11.6 Surface Physical Properties
345
5µm
Fig. 11.8 SEM image of the surface of a 4H–SiC layer on [00.1]-oriented substrate with 8 ◦ misorientation towards
[11.0]. Adapted from [1072]
100nm
(114)
(113)
(114)
(113)
10nm
Fig. 11.9 STM images of a silicon surface close to (113), 21.5 ◦ from (001) towards (111), i.e. (113)-3.7 ◦ . Adapted
from [1073]
11.6 Surface Physical Properties
11.6.1 Surface Phonons
As a model for surface vibrational states, the diatomic one-dimensional chain model discussed in
Sect. 5.2.2 can be modified to allow for a different surface spring constant C s instead of C 1 or C 2 . Such
model can be solved numerically for finite chain length. The bulk dispersion is found with an extra
mode in the gap or above the maximum frequency for a range of values of C s as depicted in Fig. 11.10.
When the spring constant of the surface atom is smaller than the larger of the two spring constants
C 1 and C 2 (in our model calculation C 2 > C 1 ), a state in the gap forms; for a surface spring constant
larger than C 2 , first the surface vibration lies within the optical bulk band and then forms a state above
the maximum optical frequency ω m , given by (5.18).
2
In order to display the dispersion of surface states together with the bulk band structure, the latter is
projected to the surface k || -vector: E n (k) is considered as E n,k ⊥ (k || ) = E ν (k || ), where ν = n, k ⊥ is a
new, continuous index. In the plot of E
surf
n
vs. k || , for each value of k || a range of energies reflects the
2 The appearance of states within the gap and above ω m resembles that of localized vibrational modes of substitutional
masses (Sect. 5.2.7).
345
5µm
Fig. 11.8 SEM image of the surface of a 4H–SiC layer on [00.1]-oriented substrate with 8 ◦ misorientation towards
[11.0]. Adapted from [1072]
100nm
(114)
(113)
(114)
(113)
10nm
Fig. 11.9 STM images of a silicon surface close to (113), 21.5 ◦ from (001) towards (111), i.e. (113)-3.7 ◦ . Adapted
from [1073]
11.6 Surface Physical Properties
11.6.1 Surface Phonons
As a model for surface vibrational states, the diatomic one-dimensional chain model discussed in
Sect. 5.2.2 can be modified to allow for a different surface spring constant C s instead of C 1 or C 2 . Such
model can be solved numerically for finite chain length. The bulk dispersion is found with an extra
mode in the gap or above the maximum frequency for a range of values of C s as depicted in Fig. 11.10.
When the spring constant of the surface atom is smaller than the larger of the two spring constants
C 1 and C 2 (in our model calculation C 2 > C 1 ), a state in the gap forms; for a surface spring constant
larger than C 2 , first the surface vibration lies within the optical bulk band and then forms a state above
the maximum optical frequency ω m , given by (5.18).
2
In order to display the dispersion of surface states together with the bulk band structure, the latter is
projected to the surface k || -vector: E n (k) is considered as E n,k ⊥ (k || ) = E ν (k || ), where ν = n, k ⊥ is a
new, continuous index. In the plot of E
surf
n
vs. k || , for each value of k || a range of energies reflects the
2 The appearance of states within the gap and above ω m resembles that of localized vibrational modes of substitutional
masses (Sect. 5.2.7).