22.3 Numerical Reproduction of Phonon Relaxation
421
oscillating dynamics. In numerical calculations for graphene, the known effective
bond length d g = 0.142 nm and z g = 5.335 for the bulk graphite, z = 3 for the single
layer graphene, and m = 2.56 for carbon were taken as input. Errors in measurements
render only the accuracy of the derivatives but not the nature and the trends of
observations.
Taking z g = 5.335 for the bulk graphite as a reference, one can obtain the reference
ω(1) and the z-dependent vibration frequency ω(z) [88]. With the known 2D peak
shifting from 2720 to 2680 cm
−1 and the D peak from 1367 to 1344 cm
−1 when the
graphite (z g ) turns to be the monolayer (z = 3) graphene [40, 62, 89], and the G mode
shifting from 1582 to 1587 cm
−1 [40, 41], one can calibrate the z-dependent relative
shift of these vibration modes. For the monolayer graphene (z = 3, m = 2.56),
C x
z, z g
=
ω x (z) − ω x (1)
ω x (z g ) − ω x (1)
=
C z
C z g
−(m/2+1)
z
z g
(D, 2D)
1 (G)
=
0.8147
0.9220
−2.28
×
3.0
5.335
= 0.7456 (D, 2D; z = 3)
1 = 1.3260 (G; z = 1)
And the reference frequency,
ω x (1) =
ω x (z) = ω x (z g )C x (z, z g )
1 − C x (z, z g )
=
ω x (3) − ω x (z g )C x (3, z g )
1 − C x (3, z g )
=
⎧
⎨
⎩
1276.8 (D)
1566.7 (G) (cm
−1
)
2562.6 (2D)
The z dependent phonon frequency for graphene,
ω(z) = ω(1) +
ω
z g
− ω(1)
× D L (z)
=
⎧
⎨
⎩
1276.8 + 90.2 × D D (z) (D)
2563.6 + 157.4 × D 2D (z) (2D)
1566.7 + 16.0 × D G (z) (G)
(22.2)
Figure 22.3 shows the BOLS-LBA reproduction of the z-dependent Raman frequencies of the D/2D modes [40, 43, 68] and the G mode. Likewise, inset b is the
z-dependence of the BP A
1
g mode [64]. The DPS for the layered BP revealed that
reduction of the number-of-layer from multiple to single transits the A 2g phonon from
466 to 470 cm
−1 and the B 2g from 438 to 441 cm
−1 . The n-resolved phonon positive
shift indicates that both the A 2g and B 2g modes arise from oscillation of an invariant
number of bonds [90]. The ω(1) for the BP was determined to be A
1
g (360.20 cm
−1 ),
B 2g (435.00 cm
−1 ), and A
2
g (462.30 cm
−1 ).
421
oscillating dynamics. In numerical calculations for graphene, the known effective
bond length d g = 0.142 nm and z g = 5.335 for the bulk graphite, z = 3 for the single
layer graphene, and m = 2.56 for carbon were taken as input. Errors in measurements
render only the accuracy of the derivatives but not the nature and the trends of
observations.
Taking z g = 5.335 for the bulk graphite as a reference, one can obtain the reference
ω(1) and the z-dependent vibration frequency ω(z) [88]. With the known 2D peak
shifting from 2720 to 2680 cm
−1 and the D peak from 1367 to 1344 cm
−1 when the
graphite (z g ) turns to be the monolayer (z = 3) graphene [40, 62, 89], and the G mode
shifting from 1582 to 1587 cm
−1 [40, 41], one can calibrate the z-dependent relative
shift of these vibration modes. For the monolayer graphene (z = 3, m = 2.56),
C x
z, z g
=
ω x (z) − ω x (1)
ω x (z g ) − ω x (1)
=
C z
C z g
−(m/2+1)
z
z g
(D, 2D)
1 (G)
=
0.8147
0.9220
−2.28
×
3.0
5.335
= 0.7456 (D, 2D; z = 3)
1 = 1.3260 (G; z = 1)
And the reference frequency,
ω x (1) =
ω x (z) = ω x (z g )C x (z, z g )
1 − C x (z, z g )
=
ω x (3) − ω x (z g )C x (3, z g )
1 − C x (3, z g )
=
⎧
⎨
⎩
1276.8 (D)
1566.7 (G) (cm
−1
)
2562.6 (2D)
The z dependent phonon frequency for graphene,
ω(z) = ω(1) +
ω
z g
− ω(1)
× D L (z)
=
⎧
⎨
⎩
1276.8 + 90.2 × D D (z) (D)
2563.6 + 157.4 × D 2D (z) (2D)
1566.7 + 16.0 × D G (z) (G)
(22.2)
Figure 22.3 shows the BOLS-LBA reproduction of the z-dependent Raman frequencies of the D/2D modes [40, 43, 68] and the G mode. Likewise, inset b is the
z-dependence of the BP A
1
g mode [64]. The DPS for the layered BP revealed that
reduction of the number-of-layer from multiple to single transits the A 2g phonon from
466 to 470 cm
−1 and the B 2g from 438 to 441 cm
−1 . The n-resolved phonon positive
shift indicates that both the A 2g and B 2g modes arise from oscillation of an invariant
number of bonds [90]. The ω(1) for the BP was determined to be A
1
g (360.20 cm
−1 ),
B 2g (435.00 cm
−1 ), and A
2
g (462.30 cm
−1 ).