16.4 Piezoelectricity
461
P pe =
⎛
⎝
2 e 15 xz
2 e 15 yz
e 31 (( xx + yy ) + e 33 zz
⎞
⎠ .
(16.17)
Values for the piezoelectric coefficients of several wurtzite semiconductors
5 are listed in Table 16.1.
The polarization (along c) for biaxial strain in heteroepitaxy (5.69) on the [00.1] surface is
P pe = 2
e 31 −
C 13
C 33
e 33
,
(16.18)
where = (a − a 0 )/a 0 is the in-plane strain. The dependence of the magnitude for GaN on the
in-plane strain is shown in Fig. 16.13 together with the polarization for uniaxial stress along [00.1] and
hydrostatic strain. In the latter two cases, the polarization is smaller.
The difference of spontaneous polarization of the constituent materials in heterostructures and piezoelectric effects in strained quantum wells lead to quantum confined Stark effect (QCSE, Sect. 15.1.2).
The spatial separation of electrons and holes leads to a larger radiative lifetime (Fig. 12.43) and thus
reduced radiative recombination rate in the presence of nonradiative channels, an effect unwanted in
light emitting diodes. In particular nitride based LEDs grown along the c-direction are prone to this
effect. Therefore growth on nonpolar surfaces such as (1 ¯
100) (m-plane) and (11 ¯
20) (a-plane) has been
investigated [1509, 1510]. However, crystal quality for these growth planes seems limited.
An alternative route are ‘semipolar’ planes, e.g. (10 ¯
1 ¯
1) or (11 ¯
22) [1511], with at least reduced
polarization effects. The (11 ¯
22)-plane is tilted about 58
◦ (for GaN) from the c-axis, close to the theoretically predicted zero of the internal electric field for InGaN/GaN quantum wells [1512] (Fig. 16.14).
Promising experimental results have been published on semipolar growth, especially for InGaN/GaN
on (11 ¯
22)-oriented GaN substrates [1513].
Fig. 16.13 Piezoelectric
polarization P pe in GaN
(Ga-face) versus in-plane
strain = (a − a 0 )/a 0 for
biaxial, uniaxial and
hydrostatic strain. The
value of the spontaneous
polarization P sp is
indicated by an arrow.
From [1508]
P
P
P
x
z
y
P
P
x
y
P z
10
10
10
10
10
2
0
.
0
-
0
-0.04
-0.06
-0.08
-0.10
P
(C/m
)
pe e
2
0
-2
-1
-3
-4
GaN
Ga-face
P sp
biaxial
uniaxial
hydrostatic
||
5 If e 14 for a zincblende material is transformed to a coordinate system along [111], the wurtzite-like piezoelectric
constants are e 33 = 2 e 14 /
√
3 and e 31 = −e 14 /
√
3 [1507].
461
P pe =
⎛
⎝
2 e 15 xz
2 e 15 yz
e 31 (( xx + yy ) + e 33 zz
⎞
⎠ .
(16.17)
Values for the piezoelectric coefficients of several wurtzite semiconductors
5 are listed in Table 16.1.
The polarization (along c) for biaxial strain in heteroepitaxy (5.69) on the [00.1] surface is
P pe = 2
e 31 −
C 13
C 33
e 33
,
(16.18)
where = (a − a 0 )/a 0 is the in-plane strain. The dependence of the magnitude for GaN on the
in-plane strain is shown in Fig. 16.13 together with the polarization for uniaxial stress along [00.1] and
hydrostatic strain. In the latter two cases, the polarization is smaller.
The difference of spontaneous polarization of the constituent materials in heterostructures and piezoelectric effects in strained quantum wells lead to quantum confined Stark effect (QCSE, Sect. 15.1.2).
The spatial separation of electrons and holes leads to a larger radiative lifetime (Fig. 12.43) and thus
reduced radiative recombination rate in the presence of nonradiative channels, an effect unwanted in
light emitting diodes. In particular nitride based LEDs grown along the c-direction are prone to this
effect. Therefore growth on nonpolar surfaces such as (1 ¯
100) (m-plane) and (11 ¯
20) (a-plane) has been
investigated [1509, 1510]. However, crystal quality for these growth planes seems limited.
An alternative route are ‘semipolar’ planes, e.g. (10 ¯
1 ¯
1) or (11 ¯
22) [1511], with at least reduced
polarization effects. The (11 ¯
22)-plane is tilted about 58
◦ (for GaN) from the c-axis, close to the theoretically predicted zero of the internal electric field for InGaN/GaN quantum wells [1512] (Fig. 16.14).
Promising experimental results have been published on semipolar growth, especially for InGaN/GaN
on (11 ¯
22)-oriented GaN substrates [1513].
Fig. 16.13 Piezoelectric
polarization P pe in GaN
(Ga-face) versus in-plane
strain = (a − a 0 )/a 0 for
biaxial, uniaxial and
hydrostatic strain. The
value of the spontaneous
polarization P sp is
indicated by an arrow.
From [1508]
P
P
P
x
z
y
P
P
x
y
P z
10
10
10
10
10
2
0
.
0
-
0
-0.04
-0.06
-0.08
-0.10
P
(C/m
)
pe e
2
0
-2
-1
-3
-4
GaN
Ga-face
P sp
biaxial
uniaxial
hydrostatic
||
5 If e 14 for a zincblende material is transformed to a coordinate system along [111], the wurtzite-like piezoelectric
constants are e 33 = 2 e 14 /
√
3 and e 31 = −e 14 /
√
3 [1507].