SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

2.10. TAILORING ELECTRONIC PROPERTIES 77

The density of states in a quantum well is

• Conduction band

N(E)=

∑

i

m∗

π^2

θ(E−Ei) (2.10.6)

whereθis the heavyside step function (unity ifE>Ei; zero otherwise) andEiare the subband
energy levels.

• Va l e n c e b a n d

N(E)=

∑

i

∑^2

j=1

m∗j

π^2

θ(Eij−E) (2.10.7)

whereirepresents the subbands for the heavy hole(j=1)and light holes(j=2). The density
of states is shown in figure 2.33 and has a staircase-like shape.

The differences between the density of states in a quantum well and a three-dimensional semi-
conductor is one of the important reasons why quantum wells are useful for optoelectronic de-
vices. The key difference is that the density of states in a quantum well is large and finite at the
effective bandedges (lowest conduction subband and highest valence subband). As a result the
carrier distribution is highest at the bandedges.
The relationship between the electron or hole density (areal density for 2D systems) and the
Fermi level is different from that in three-dimensional systems because the density of states
function is different. The 2D electron density in a single subband starting at energyE 1 eis

n =

m∗e

π^2

∫∞

Ee 1

dE

exp

(

E−EF

kBT

)

+1

=

m∗ekBT

π^2

[

ln

{

1+exp

(

EF−Ee 1

kBT

)}]

or EF = Ee 1 +kBTln

[

exp

(

nπ^2

m∗ekBT

)

− 1

]

(2.10.8)

If more than one subband is occupied we can add their contribution similarly. For the hole
density we have (considering both the HH and LH ground state subbands)

p=

m∗hh

π^2

∫−∞

Ehh 1

dE

exp

(

EF−E

kBT

)

+1

+

m∗h

π^2

∫−∞

Eh 1

dE

exp

(

EF−E

kBT

)

+1

(2.10.9)

wherem∗hhandm∗hare the in-plane density of states masses of the HH and LH subbands. We
then have

p =

m∗hhkBT

π^2

[

ln

{

1+exp

(E 1 hh−EFp)

kBT

}]

+

m∗hkBT

π^2

[

ln

{

1+exp

(E 1 h−EFp)

kBT

}]

(2.10.10)

If Ehh 1 −Eh 1 >kBT