116
5 Mechanical Properties
Fig. 5.24 Density of states versus energy (in units of maximum phonon energy E m = ω max according to (5.18)) for
the diatomic linear chain model (M 1 = M 2 , C 2 = 2 C 1 , 2 7 dimers, average over 2 12 configurations) for various levels of
random relative variations of the masses (solid lines). The forbidden energy ranges for the ideal chain are shown in light
grey. The DOS has been normalized such that its integral over all energies is 1 (the energy bins have a width of 0.01 E m )
Fig. 5.25 a Diatomic linear chain model (DLCM) with A–B dimers (one unit cell shown in grey) in an infinite chain;
A-sites (B-sites) are shown in red (blue); the connecting springs have force constants C 1 (green) and C 2 (orange),
respectively. b DLCM with finite number of dimers (shown for N = 6) with both end sites fixed (shown in black). The
movable sites now consist of B–A dimers (one unit cell shown in grey)
(i.e. for periodic boundary conditions) where a ‘clean’ gap is present; for C 1 = C 2 of course no gap
appears. In case (ii), two states appear within the gap with energy ω g = ω max /
√
2. The eigenvalues of
the problem are shown as a function of C 2 /C 1 in Fig. 5.26a. The mode pattern for the two gap states
are depicted in Fig. 5.27a (for C 2 /C 1 = 1/2). All other states follow bulk like vibration patterns as
expected from a vibrating string. For M 1 = M 2 the gap states are degenerate, otherwise split as shown
in Fig. 5.26c. For the case of C 2 > C 1 , the patterns for the highest mode of the lower band and the
lowest mode of the top band are depicted in Fig. 5.27b; the are obviously bulk-like as all others and no
localized states exist.
The appearance of end states in one of the cases is the expression of the fact that the two bulk band
structures for C 1 > C 2 and C 1 < C 2 are topologically different
7 (cmp. Sect. 5.2.3). The general effect
is the so-called ‘bulk-boundary’ correspondence where edge states appear at the interface between
different topological phases (one of these phases can be the topologically trivial vacuum outside the
solid). Such edge states are termed more precisely end-states for (quasi-) one-dimensional systems.
7 Since the two outer atoms are fixed, the movable parts of the chain consists of B–A dimers (Fig. 5.25b), thus the
topologically non-trivial case occurs here for C 2 < C 1 .
5 Mechanical Properties
Fig. 5.24 Density of states versus energy (in units of maximum phonon energy E m = ω max according to (5.18)) for
the diatomic linear chain model (M 1 = M 2 , C 2 = 2 C 1 , 2 7 dimers, average over 2 12 configurations) for various levels of
random relative variations of the masses (solid lines). The forbidden energy ranges for the ideal chain are shown in light
grey. The DOS has been normalized such that its integral over all energies is 1 (the energy bins have a width of 0.01 E m )
Fig. 5.25 a Diatomic linear chain model (DLCM) with A–B dimers (one unit cell shown in grey) in an infinite chain;
A-sites (B-sites) are shown in red (blue); the connecting springs have force constants C 1 (green) and C 2 (orange),
respectively. b DLCM with finite number of dimers (shown for N = 6) with both end sites fixed (shown in black). The
movable sites now consist of B–A dimers (one unit cell shown in grey)
(i.e. for periodic boundary conditions) where a ‘clean’ gap is present; for C 1 = C 2 of course no gap
appears. In case (ii), two states appear within the gap with energy ω g = ω max /
√
2. The eigenvalues of
the problem are shown as a function of C 2 /C 1 in Fig. 5.26a. The mode pattern for the two gap states
are depicted in Fig. 5.27a (for C 2 /C 1 = 1/2). All other states follow bulk like vibration patterns as
expected from a vibrating string. For M 1 = M 2 the gap states are degenerate, otherwise split as shown
in Fig. 5.26c. For the case of C 2 > C 1 , the patterns for the highest mode of the lower band and the
lowest mode of the top band are depicted in Fig. 5.27b; the are obviously bulk-like as all others and no
localized states exist.
The appearance of end states in one of the cases is the expression of the fact that the two bulk band
structures for C 1 > C 2 and C 1 < C 2 are topologically different
7 (cmp. Sect. 5.2.3). The general effect
is the so-called ‘bulk-boundary’ correspondence where edge states appear at the interface between
different topological phases (one of these phases can be the topologically trivial vacuum outside the
solid). Such edge states are termed more precisely end-states for (quasi-) one-dimensional systems.
7 Since the two outer atoms are fixed, the movable parts of the chain consists of B–A dimers (Fig. 5.25b), thus the
topologically non-trivial case occurs here for C 2 < C 1 .