2.9. DOPING IN POLAR MATERIALS 71
(a) (b)(^0204060)
1x10^18
2x10^18
3x10^18
Charge (cm
-3)
Distance (nm)
30%
(^020406080)
2x10^18
4x10^18
Charge (cm
-3)
Distance (nm)
(^020406080)
2x10^19
4x10^19
6x10^19
8x10^19
1x10^20
Charge (cm
-3)
Distance (nm)
Al Comp Al Comp Al Comp
HEMT Linear Grading Parabolic Grade
(c)
Figure 2.28: A 3-dimensional charge distribution can be induced in polar materials via bandgap
grading. (a) 2-dimensional charge distribution induced via an abrupt interface. (b) Linear grade
and (c) parabolic grade result in the displayed 3-dimensional charge distributions. (Figure cour-
tesy S. Rajan, UCSB)
Piezoelectric effect can be exploited to create interface charge densities as high as 1013 cm−^2
in materials. In Table 2.4 we provide the values of piezoelectric constants for some semicon-
ductors. In addition to the polarization induced by strain, the cation and anion sublattices are
spontaneously displaced with respect to each other producing an additional polarization. For
heterostructures the difference of the spontaneous polarization appears at the interfaces, as noted
earlier. In Chapter 1 we have provided the values of spontaneous polarization for AlN, GaN, and
InN.
EXAMPLE 2.5A thin film of Al 0. 3 Ga 0. 7 N is grown coherently on a GaN substrate. Calculate the polar
charge density and electric field at the interface.
The lattice constant of Al 0. 3 Ga 0. 7 N is given by Vegard’s law
aall=0. 3 aAlN+0. 7 aGaN=3. 111 A ̊
The strain tensor is
xx=0. 006
Using the elastic constant values from Chapter 1
zz=− 0. 6 × 0 .006 = 0. 0036
The piezoelectric effect induced polar charge then becomes
Ppz=0.0097 C/m^2