11.3 Surface Energy
341
11.3 Surface Energy
The surface energy, more precisely the surface energy per area γ, is related to the work that is necessary
to split a crystal in two parts. Such process will leave broken (‘dangling’) bonds. This energy will
depend on the crystal orientation as is already clear from the fact that there are easy cleaving planes
(Sect. 5.4.2). The surface energy will also depend on the surface reconstruction (see Sect. 11.4), i.e. the
rearrangement of surface bonds and atoms. Generally, the reduction of the number of dangling bonds
on a surface lowers its energy, while the distortion of bonds increases its energy.
The anisotropy of the surface energy for a given orientation hkl, γ(hkl), determines the equilibrium
crystal shape (ECS) at a given temperature (below melting temperature). The crystallite is assumed
to be of at least mesoscopic size such that energy effects due to edges and apexes can be neglected
compared to the surface energy terms. As an example theoretically calculated surface energy values for
covalent semiconductors are listed in Table 11.2. The equilibrium shape of silicon has been observed
for µm-sized bulbs as shown in Fig. 11.2a for T = 1323 K ; the experimental anisotropy of surface
energy of silicon is shown in Fig. 11.2b.
11.4 Surface Reconstruction
When in a Gedankenexperiment the bulk crystal is split such that a surface with defined orientation develops, many bonds are cut. These dangling bonds can be saturated with other atoms such as
hydrogen. In particular under vacuum conditions, the dangling bonds will rearrange and form new
Table 11.2 Surface energy (in J/m 2 ) for various C, Si and Ge surfaces. Data from [1060]
Material
{111}
{110}
{100}
{311}
C
8.12
7.48
9.72
8.34
Si
1.82
2.04
2.39
2.12
Ge
1.32
1.51
1.71
1.61
(a)
(111) ( 1 1 3 )
(
0
0
1
)
(110)
( 1 1 0 )
( 1 1 1 )
(1
1
1
) (11
3)
(b)
1.02
1.01
1.00
0.99
0.98
0.97
0.96
/ (111)
0
3
-
0
0
6
-
0
9
-
0
9
0
3
0
6
Si
(111) (113) (001) (113)
(110)
(111)
Angle (degree)
Fig. 11.2 a Equilibrium crystal shape (cross section in 110 azimuth) of 1.06 µm diameter Si bulb at T = 1323 K with
facet orientations as labelled. a Anisotropy of surface energy (relative to γ(111)) for Si, the dashed line is guide to the
eye. Adapted from [1061]
341
11.3 Surface Energy
The surface energy, more precisely the surface energy per area γ, is related to the work that is necessary
to split a crystal in two parts. Such process will leave broken (‘dangling’) bonds. This energy will
depend on the crystal orientation as is already clear from the fact that there are easy cleaving planes
(Sect. 5.4.2). The surface energy will also depend on the surface reconstruction (see Sect. 11.4), i.e. the
rearrangement of surface bonds and atoms. Generally, the reduction of the number of dangling bonds
on a surface lowers its energy, while the distortion of bonds increases its energy.
The anisotropy of the surface energy for a given orientation hkl, γ(hkl), determines the equilibrium
crystal shape (ECS) at a given temperature (below melting temperature). The crystallite is assumed
to be of at least mesoscopic size such that energy effects due to edges and apexes can be neglected
compared to the surface energy terms. As an example theoretically calculated surface energy values for
covalent semiconductors are listed in Table 11.2. The equilibrium shape of silicon has been observed
for µm-sized bulbs as shown in Fig. 11.2a for T = 1323 K ; the experimental anisotropy of surface
energy of silicon is shown in Fig. 11.2b.
11.4 Surface Reconstruction
When in a Gedankenexperiment the bulk crystal is split such that a surface with defined orientation develops, many bonds are cut. These dangling bonds can be saturated with other atoms such as
hydrogen. In particular under vacuum conditions, the dangling bonds will rearrange and form new
Table 11.2 Surface energy (in J/m 2 ) for various C, Si and Ge surfaces. Data from [1060]
Material
{111}
{110}
{100}
{311}
C
8.12
7.48
9.72
8.34
Si
1.82
2.04
2.39
2.12
Ge
1.32
1.51
1.71
1.61
(a)
(111) ( 1 1 3 )
(
0
0
1
)
(110)
( 1 1 0 )
( 1 1 1 )
(1
1
1
) (11
3)
(b)
1.02
1.01
1.00
0.99
0.98
0.97
0.96
/ (111)
0
3
-
0
0
6
-
0
9
-
0
9
0
3
0
6
Si
(111) (113) (001) (113)
(110)
(111)
Angle (degree)
Fig. 11.2 a Equilibrium crystal shape (cross section in 110 azimuth) of 1.06 µm diameter Si bulb at T = 1323 K with
facet orientations as labelled. a Anisotropy of surface energy (relative to γ(111)) for Si, the dashed line is guide to the
eye. Adapted from [1061]