Optical Selection Rules for Zigzag Graphene Nanoribbons

This Demonstration presents a complex analysis of the wavefunction parity and optical selection rules for zigzag graphene nanoribbons (ZGNRs). Selection rules are illustrated for optical transition matrix elements of a linearly polarized light. The plane of polarization of the incident light is parallel to the ribbon's longitudinal axis.

Details

This Demonstration is based on the work in[1], which is an analytical extension of the work in[2]. All the notations are adopted from there. In particular, the energy bands of the ribbon are labeled by
J(s)
, where
J
labels the band number and
s
, the band type. Therefore,
s=c
stands for the conduction band,
s=v
for the valence band.
The normalized wavefunctions
Ψ
J
1
(
s
1
)
(blue) and
Ψ
J
2
(
s
2
)
(red) are plotted in the upper-left as functions of the normalized transverse coordinate
x
i
/W
, where
x
i
is the
x
coordinate of the
th
i
atom in the ribbon unit cell and
W
is the width of the ribbon. The
x
coordinates of the atoms from the A and B sublattices forming a hexagonal structure of a zigzag graphene nanoribbon (ZGNR) are
x
2p-1
=
3
a
2
(p-1)
and
x
2p
=
a
2
3
+
x
2p-1
, where
p=1,…,w
, with
w
being the number of zigzag chains specifying the width of the ribbon. The number of zigzag chains is
w=N/2
, with
N
being the number of carbon atoms in the zigzag ribbon unit cell. Then, the ribbon width is
W=
x
2w
. Thus, the ribbon with a certain width can be labeled as
ZGNR(w)
. The wavefunctions
Ψ
J
1
(
s
1
)
(blue) and
Ψ
J
2
(
s
2
)
(red) are offset for clarity by
0.3
and
-0.3
, respectively.
The energy bands of a chosen
ZGNR(w)
presented in the upper-right band structure plot are normalized by the hopping integral
γ≈3eV
. The red and blue points in the band structure plot represent the states with wavefunctions
Ψ
J
2
(
s
2
)
and
Ψ
J
1
(
s
1
)
, respectively.
The lower-left plot shows the
Ψ
J
1
(
s
1
)
and
Ψ
J
2
(
s
2
)
wavefunctions overlapping for chosen
J
2
(
s
2
)
and
J
1
(
s
1
)
.
The optical matrix elements
M
J
2
(
s
2
)
J
1
(
s
1
)
for a chosen transition
J
2
(
s
2
)
J
1
(
s
1
)
are presented in the lower right plot as functions of the electron wave number
k
. These matrix elements are velocity matrix elements normalized by the Fermi velocity of electrons in graphene,
v
F
=
3
aγ
2ℏ
, where
a=2.46Å
is the graphene lattice constant,
γ
is the hopping integral, and
ℏ
is the reduced Planck's constant. The black point denotes the matrix element value for the transition depicted in the band structure plot.
The point
k=2π/3
corresponding to the Dirac point in graphene is marked by the vertical line labeled as
K
in the energy band and matrix element plots. Similar marking by the vertical line is used for the transition point
k
t
=2arccos
w
2(w+1)
, where the bulk states meet the edge states in the subbands
1(c)
and
1(v)
.
Snapshot 1: the wavefunction (red) of the bulk state in the subband
1(v)
of
ZGNR(11)
Snapshot 2: the wavefunction (red) of
ZGNR(11)
at the transition point
k
t
, where the bulk states meet the edge states in the subband
1(v)
Snapshot 3: the wavefunction (red) of the subband
1(v)
edge states localized at the ribbon edges for
ZGNR(11)
Snapshot 4: forbidden transition
1(v)5(c)
between valence and conduction subbands of
ZGNR(11)
Snapshot 5: allowed transition
1(v)6(c)
between valence and conduction subbands of
ZGNR(11)
Snapshot 6: forbidden transition
1(c)6(c)
between conduction subbands of
ZGNR(11)
Snapshot 7: allowed transition
1(c)5(c)
between conduction subbands of
ZGNR(11)
Snapshots 1–3 show the transformation of the electron wavefunction (red) as one moves from the bulk to the edge states within the
1(v)
subband. Snapshots 4 and 5 demonstrate the odd selection rule
ΔJ=
J
2
(c)-
J
1
(v)
for allowed transitions between the conduction and valence subbands. Snapshots 6 and 7 demonstrate the even selection rule
ΔJ=
J
2
(s)-
J
1
(s)
for allowed transitions between the conduction (valence) subbands only.

References

[1] V. A. Saroka, M. V. Shuba and M. E. Portnoi, "Optical Selection Rules of Zigzag Graphene Nanoribbons," Physical Review B, 95(15), 2017 155438. doi:10.1103/PhysRevB.95.155438.
[2] H. C. Chung, M. H. Lee, C. P. Chang and M. F. Lin, "Exploration of Edge-Dependent Optical Selection Rules for Graphene Nanoribbons," Optics Express, 19(23), 2011 pp. 23350–23363. doi:10.1364/OE.19.023350.

External Links

Electronic Band Structure of Armchair and Zigzag Graphene Nanoribbons
Electronic Structure of a Single-Walled Carbon Nanotube in Tight-Binding Wannier Representation
Electronic Band Structure of a Single-Walled Carbon Nanotube by the Zone-Folding Method
Optical Matrix Elements of Single-Walled Carbon Nanotubes for Longitudinal Polarization of Light
Optical Properties of Graphene

Permanent Citation

Vasil Saroka
​
​"Optical Selection Rules for Zigzag Graphene Nanoribbons"​
​http://demonstrations.wolfram.com/OpticalSelectionRulesForZigzagGrapheneNanoribbons/​
​Wolfram Demonstrations Project​
​Published: August 3, 2017