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

, where

labels the band number and

, the band type. Therefore,

stands for the conduction band,

for the valence band.

The normalized wavefunctions

(blue) and

(red) are plotted in the upper-left as functions of the normalized transverse coordinate

, where

is the

coordinate of the

atom in the ribbon unit cell and

is the width of the ribbon. The

coordinates of the atoms from the A and B sublattices forming a hexagonal structure of a zigzag graphene nanoribbon (ZGNR) are

and

, where

, with

being the number of zigzag chains specifying the width of the ribbon. The number of zigzag chains is

, with

being the number of carbon atoms in the zigzag ribbon unit cell. Then, the ribbon width is

. Thus, the ribbon with a certain width can be labeled as

. The wavefunctions

(blue) and

(red) are offset for clarity by

and

, respectively.

The energy bands of a chosen

presented in the upper-right band structure plot are normalized by the hopping integral

. The red and blue points in the band structure plot represent the states with wavefunctions

and

, respectively.

The lower-left plot shows the

and

wavefunctions overlapping for chosen

and

.

The optical matrix elements

for a chosen transition

are presented in the lower right plot as functions of the electron wave number

. These matrix elements are velocity matrix elements normalized by the Fermi velocity of electrons in graphene,

, where

is the graphene lattice constant,

is the hopping integral, and

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

corresponding to the Dirac point in graphene is marked by the vertical line labeled as

in the energy band and matrix element plots. Similar marking by the vertical line is used for the transition point

, where the bulk states meet the edge states in the subbands

and

.

Snapshot 1: the wavefunction (red) of the bulk state in the subband

of

Snapshot 2: the wavefunction (red) of

at the transition point

, where the bulk states meet the edge states in the subband

Snapshot 3: the wavefunction (red) of the subband

edge states localized at the ribbon edges for

Snapshot 4: forbidden transition

between valence and conduction subbands of

Snapshot 5: allowed transition

between valence and conduction subbands of

Snapshot 6: forbidden transition

between conduction subbands of

Snapshot 7: allowed transition

between conduction subbands of

Snapshots 1–3 show the transformation of the electron wavefunction (red) as one moves from the bulk to the edge states within the

subband. Snapshots 4 and 5 demonstrate the odd selection rule

for allowed transitions between the conduction and valence subbands. Snapshots 6 and 7 demonstrate the even selection rule

for allowed transitions between the conduction (valence) subbands only.

[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.