, Ge, and
GaAs The values given are those accepted from experiments. These values are lower than the
ones we get by using the equations derived in this section. The reason for this difference is due
to inaccuracies in carrier masses and the approximate nature of the analytical expressions.
We note that the carrier concentration increases exponentially as the bandgap decreases. Re-
sults for the intrinsic carrier concentrations for Si, Ge, GaAs, and GaN are shown in figure 2.18.

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