Page 86 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
Box 2.5. The characteristic surfaces of waves in anisotropic media [2.11].
For a general anisotropic medium we can write the k  δδ δ δ k ∇ c k()⋅

k
∇ L = ∇ --------- = --------- – ------------------------ (B 2.5.8)
orientation-dependent strength of a plane wave as k k  c k() c k() 2
c k()
ck() = c k()k , k = 1 (B 2.5.1) But ∇ c k() is nothing else than the group veloc-
k
Thus ity of the wave c G , which in turn equals to
T
2
⋅
⁄
(
⋅
ck() k = c k() (B 2.5.2) ( 2 A ⋅ C):Ak) ( c k()ρ A ) , so that we
E
It is usual to set deduce that c k() dLk()⋅ = 0 . This means that
the energy velocity is normal to the slowness sur-
,
,
θ
k = ( sin θ cos φ sin sin φ cos θ) (B 2.5.3)
face.
Equation (B 2.5.1) describes a closed surface
called the velocity surface. We have seen that a Complementarity. The scalar product of the
general anisotropic medium supports three waves, energy velocity and the direction vector gives the
two of which are quasi-shear transverse waves and magnitude of the wave velocity in that direction,
E
i.e., c k() k⋅ = c k() . This follows from the
one that is a quasi-parallel normal wave. Thus we
have a family of three wave surfaces. Inverting Christoffel equation, (2.79). Hence we see that
E
⋅
equation (B 2.5.1) we obtain c k() Lk() = 1 . Now consider the following
construction
⁄
Lk() = k c k() (B 2.5.4)
E
so that ck() Lk()⋅ = 1 . Equation (B 2.5.4) d c k() Lk()⋅( E ) = dc k() Lk() +
⋅
(B 2.5.9)
describes a closed surface called the slowness sur- E E
⋅
c k() dLk()⋅ = dc k() Lk() = 0
face. Again there are three surfaces corresponding
Inserting the definition for Lk() , we obtain that
to the different wave mode solutions. E
dc k() k⋅ = 0 , or, that the tangent of the veloc-
E
Normality. We now wish to form c k() dLk()⋅ ,
ity surface is normal to the propagation direction.
for which we first define the two arguments. The
The above findings are now summarized in.
modal wave’s Poynting vector is
(
C: A ⊗ A) k ∂f  2
⋅ 
⋅
P = ( – σ u ˙) = ----------------------------------- ------ (B 2.5.5)
c k()  ∂τ c E dc E
dL k
Together with the wave’s energy density
1 2 ∂f  2 L

E = ---ρ A ------ (B 2.5.6)
0 2  ∂τ
E Slowness
we form the energy velocity vector c by
Surface
(
⋅
E P 2C: A ⊗ A) k
c k() = ------ = --------------------------------------- (B 2.5.7) Wave
E 2
0 c k()ρ A Surface
⋅
Since dL = ∇ L dk , we obtain
k Figure B2.5.1: The basic geometric relations of
the characteristic wave surfaces. Also refer to
Figure 2.27 for the slowness surface of silicon.
Vibrational energy quanta follow the statistics of bosons, i.e., they are not
subject to the Pauli exclusion principle that holds for fermions such as
the electrons, so that
Semiconductors for Micro and Nanosystem Technology 83