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Anisotropic scattering effect of the inclined misfit dislocation on the two-dimensional
electron gas in Al(In)GaN/GaN heterostructures
Dong-Dong Jin, Lian-shan Wang, Shao-Yan Yang, Liu-Wan Zhang, Hui-jie Li, Heng Zhang, Jian-xia Wang, Ruofei Xiang, Hong-yuan Wei, Chun-mei Jiao, Xiang-Lin Liu, Qin-Sheng Zhu, and Zhan-Guo Wang
Citation: Journal of Applied Physics 115, 043702 (2014); doi: 10.1063/1.4862803
View online: http://dx.doi.org/10.1063/1.4862803
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/4?ver=pdfcov
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JOURNAL OF APPLIED PHYSICS 115, 043702 (2014)
Anisotropic scattering effect of the inclined misfit dislocation
on the two-dimensional electron gas in Al(In)GaN/GaN heterostructures
Dong-Dong Jin,1,2 Lian-shan Wang,1,a) Shao-Yan Yang,1,b) Liu-Wan Zhang,2 Hui-jie Li,1
Heng Zhang,1 Jian-xia Wang,1 Ruo-fei Xiang,1 Hong-yuan Wei,1 Chun-mei Jiao,1
Xiang-Lin Liu,1 Qin-Sheng Zhu,1,c) and Zhan-Guo Wang1
1
Key Laboratory of Semiconductor Materials Science, and Beijing Key Laboratory of Low Dimensional
Semiconductor Materials and Devices, Institute of Semiconductors, Chinese Academy of Sciences,
P.O. Box 912, Beijing 100083, People’s Republic of China
2
Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China
(Received 10 December 2013; accepted 8 January 2014; published online 23 January 2014)
In this paper, a theory is developed to study the anisotropic scattering effect of the inclined misfit
dislocation on the two-dimensional electron gas in Al(In)GaN/GaN heterostructures. The inclined
misfit dislocation, which differs from the well-known vertical threading dislocation, has a
remarkable tilt angle from the vertical. The predicted electron mobility shows a remarkable
anisotropy. It has a maximum mobility value along the direction perpendicular to the projection of
the inclined dislocation line, and a minimum mobility value along the direction parallel to the
projection. The degree of the anisotropic scattering effect will be even greater with the increase of
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862803]
the tilt angle. V
I. INTRODUCTION
Over the past decade or so, Al(In)GaN/GaN heterostructures have been an area of active research, owing to the
demonstration of high power microwave high electron
mobility transistors (HEMTs). These devices often require
high quality hetero-epitaxial layers on sapphire, SiC or
Si(111) having different compositions in order to optimize
electrical and optical performance.1–6 However, due to the
large lattice mismatch with the present substrate of choice,
hetero-epitaxial Al(In)GaN layers experience significant
mechanical stresses.
Many laboratories7–9 had shown that these stresses
could be relaxed by kinds of misfit dislocations both in the
polar and non-polar heterostructures. Different from our
intuition, most of the dislocations are not vertically passing
through the interfacial plane, but they have different levels
of inclination.10–15 For example, Floro et al.10 found experimentally the inclined dislocations in (1122) glid planes
by about 60 from the vertical in polar (0001) AlGaN/GaN
heterostructures. Meanwhile, the other typical inclined dislocations has been observed by many other authors11–15 in the
Al(In)GaN layers with different tilt angles ranged from 15
to 60 .
Up to now, however, the anisotropic scattering effect of
the inclined dislocations on two-dimensional electron gas
(2DEG) has never been noted. At low temperatures, the
mobility is limited by the structural defects such as dislocations, and interface roughness. Though a large amount of
effort16–20 has been made to analyze the dislocation scattering, the dislocation in most of the models is simply treated as
a vertical line passing though the 2DEG. Actually, if the tilt
a)
[email protected]
[email protected]
c)
[email protected]
b)
0021-8979/2014/115(4)/043702/4/$30.00
angle of the dislocation is large enough, 60 for example, the
anisotropic scattering effect is not ignorable.
It is well known that the dislocations can be modeled as
a line of charge.16–20 Therefore, the coulomb scattering due
to these charged dislocation lines is found to occur, resulting
in the reduction of the electron transport mobility. In this
Letter, a theory is developed to study the anisotropic scattering effect of the inclined misfit dislocation on the 2DEG.
II. MODEL DESCRIPTION
For simplicity, here we consider a perfect 2DEG, i.e.,
there is no spatial extent of carriers in the growth direction.
Figure 1 shows schematically an individual charged inclined
dislocation line passing through the perfect 2DEG. Assuming
that each charged dislocation line is infinitely extended along
the y-z plane with an inclined angle a, so the y-axis is exactly
the projection of the inclined dislocation line in-plane. The
vertical distance from any point ~
r ðx; yÞ in plane to the
inclined dislocation line is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(1)
d ¼ x2 þ ðy cos aÞ2 :
The coulombic potential due to an infinite charged line is
given by
Vðx; yÞ ¼
eqL
d0
ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
2pee0
x2 þ ðy cos aÞ2
(2)
where ee0 is the dielectric constant. In the AlGaN/GaN heterostructures, because the dielectric constants of GaN and AlN
are very close to each other (in the static, 8.9 for GaN and
8.5 for AlN), the dielectric constant is always assumed to be
same in the 2DEG and outside it.16 Even in the InGaN/GaN
heterostructures, the present InGaN materials usually has
very little Indium composition (no more than 10%) because
115, 043702-1
C 2014 AIP Publishing LLC
V
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043702-2
Jin et al.
J. Appl. Phys. 115, 043702 (2014)
FIG. 1. Schematic diagram of an individual inclined dislocation line passing
through a perfect 2DEG. The Line dislocation can be modeled as a line of
charge. Assuming that each charged dislocation line is infinitely extended
along the y-z plane and the inclined angle is a.
of growth difficulties, so, for simplicity, the dielectric
constants could also be assumed to be uniform. Therefore,
the image effect of charged inclined dislocation line is
ignored here. e is the electronic charge, d0 is any selected
zero potential position. qL is the line charge density, which is
given by
qL ¼
ef
;
c
(3)
where c is the wurtzite GaN lattice spacing in the dislocation
direction, f is the fraction of filled states. Thus, the
Hamiltonian matrix element hk0 jHjki can be obtained from
ðð
1
Vðx; yÞeiqx xþiqy y dxdy;
(4)
hk0 jHjki ¼
A
FIG. 2. Scattering time limited by the inclined dislocations is plotted as a
function of the angle h between the initial wave vector k and the x-axis, with
the dislocation density Ndis ¼ 1 109 cm2 and the 2DEG sheet density
n2D ¼ 5 1012 cm2. The inset shows the process of the elastic scattering in
plane, and h0 is the scattering angle between k and k0 .
1
2Ndis m e4 f 2 ðcos aÞ2
¼
sðhÞ
ph3 e2 e20 c2
2p
ð
0
1
!2
q2x ðcos aÞ2 þ q2y
Sc ð1 cos h0 Þdh0 ;
(7)
where m is the effective mass, and h is the reduced Planck
constant. In terms of the scattering time, the 2DEG mobility
can be expressed as following:21
e
lðuÞ ¼
pm
2ðp
sðhÞcos2 ðu hÞdh;
(8)
0
where A is the area of the system. Using the two-dimensional
Fourier function, the Hamiltonian matrix element can be
transformed as
hk0 jHjki ¼
2eqL cos a
1
:
2
Aee0 qx ðcos aÞ2 þ q2y
(5)
As shown in the inset of Figure 2, h is the angle of the initial
wave vector k with respect to the x axis, and h0 is the scattering
0
0
angle
between k and k 0 , qx and qy can0 be got as qx ¼ kx 0 kx
¼ k cos h cosðh þ h Þ , qy ¼ky ky ¼k sinhsinðhþh Þ .
The final expression for scattering time is then given by
ð
1
A 2p
2
¼ ANdis jhk0 jHjkij
sðhÞ
ð2pÞ2 h
Sc ð1 cos h0 ÞdðEk0 Ek Þdk~0 ;
(6)
where Ek and Ek0 are the energies corresponding to wave
vectors k and k0 , ANdis is the total inclined dislocation number, with A and Ndis being the system area and inclined disloh
i2
cation density, respectively. Sc ¼ qþqq TF is the screening
factor, where qTF ¼ a2 is the Thomas-Fermi wave vector,
with a* being the effective Bohr radius.
Introducing Eq. (5) into Eq. (6), and integrating over k0
by using the property of the delta function yields
where u is the angle of the applied electric field with respect
to the x-axis.
For a degenerate gas as in a 2DEG, scattering takes
place mainly among electrons
with wave vectors near the
pffiffiffiffiffiffiffiffiffiffiffiffi
Fermi wave vector kF ¼ 2pn2D , where n2D being the
2DEG sheet density. So the k in the integral can be replaced
by kF . For clarity, the fraction of filled states f in Eq. (3) is
set to be unity. That is the extreme case in which the predicted mobility suffers considerably from the heavily
charged dislocation lines.
III. RESULTS AND DISCUSSION
The sample in our calculation is a polar AlGaN/GaN
heterostructur according to Ref. 10 and the tilt angle a is 60
according to Ref. 10. Figure 2 shows the results of the h dependent scattering time sðhÞ with the angle h ranged from 0
to 2p, with the dislocation density Ndis ¼ 1 109 cm2 and
the 2DEG sheet density n2D ¼ 5 1012 cm2. From Fig. 2,
one can observe obviously the anisotropy of the scattering
time. The maximum value of the scattering time is found at
the angle h ¼ 0, or p, which corresponds to the case of
2DEG moving along x-axis, i.e., the direction perpendicular
to the projection of the inclined dislocation line. The minimum value of the scattering time is found at the angle
h ¼ p=2, or 3p=2, which corresponds to the case of 2DEG
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043702-3
Jin et al.
moving along y-axis, i.e., the direction parallel to the projection of the inclined dislocation line. The result of Fig. 2
depicts that the 2DEG will be more easily scattered as they
move in the direction parallel to the projection of the
inclined dislocation line than perpendicular to the projection.
It is a very interesting phenomenon, for which we may give
a simple qualitative interpretation. From Fig. 1, we may
choose any two points in-plane at the same distance from the
coordinate center, for example, point (l, 0) in x-axis and
point (0, l) in y-axis. However, the actual vertical distances
to the inclined dislocation line for the two points are jlj and
jljcosðaÞ, respectively. Since jljcosðaÞ is much less than jlj,
the coulombic potential in point (0, l) is greater than that in
point (l, 0) although they have the same distance from the
coordinate center. Qualitatively, at the same distance from
the coordinate center, the coulombic scattering potential in
y-axis is greater than that in x-axis, so that the 2DEG will be
more easily scattered as they move along y-axis, i.e., the
direction parallel to the projection of the inclined dislocation
line.
Figure 3 illustrates the dependence of the 2DEG mobility on the angle u, with u being the applied electric field
with respect to the x-axis. From the curve, one can observe
obviously the anisotropic mobility that is similar to the sðhÞ
curve in Fig. 2. The maximum mobility is found at the angles
u ¼ 0 and u ¼ p, while the minimum mobility is found at
the angles u ¼ p=2 and u ¼ 3p=2. The results indicate that
the 2DEG will have a more well mobility as it moves along
the direction perpendicular to the projection of the inclined
dislocation line. The maximum value of the mobility is about
1.5 times as the minimum value, which is not ignorable for
the actual production of the Al(In)GaN/GaN high electron
mobility transistors.
In Figure 4, anisotropic mobilities lxx (u ¼ 0) and
lyy (u ¼ p=2) are plotted as a function of the 2DEG sheet
density for different inclined dislocation density Ndis at
1 108 cm2, 1 109 cm2, and 1 1010 cm2, respectively.
The predicted mobility varies in the range of 102–105 cm2/Vs,
which also increases with the increasing n2D and decreases
FIG. 3. Anisotropic mobility is plotted as a function of the angle u between
the applied electric field and the x-axis, with the dislocation density
Ndis ¼ 1 109 cm2 and the 2DEG sheet density n2D ¼ 5 1012 cm2.
J. Appl. Phys. 115, 043702 (2014)
FIG. 4. Anisotropic mobilities lxx (u ¼ 0) and lyy (u ¼ p=2) are plotted as a
function of the 2DEG sheet density for different inclined dislocation density
Ndis at 1 108 cm2, 1 109 cm2, and 1 1010 cm2, respectively.
with the decreasing Ndis . Jena et al.16 had calculated the 2DEG
mobility limited by the vertical threading dislocation having a
value ranged from 102 to 105 cm2/Vs for threading dislocation
density 108–1010 cm2. Thus, the resulting value of the 2DEG
mobility limited by the inclined dislocation scattering is
predicted to be about the same order of magnitude with that
limited by the vertical threading dislocation scattering.
In order to clarify how large the anisotropy is, the mobility ratio lxx =lyy was studied as a function of the 2DEG
density for different tilt angle a in Figure 5. From this figure
one can see that the mobility ratio lxx =lyy is sensitive to the
tilt angle a. Large tilt angle leads to large mobility ratio.
When the tilt angle a equals to zero, the scattering is isotropic. This characteristic is consistent with what is expected
from our qualitative intuition.
IV. SUMMARY
In conclusion, we have studied theoretically the 2DEG
mobility limited by the inclined misfit dislocation scattering
in Al(In)GaN/GaN heterostructures and performed a
FIG. 5. Anisotropic mobility ratio lxx =lyy is plotted as a function of the
2DEG density n2D for different values of tilt angle a.
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043702-4
Jin et al.
quantitative analysis on the scattering strength. The results
show that the 2DEG will be more easily scattered as they
move in the direction parallel to the projection of the inclined
dislocation line than perpendicular to the projection, i.e., the
electron mobility has a maximum value along the direction
perpendicular to the projection of the inclined dislocation
line, and vice versa. The results are meaningful and should be
noted for the production of the Al(In)GaN/GaN HEMTs.
ACKNOWLEDGMENTS
This work was supported by National Science
Foundation of China (Grant Nos. 91233111, 61274041,
11275228, 61006004, and 61076001), and by Special Funds
for Major State Basic Research Project (973 program) of
China (Grant No. 2012CB619305), and by the 863 High
Technology R&D Program of China (Grant No.
2011AA03A101), and also by Guangdong Provincial Special
Fund for LED Industrial Development 2012A080302003.
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