21.4 Formulation of Multifield Perturbation
405
21.4.2 Thermal Excitation: Debye Thermal Decay
21.4.2.1 Lattice Debye Thermal Expansion
In place of the Grüneisen notion description, one may consider the thermal expansion
coefficient α(t) for the LBA as follows [28]. From the definition, one can translate
L = L 0
1 +
T
0 α(t)dt
to
α(t) ∼ =
d L
L 0 dt
=
1
L 0
∂ L
∂u
du
dt
∝ −
η 1 (t)
L 0 F(r )
= A(r )η 1 (t)
(21.14)
where (∂ L/∂u) = −F
−1 is the inverse of the gradient of the interatomic potential u(r
≈ d) nearby equilibrium. The du/dt is the specific heat of Debye approximation. The
dynamic A(r) = (−L 0 F(r))
−1 approaches the inverse of binding energy closes to but
not equals equilibrium. Compared to the Grüneisen’s volumetric thermal expansion
coefficient (TEC) [29], the α(t) = (VB T )
−1
γη 1 (t), γ/(VB T ) = [L 0 F(r)]
−1 is almost
a constant in the dimension of inverse energy.
Normally, the α(T > θ D ) is in the order of 10
−(6~7) K
−1 . The smaller expansion
coefficient for nanoparticles [30–34] indicates an increase of the potential gradient,
or stronger bond near the equilibrium—a narrowed shape of the interatomic potential
for the contracted bond [13]. Thermal expansion is harder for the contracted bonds
of a nanograin than those in the bulk standard. Approximating A(r ≈ d) to be a
constant, the α(t) follows closely the single-bond Debye specific heat, η(t). The current approach covers the full-range of T-dependent α(T ) showing exceedingly good
agreement with the measured data for AlN, Si 3 N 4 , and GaN as shown in Fig. 21.3
and Table 21.2 [28].
0
400
800 1200 1600 2000
0.0
1.5
3.0
4.5
6.0
7.5
GaN-c
GaN-a
Si 3 N 4
( 10
-6
K
-1
)
T (K)
AlN
0
400
800
1200
1600
0.0
1.5
3.0
4.5
6.0
7.5
9.0
(x10
-6
K
-1
)
T (K)
Diamond
Si
Ge
(a)
(b)
Fig. 21.3 LBA reproduction (solid lines [28]) of the measured (symbols) temperature dependence
of the TECs for a AlN [41], Si 3 N 4 [41], and GaN [34]), b Si [42, 43], Ge [44], and Diamond [33,
45–47]. Table 21.2 lists information derived from the fitting. (Reprinted with permission from [28])
405
21.4.2 Thermal Excitation: Debye Thermal Decay
21.4.2.1 Lattice Debye Thermal Expansion
In place of the Grüneisen notion description, one may consider the thermal expansion
coefficient α(t) for the LBA as follows [28]. From the definition, one can translate
L = L 0
1 +
T
0 α(t)dt
to
α(t) ∼ =
d L
L 0 dt
=
1
L 0
∂ L
∂u
du
dt
∝ −
η 1 (t)
L 0 F(r )
= A(r )η 1 (t)
(21.14)
where (∂ L/∂u) = −F
−1 is the inverse of the gradient of the interatomic potential u(r
≈ d) nearby equilibrium. The du/dt is the specific heat of Debye approximation. The
dynamic A(r) = (−L 0 F(r))
−1 approaches the inverse of binding energy closes to but
not equals equilibrium. Compared to the Grüneisen’s volumetric thermal expansion
coefficient (TEC) [29], the α(t) = (VB T )
−1
γη 1 (t), γ/(VB T ) = [L 0 F(r)]
−1 is almost
a constant in the dimension of inverse energy.
Normally, the α(T > θ D ) is in the order of 10
−(6~7) K
−1 . The smaller expansion
coefficient for nanoparticles [30–34] indicates an increase of the potential gradient,
or stronger bond near the equilibrium—a narrowed shape of the interatomic potential
for the contracted bond [13]. Thermal expansion is harder for the contracted bonds
of a nanograin than those in the bulk standard. Approximating A(r ≈ d) to be a
constant, the α(t) follows closely the single-bond Debye specific heat, η(t). The current approach covers the full-range of T-dependent α(T ) showing exceedingly good
agreement with the measured data for AlN, Si 3 N 4 , and GaN as shown in Fig. 21.3
and Table 21.2 [28].
0
400
800 1200 1600 2000
0.0
1.5
3.0
4.5
6.0
7.5
GaN-c
GaN-a
Si 3 N 4
( 10
-6
K
-1
)
T (K)
AlN
0
400
800
1200
1600
0.0
1.5
3.0
4.5
6.0
7.5
9.0
(x10
-6
K
-1
)
T (K)
Diamond
Si
Ge
(a)
(b)
Fig. 21.3 LBA reproduction (solid lines [28]) of the measured (symbols) temperature dependence
of the TECs for a AlN [41], Si 3 N 4 [41], and GaN [34]), b Si [42, 43], Ge [44], and Diamond [33,
45–47]. Table 21.2 lists information derived from the fitting. (Reprinted with permission from [28])