SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

5.6. SEMICONDUCTOR HETEROJUNCTIONS 237

smaller than that seen by holes by an amountΔEv, and so the corresponding ratio of electron
current to hole current becomes
(
In
Ip

)

het

=

DnNdLp

DpNaLn

exp

(

ΔEv

kBT

)

(5.6.19)

We can see that even ifNais kept high, the ratio of electron current to hole current in a hetero-
junction can still be kept large, even for a relatively modest bandgap discontinuity.

5.6.2 Gradedp-nheterojunction .........................

Although the abruptp-nheterostructure discussed in the previous section did result in an
increase in the barrier to hole injection, the notch in the conduction band at the interface also
caused an undesirable increase in the barrier to electron injection. While the net effect was still
an increase in the ratioIn/Ip, eliminating this notch further increasesIn/Iptoavalue

(

In

Ip

)

=

DnNdLp

DpNaLn

exp

(

ΔEg

kBT

)

(5.6.20)

In order to reduce this notch, the bandgap of thepmaterial can be graded upwards from the
junction, as shown in figure 5.11b. For example, in ann-AlGaAs/p-GaAs graded heterostructure,
thenmaterial is GaAs at the junction and is graded to the final AlGaAs composition over a short
distance. The final shape of the notch depends on the length and profile of the grade; longer
grading typically gives a smaller notch. However, it is important that the grade is contained well
within the depletion region. If the grade ends outside the depletion region, then the barrier seen
by holes decreases, thus reducing the benefits of the heterojunction. Note that the barrier to holes
in both abrupt and graded heterojunctions is the same. It is just the barrier for electron flow that
is reduced in the graded structures, allowing for the increased ratio ofIn/Ip.

Example 5.3 Designing ap-nheterojunction grade

Consider four different n-p+Al 0. 3 Ga 0. 7 As/GaAs heterojunctions with ND=10^17 and

NA=5× 1018. The AlGaAs in these junctions is graded fromx=0tox=0. 3 over

XGrade=0(abrupt),XGrade= 100A ̊,XGrade= 300A ̊,andXGrade=1μ. Calculate

and plot the energy band diagrams for the above four cases.

Assume the dielectric constant of AlGaAs to be the same as that of GaAs.

Eg=1.8eVforAl 0. 3 Ga 0. 7 As, and Eg= 1.4 eV for GaAs.ΔEg= 0.374 eV,

ΔEC= 0.237 eV, andΔEV= 0.137 eV. On the AlGaAs emitter side, far away from the

junction,

φn=EC−EF=kBTln

(

NC

n

)

=0.0323 eV (5.6.21)

Since the thep-GaAs is degenerately doped, Joyce-Dixon statistics must be applied:

φp=EV−EF=