102
5 Mechanical Properties
In the zone center, we find v −,2 (() = +v −,1 (() in the lower branch and v +,2 (() = −v +,1 (() in the
upper branch. Thus the two atoms in the base oscillate in phase (out of phase) for long wavelengths in
the acoustic (optical) branch as said before.
At the border of the Brillouin zone, k = ±π/a (and for C 1 = C 2 ),
v −,2 (X ) = +v −,1 (X ) sgn(C 1 − C 2 )
(5.25)
v +,2 (X ) = −v +,1 (X ) sgn(C 1 − C 2 ) .
(5.26)
Now, the relative phase of the two atoms in the base depends on the ratio of C 2 /C 1 ≷ 1! For C 1 > C 2 ,
the top of the acoustic branch has v −,2 (X ) = v −,1 (X ) and thus the same as at the -point, preserving
the ‘acoustic’ nature of the band. For C 2 > C 1 , however, v −,2 (X ) = −v −,1 (X ) and thus out-of-phase,
resembling an optical mode. For the upper branch the situation is vice versa.
The situation can be visualized by plotting the relative phase of v 2 and v 1 , as expressed by
δφ ± (k) = arg
v ±,2 (k)
v ±,1 (k)
,
(5.27)
for the two cases C 2 /C 1 ≷ 1 (note that |v 2 | = 1 for all cases) for the lower branch in Fig. 5.7. The
dispersions for the two cases C 2 /C 1 ≷ 1 with the phase δφ ± depicted as color are compared in Fig. 5.8.
For all cases of C 2 /C 1 > 1, v ±,2 (π/a) − v ±,2 (−π/a) = 0. For all cases of C 2 /C 1 < 1, v ±,2 (π/a) −
v ±,2 (−π/a) = 2 π . Thus this number is a ‘topological invariant’ since it cannot be changed by smooth
changes of the system, i.e. small and slow variations of C 1 /C 2 , unless one goes through the gapless
state (C 1 /C 2 = 1). The case C 1 = C 2 , δφ − = k a/2 and δφ + = π + k a/2 is shown as dotted lines in
Fig. 5.7. It should be noted that the phase difference in (5.27) and the curves in Fig. 5.7 are independent
of the mass ratio.
Generally, the symmetry of a mode v with respect to the operation X that exchanges A- and B-sites,
X v = X
v 1
v 2
=
v 2
v 1
= exp(ı δφ)
v 1
v 2
,
(5.28)
can be expressed by the phase δφ. X is a unitary transformation since X
H X = 1. For a well defined
parity-like symmetry, we demand X
2
= 1; it is present for a phase factor of 1 (δ = 0, positive
(a)
(b)
Fig. 5.7 Relative phase of the B-site to the A-site in the a lower (acoustic, δφ − ) branch and b upper (optical, δφ + ) branch,
expressed by arg(v 2 /v 1 ) from (5.27). The case C 1 /C 2 = 2 is shown as solid line; the inverted case of C 1 /C/2 = 1/2 is
shown as dashed line. The case C 1 = C 2 is shown as dotted lines
5 Mechanical Properties
In the zone center, we find v −,2 (() = +v −,1 (() in the lower branch and v +,2 (() = −v +,1 (() in the
upper branch. Thus the two atoms in the base oscillate in phase (out of phase) for long wavelengths in
the acoustic (optical) branch as said before.
At the border of the Brillouin zone, k = ±π/a (and for C 1 = C 2 ),
v −,2 (X ) = +v −,1 (X ) sgn(C 1 − C 2 )
(5.25)
v +,2 (X ) = −v +,1 (X ) sgn(C 1 − C 2 ) .
(5.26)
Now, the relative phase of the two atoms in the base depends on the ratio of C 2 /C 1 ≷ 1! For C 1 > C 2 ,
the top of the acoustic branch has v −,2 (X ) = v −,1 (X ) and thus the same as at the -point, preserving
the ‘acoustic’ nature of the band. For C 2 > C 1 , however, v −,2 (X ) = −v −,1 (X ) and thus out-of-phase,
resembling an optical mode. For the upper branch the situation is vice versa.
The situation can be visualized by plotting the relative phase of v 2 and v 1 , as expressed by
δφ ± (k) = arg
v ±,2 (k)
v ±,1 (k)
,
(5.27)
for the two cases C 2 /C 1 ≷ 1 (note that |v 2 | = 1 for all cases) for the lower branch in Fig. 5.7. The
dispersions for the two cases C 2 /C 1 ≷ 1 with the phase δφ ± depicted as color are compared in Fig. 5.8.
For all cases of C 2 /C 1 > 1, v ±,2 (π/a) − v ±,2 (−π/a) = 0. For all cases of C 2 /C 1 < 1, v ±,2 (π/a) −
v ±,2 (−π/a) = 2 π . Thus this number is a ‘topological invariant’ since it cannot be changed by smooth
changes of the system, i.e. small and slow variations of C 1 /C 2 , unless one goes through the gapless
state (C 1 /C 2 = 1). The case C 1 = C 2 , δφ − = k a/2 and δφ + = π + k a/2 is shown as dotted lines in
Fig. 5.7. It should be noted that the phase difference in (5.27) and the curves in Fig. 5.7 are independent
of the mass ratio.
Generally, the symmetry of a mode v with respect to the operation X that exchanges A- and B-sites,
X v = X
v 1
v 2
=
v 2
v 1
= exp(ı δφ)
v 1
v 2
,
(5.28)
can be expressed by the phase δφ. X is a unitary transformation since X
H X = 1. For a well defined
parity-like symmetry, we demand X
2
= 1; it is present for a phase factor of 1 (δ = 0, positive
(a)
(b)
Fig. 5.7 Relative phase of the B-site to the A-site in the a lower (acoustic, δφ − ) branch and b upper (optical, δφ + ) branch,
expressed by arg(v 2 /v 1 ) from (5.27). The case C 1 /C 2 = 2 is shown as solid line; the inverted case of C 1 /C/2 = 1/2 is
shown as dashed line. The case C 1 = C 2 is shown as dotted lines