Corresponding<EMAIL_ADDRESS>
# Tuning of energy levels and optical properties of graphene quantum dots

Z. Z. Zhang  Kai Chang  [  SKLSM, Institute of Semiconductors, Chinese Academy
of Sciences, P. O. Box 912, Beijing 100083, China  F. M. Peeters  Departement
Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen,
Belgium

###### Abstract

We investigate theoretically the magnetic levels and optical properties of
zigzag- and armchair-edged hexagonal graphene quantum dots (GQDs) utilizing
the tight-binding method. A new bound edge state at zero energy appears for
the zigzag GQDs in the absence of a magnetic field. The magnetic levels of
GQDs exhibit a Hofstadter-butterfly spectrum and approach the Landau levels of
two-dimensional graphene as the magnetic field increases. The optical
properties are tuned by the size, the type of the edge, and the external
magnetic field.

###### pacs:

73.22.-f, 78.67.-n, 75.75.+a, 81.07.Nb

Graphene is a single atomic layer consisting of a two-dimensional honeycomb
lattice of carbon atoms. This novel system has attracted intense attention
because of new fundamental physics and promising applications in
nanoelectronics Novoselov ; Novoselov2 . It exhibits high crystal quality, an
exotic Dirac-type spectrum, and ballistic transport properties on a submicron
scale. Graphene samples are usually fabricated by micromechnical cleavage of
graphite and have excellent mechanical properties that make it possible to
sustain huge electric currents. The lateral confinement of Dirac fermions in
graphene is still an enigmatic and extremely challenging task due to the well-
known Klein paradox. The Klein paradox makes it impossible to localize the
carriers in a confined region utilizing an electrostatic gate. The confinement
of Dirac fermions at a nanometer scale is one of the central goals of
graphene-based electronics and has attracted increasing interestEfetov ; Egger
; HChen ; Peeters ; Trauzettel ; Antidot . Recently it was demonstrated
experimentally that graphene can be cut in the desired shape and sizeNovoselov
; Novoselov2 . Recent progresses in fabricating and characterizing stable
graphene nanostructures provides the opportunity to explore the various
remarkable opticalFalko ; Carbotte ; Reichl  and transport propertiesKim  of
these structures.

In this work, we investigate theoretically the electronic structure and
optical properties of zigzag- and armchair-edged hexagonal graphene quantum
dots (GQDs) (see Fig. 1) utilizing the nearest-neighbor tight-binding model.
The dangling bonds at the edges are passivated by hydrogen atoms. The model
has been successfully used for fullerene molecules, carbon naotubes, and other
carbon-related materials Nakada ; Waka ; Saito ; Ezawa . The Hamiltonian of
GQDs can be written as
$H=\sum\limits_{i}\varepsilon_{i}c_{i}^{{\dagger}}c_{i}+\sum\limits_{\left\langle
i,j\right\rangle}t_{i,j}c_{i}^{{\dagger}}c_{j}$, where $\varepsilon_{i}$ is
the site energy, $t_{ij}$ is the transfer energy between the nearest-neighbor
sites, and $c_{i}^{{\dagger}}$ ($c_{i}$) is the creation (annihilation)
operator of the $\pi$ electron at the site $i$. When considering a magnetic
field $B$ applied perpendicularly to the plane of a GQD, the transfer integral
$t_{ij}$ becomes $t_{ij}=te^{i2\pi\phi_{i,j}},$ where
$\phi_{ij}=\frac{e}{h}\int_{r_{i}}^{r_{j}}d\mathbf{l}\cdot\mathbf{A}$ is the
Peierls phase. $\mathbf{A}=\left(0,Bx,0\right)$ is the vector potential
corresponding to the magnetic field B along the z axis, which is perpendicular
to the graphene plane. In our calculation, we take $\Phi_{0}=h/e$ as the unit
of the magnetic flux and $\Phi=\sqrt{3}Ba_{0}^{2}/2$ as the magnetic flux
through a plaquette, where $a_{0}=2.46\mathring{A}$ is the lattice constant of
graphite. The difference between the values of $\varepsilon_{i}$ and $t_{ij}$
for the atoms at the edge and the center is neglected. The relevant parameters
used in our calculation are $\varepsilon=0$, $t=-3.033$Saito . The eigenvalues
and eigenstates can be obtained from the secular equation
$\det\left|\varepsilon-H\right|=0$, where $H_{ii}=0,H_{\left\langle
i,j\right\rangle}=te^{i2\pi\phi_{i,j}}$.

Figure 1:  (Color online) Electronic density distributions of the highest
valence level (HVL) and lowest conduction level (LCL) in the absence of
magnetic field. Panels (a) and (b): HVL and LCL for the $N_{z}=2$ zigzag-edged
graphene quantum dot (ZGQD). Panels (c) and (d) : the same with $N_{z}=12$.
Panels (e) and (f): the same with armchair-edged graphene quantum dot (AGQD).

Figs. 1 show electronic density distributions of the zigzag and armchair-edged
graphene quantum dots (ZGQD and AGQD, respectively), in the absence of a
magnetic field. The size of a dot is characterized by N, the number of
hexagonal units along an edge. Figure 1(a) and 1(b) show the probability
distributions of the highest valence level (HVL) and lowest conduction level
(LCL) for the ZGQD with small size ($N=2$). The probability distributions of
the HVL and LCL correspond to the bonding and anti-bonding states that are
localized at the corner of the hexagonal GQD. In contrast to conventional
semiconductor quantum dots where the ground state is localized at the center
of dot, the ground states for the conduction and valence bands, i.e., HVL and
LCL, localize at the middle of each edge in the ZGQD(see Fig. 1(c) and (d)) as
the size of the ZGQD increases. This feature can be understood as follows: the
Dirac fermion in a ZGQD behaves like a confined photon in a cavity, and the
lowest mode is the whispering gallery mode, which also localizes at the
boundary of the cavity. The difference between the bonding and anti-bonding
states becomes smaller as the size of the ZGQD increases. The difference
between the edge states of a ZGQD and a graphene nanoribbon is that the edge
state of the ZGQD localizes at the middle of the edge of GQD, in contrast to
the homogeneously distributed edge state of a zigzag graphene nanoribbonWaka ;
Brey  or a zigzag triangular GQDYamamoto . This occurs because the
contribution of each carbon atom at the edge of a ZGQD to the edge state is
different, while it is the same for a zigzag nanoribbon or a zigzag triangular
GQD. The density distributions of the LCL and HVL in an AGQD (see Figs. 1(e)
and (f)) extend more completely over the whole GQD region and are very
different from that in a ZGQD. This difference is indeed caused by the
different topological geometry of the boundary of the graphene nanostructures.

Figure 2: (Color online) Density of states of ZGQD (a) and AGQD (b). We use a
Gaussian function $f(E)=e^{-(E-E_{0})^{2}/\Gamma^{2}}$ with a broadening
factor $\Gamma=0.05$ eV to smooth the discontinuous energy spectra. (c) The
gap of ZGQD and AGQD as a function of the size, and the function of the fit
line is $a/N$ with $a=4.9$ eV.

Figs. 2(a) and (b) show the density of states (DOS) of ZGQDs and AGQDs,
respectively, with different sizes in the absence of a magnetic field. The
total number of the carbon atoms in ZGQD and AGQD are $6N_{z}^{2}$ and
$6(3N_{a}^{2}-3N_{a}+1)$, respectively. From the figures, we find that there
is no edge state in a small ZGQD, and the edge state appears when the size of
ZGQD increases according to the states at zero energy. Meanwhile, there is
never an edge state for the AGQD. To demonstrate how the edge state appears,
we plot the energy gap, i.e., the energy difference between the lowest
conduction band level (LCL) and the highest valence band level (HVL), as a
function of the size ($N$) of the GQD in Fig. 2(c). The energy gap decreases
as the size of the GQD increases. Interestingly, the energy gap of the zigzag
(armchair) GQD decays to zero quickly(slowly) as the size of the GQD
increases. When the size of the AGQD approaches infinity, the gap decreases to
zero, i.e., we recover the two-dimensional graphene case. The calculated
energy gap for the AGQD falls off as $1/N$ $\propto 1/L$ (see the solid line
in Fig. 2(c)), where $L$ is the length of each edge of the hexagonal GQD. This
dependence of the band gap on the size of GQD is very different from that of a
conventional semiconductor QD, which behaves as $1/L^{2}$.

Figs. 3(a) and 4(a) depict the magnetic field dependence of the energy
spectrum of a ZGQD and AGQD exhibiting a clear Hofstadter butterfly
characteristic, which is fractal and exhibits self-similarityHosfstadter ;
Waka ; Aoki ; Nemec . As the magnetic flux increases, the magnetic levels in
the GQD, i.e., the so-called Fock-Darwin levels, approach the Landau levels
(see the red lines in Figs. 3(b) and 4(b)) in graphene
$E_{n}=sgn\left(n\right)(\sqrt{3}ta_{0}/2l_{B})\sqrt{2\left|n\right|}$, where
$l_{B}=\sqrt{\hbar/eB}$ is the cyclotron radius, $n$ is an integer, and $sgn$
is the sign function.

Figure 3:  (Color online) (a) The spectrum of the $N_{z}=12$ ZGQD in a
magnetic field. We use a Gauss function with a broadening factor of 0.1 eV to
smooth discontinuous energy spectra. (b) and (c) the magnetic energy level fan
near the Dirac point, i.e., the zero energy point. The red lines in (b)
correspond to the Landau level of two-dimensional graphene. (d) the DOS at the
Dirac point as a function of the inverse flux $\Phi/\Phi_{0}$, where we use a
Gauss function with a small broadening factor of 0.01 meV. Figure 4:  (Color
online) The same as Fig.3, but for the $N_{a}=9$ AGQD

Figs. 3(c) and 4(c) show in detail how the magnetic levels of ZGQD and AGQD
approach the zero-th Landau level at small magnetic flux. As magnetic flux
increases, more energy levels approach the zero-th Landau level in pairs. The
degeneracy of the energy level at zero energy will reach its maximum value
$2N$ for $\Phi/\Phi_{0}=1/2N$. When the magnetic flux increases further, the
degeneracy of the energy level at zero energy is lifted fast. This feature can
also be seen in Fig. 3(d), which plots the DOS at the Dirac point. This figure
indicates that the degeneracy, i.e., the number of energy levels at the zero
energy, _approximately_ decreases inverse linearly with the magnetic flux
$\Phi/\Phi_{0}$. These figures clearly demonstrate that the energy spectrum of
the GQD possesses electron-hole symmetry when we neglect the second-nearest-
neighbor interaction. The DOS and the magnetic level fan of the AGQD are
similar to that of the ZGQD except at small magnetic flux. Comparing Fig. 3(c)
to Fig. 4(c), the magnetic levels in the AGQD are distinct from those in the
ZGQD at small magnetic flux, because the ZGQD shows the edge state and AGQD
does not for the levels near the Dirac point in absense of magnetic field.
Therefore, the magnetic levels exhibit distinct behavior as the magnetic flux
increases. The DOS of the AGQD (see Fig. 4(d)) also shows a step-like feature
as the magnetic flux at the Dirac point increases.

Fig. 5 describes the density distributions of the LCL and HVL in the ZGQD and
AGQD at small magnetic flux $\Phi/\Phi_{0}=0.01$. Interestingly, the density
distributions of the LCL and HVL penetrate into the center of the GQD for the
ZGQDs, which is very different from the AGQD case where both the electron and
hole are dominantly localized in the center of the GQD. The density
distributions for the ZGQD and AGQD show $C_{6v}$ symmetry. This
characteristic is caused by the magnetic confinement when the magnetic length
$l_{B}$ becomes comparable with the size of the GQD. In addition to those
differences, the LCL and HVL of the zigzag GQD show opposite symmetry order
with respect to that of the armchair GQD, i.e., the LCL (HVL) and HVL(LCL) of
the ZGQD (AGQD) belong to the $E_{1}$($E_{2}$) and $E_{2}$($E_{1}$)
representations at zero magnetic field (see Fig. 6).

Figure 5:  (Color online) (a) and (b) show the density distributions of the
HVL and LCL for the $N_{z}=12$ ZGQD in the presence of the magnetic flex
$\Phi/\Phi_{0}=0.01$, respectively. (c) and (d) the same as (a) and (b), but
now for $N_{a}=9$ AGQD.

The optical properties of GQDs are promising for potential applications in
optic-electronic devices based on graphene. Therefore, we calculate the
absorption spectra of GQD $\alpha(\hslash\omega)=\frac{\pi
e^{2}}{m_{0}^{2}\varepsilon_{0}cn\omega
V}\sum\limits_{c,v}|\vec{\varepsilon}\cdot
P_{cv}|^{2}\times\delta\left(E_{c}-E_{v}-\hbar\omega\right),$ where $n$ is the
refractive index, $c$ the speed of light in vacuum, $\varepsilon_{0}$ the
permittivity of vacuum, $m_{0}$ the free-electron mass, and
$\vec{\varepsilon}$ is the polarization vector of the incident light along the
$x$ direction. The coupling between the $sp_{2}$ states and the $p_{z}$ state
is neglected since we are only interested in the optical properties of the GQD
near the Dirac point, i.e., at the low energy regime. The momentum
matrixPedersen  is $\left\langle
n\right|\mathbf{p}\left|m\right\rangle=im_{0}/\hbar\sum\limits_{\mathbf{r}}\sum\limits_{\mathbf{r}^{\prime}}c_{\mathbf{r}}^{\ast}c_{\mathbf{r}^{\prime}}\left(\mathbf{r}^{\prime}-\mathbf{r}\right)\left\langle
p_{z},\mathbf{r}\right|H\left|p_{z},\mathbf{r}^{\prime}\right\rangle$. The
momentum operator $p_{x}(p_{y})$ has $E_{2}$ symmetry and its direct product
with all the irreducible representations of the C6v group can be found in
Table 1. We divide the levels of the GQD into two different families:
$A_{1},A_{3},E_{1}\in\Omega_{1}$ and $A_{2},A_{4},E_{2}\in\Omega_{2}$. The
symmetry requires that only transitions between the valence band levels and
the conduction band levels belonging to the different families $\Omega_{1}$
and $\Omega_{2}$ are allowed. Notice that the initial or final states of the
transition should belong to the $E_{1}$ or $E_{2}$ representations.

Table 1:  Direct products of the $E_{2}$ representation for momentum operator $p_{x}(p_{y})$ with all the irreducible representations of the $C_{6}v$ group. The results are presented as direct sums of all possible irreducible representations of the $C_{6}v$ group. The notations of symmetries are adopted from Ref. Group,  Direct product | Direct sum
---|---
$E_{2}$$\otimes$$A_{1}$ | $E_{2}$
$E_{2}$$\otimes$$A_{2}$ | $E_{1}$
$E_{2}$$\otimes$$A_{3}$ | $E_{2}$
$E_{2}$$\otimes$$A_{4}$ | $E_{1}$
$E_{2}$$\otimes$$E_{1}$ |  $A_{2}$$\oplus$$A_{4}$$\oplus$$E_{2}$
$E_{2}$$\otimes$$E_{2}$ |  $A_{1}$$\oplus$$A_{3}$$\oplus$$E_{1}$

In Fig. 6(a) and (d), we label the level structure of a $N_{z}=12$ and
$N_{a}=9$ GQD near the Dirac point as $C_{1}-C_{n}$ for conduction bands with
ascending order and $V_{1}-V_{n}$ for valence bands with descending order,
respectively. The conduction band levels $C_{i}$ and valence band levels
$V_{i}$ belong to the distinct families $\Omega_{1}$ and $\Omega_{2}$,
respectively. For example, if $C_{i}$ belongs to the family $\Omega_{1}$,
i.e., $A_{1}$, $A_{3}$ or $E_{1}$, $V_{i}$ must belong to the family
$\Omega_{2}$, i.e., $A_{2},$ $A_{4}$ or $E_{2}$, or vice versa. For zigzag
GQDs with even $N_{z}$, the conduction band levels, from bottom to top,
exhibit different symmetries, i.e., $E_{2}$, $A_{3}$, $E_{1}$, $A_{2}$
$\cdots$, the corresponding valence band levels show $E_{1}$, $A_{4}$,
$E_{2}$, $A_{1}$ $\cdots$. For zigzag GQDs with odd $N_{z}$, the conduction
band levels display the opposite (same) symmetries $E_{1}$, $A_{4}$, $E_{2}$,
$A_{1}$, $\cdots$ to the conduction (valence) band levels of zigzag GQDs with
even $N_{z}$. For armchair GQDs, the lowest conduction band level always shows
the symmetries $E_{1}$, $A_{4}$, $A_{2}$, $E_{2}$, $\cdots$ from bottom to top
and this order is independent of the size ($N_{a}$) of the armchair GQD.

Figure 6:  (Color online) (a) and (d) are the level diagram for $N_{z}=12$
ZGQD and $N_{a}=9$ AGQD without the magnetic field, where different symmetries
are represented by different colors and lines: black solid, black dashed, red
solid, red dashed, green solid, and green dashed lines for the $E_{1}$,
$E_{2}$, $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ irreducible representations of
the $C_{6v}$ symmetry, respectively. (b) and (c) the JDOS and the optical
absorption spectrum $\alpha$ for $N_{z}=12$ ZGQD. We used a Gauss function
with different broadening factors: 0.02 and 0.005 eV for the black and red
line. (e) and (f) are the same as (b) and (c), but for $N_{a}=9$ AGQD.

For $N_{z}=12$ ZGQD, the lowest optical-absorption peak (peak A) corresponds
to the transition between the lowest conduction band level $C_{1}$ with
$E_{2}$ symmetry and the highest valence band level $V_{1}$ with $E_{1}$
symmetry. The second and third lowest transitions correspond to the transition
between the level $C_{2}\left(C_{3}\right)$ with $A_{3}\left(E_{1}\right)$
symmetry and the level $V_{3}\left(V_{2}\right)$ with
$E_{2}\left(A_{4}\right)$ symmetry and the level $C_{1}\left(C_{4}\right)$ and
$V_{4}\left(V_{1}\right)$, respectively. But the strengths of these three
transitions are very small, therefore these transitions are not clearly seen
in the contour spectrum in Fig. 7 at $\Phi/\Phi_{0}=0$. The strong absorption
peak (peak D) appears at $E=0.26$ eV, corresponding to the transition between
the level $C_{3}$ with $E_{1}$ symmetry and the level $V_{3}$ with $E_{2}$
symmetry. This strong absorption arises from the large moment matrix
$\left\langle n\right|\mathbf{p}\left|m\right\rangle$ between these states.

Figure 7:  (Color online) The contour plot of the magneto-optical spectra of
zigzag (a) and armchair (b) GQD, respectively.

For $N_{a}=9$ AGQD, the lowest peak (peak A) is similar to that in the zigzag
GQD, corresponding to the transition between $C_{1}$ and $V_{1}$. But the
second peak (peak B) is different from those of the zigzag GQD. This peak
corresponds to the transition between the level $C_{2}\left(C_{4}\right)$ with
$A_{4}\left(E_{2}\right)$ symmetry and the level $V_{4}\left(V_{2}\right)$
with $E_{1}\left(A_{3}\right)$ symmetry. The third strong peak (peak C)
indicates the transition between the level $C_{4}$ with $E_{2}$ symmetry and
the level $V_{4}$ with $E_{1}$ symmetry. Strong absorption takes place when
the initial ($V_{i}$) and final states ($C_{i}$) have either $E_{1}$ symmetry
or $E_{2}$ symmetry. As the size of the GQD increases, the absorption peaks
shift to long wavelength for both ZGQD and AGQD. The absorption peaks of the
ZGQD shift to the long wavelength faster than those of the AGQD. The relative
strength between the peak D and A increases as the size of the GQD increases
for ZGQDs. But for AGQDs, the relative strength between the peak C and A is
almost independent of the size.

Next, we discuss the effect of a magnetic field on the optical spectrum of a
GQD. Here, we only focus on the small magnetic flux case (see Fig. 7). The
spectra of two distinct GQDs, zigzag and armchair GQD, exhibit quite different
behavior due to their different level structures and the oscillator strengths
determined by the boundary, especially for the LCL and HVL which localize at
the edge of ZGQD. The spectra of two distinct GQDs show that the strengths of
the transitions vary as the magnetic field increases. In particular, the
strong absorption lines exhibit $\sqrt{B}$ asymptotic behavior corresponding
to the transitions between the conduction and valence band Landau levels at
high magnetic field. We also find anti-crossings in the spectra, since the
magnetic field induces the mixing of the levels belonging to the different
families.

In summary, we investigated theoretically the magnetic levels and the optical
spectrum in GQDs. In contrast to conventional semiconductor QDs, the LCL and
HVL exhibit an edge-state feature, i.e., a non-zero probability of being at
the edge of the sample, and the density distribution depends sensitively on
the type of boundary of GQDs and the magnetic field strength. The magnetic
levels of GQD display a Hoftstadter butterfly characteristic, and approach the
Landau levels of two-dimensional graphene as the magnetic field increases. The
magneto-optical spectrum of a graphene quantum dot in the interesting energy
range (0-3 eV) is promising for carbon-based electronics applications. The
position and strength of the absorption peaks can be tuned by the size of the
GQD, the type of the edge of the GQD, and the external magnetic field.

###### Acknowledgements.

This work is supported by the NSF of China Grant No. 60525405 and the Flander-
China bilateral programme.

## References

  * (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science 306, 666 (2004); H. Hiura, Appl. Surf. Sci. 222, 374 (2004); Y. Zhang, J. W. Tan, H. L. Stormer and P. Kim, Nature (London) 438, 201 (2005); A. K. Geim, and K. S. Novoselov, Nature Materials 6, 183 (2007).
  * (2) K. S. Novoselov, Nature (London) 438, 197 (2005).
  * (3) P. G. Silvestrov and K. B. Efetov, Phys. Rev. Lett. 98, 016802 (2007).
  * (4) A. DeMartino, L. Dell’Anna, and R. Egger, Phys. Rev. Lett. 98, 066802 (2007).
  * (5) H. Y. Chen, V. Apalkov, and T. Chakraborty, Phys. Rev. Lett. 98, 186803 (2007).
  * (6) J. M. Pereira, Jr., P. Vasilopoulos, and F. M. Peeters, Nano. Lett. 7, 946 (2007).
  * (7) B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nature Phys. 3, 192 (2007).
  * (8) T. G. Pedersen, C. Flindt, J. Pedersen, N. A. Mortensen, A.-P. Jauho, and K. Pedersen, Phys. Rev. Lett. 100, 136804 (2008).
  * (9) M. O. Goerbig, J. N. Fuchs, K. Kechedzhi, and V. I. Fal’ko, Phys. Rev. Lett. 99, 087402 (2007).
  * (10) V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 98 157402 (2007).
  * (11) H. Hsu and L. E. Reichl, Phys. Rev. B 76, 045418 (2007).
  * (12) B. Ozyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov, and P. Kim, Phys. Rev. Lett. 99, 166804 (2007).
  * (13) K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
  * (14) K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phys. Rev. B 59, 8271 (1999).
  * (15) R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998).
  * (16) M. Ezawa, Phys. Rev. B 73, 045432 (2006).
  * (17) L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006).
  * (18) T. Yamamoto, T. Noguchi, and K. Watanabe, Phys. Rev. B 74, 121409(R) (2006).
  * (19) T. G. Pedersen, K. Pedersen, and T. B. Kriestensen, Phys. Rev. B 63, 201101(R) (2001).
  * (20) D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976); R. Rammal, J. Phys. (Paris) 46, 1345 (1985).
  * (21) Y. Hatsugai, T. Fukui, and H. Aoki, Phys. Rev. B 74, 205414 (2006).
  * (22) N. Nemec and G. Cuniberti, Phys. Rev. B 75, 201404 (2007).
  * (23) G. L. Bir and G. E. Pikus, Symmetry and Strain Induced Effects in Semiconductors (Wiley, New York, 1974); G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Slatz, Properties of the Thirty-Two Point Groups (MIT Press, Cambridge, 1966).