n
1
ð Þ
¼
1
qI
½ ε 33 Iλ
2
1 À ε 11 A
À
Á
C 1 sinh λ 1 x 3 þ C 2 cosh λ 1 x 3
ð
Þ
þ ε 33 Iλ
2
3 À ε 11 A
À
Á
C 3 sinh λ 3 x 3 þ C 4 cosh λ 3 x 3
ð
Þ
À
n 0 μ
n
11 I e 33 c 44 À e 15 c 33
ð
Þ
qn 0 μ n
11 I þ D
n
11 ε 11 A
À
Á
c 44
C 5 ,
ð4:41Þ
v ¼
e 33
c 33
À
e 15
c 44
1
λ 1
C 1 cosh λ 1 x 3 þ C 2 sinh λ 1 x 3
ð
Þ
þ
1
λ 3
C 3 cosh λ 3 x 3 þ C 4 sinh λ 3 x 3
ð
Þ
!
À
C 5
6
x
3
3 À
C 6
2
x 3
2
þ
c 33 I
c 44 A
À
D
n
11 A e 33 c 44 À e 15 c 33
ð
Þ
2
D
n
33 ε 33 Ic 33 c
2
44 a 2
!
C 5 À C 7
"
#
x 3 þ C 8 ,
ð4:42Þ
where C 1 through C 8 are eight arbitrary constants. λ 1 through λ 4 are the four roots of
the following equation:
λ
4
À a 1 λ
2
þ a 2 ¼ 0,
ð4:43Þ
and
a 1 ¼
qμ
n
33 n 0
D
n
33 ε 33
þ
D
n
33 ε 11 þ D
n
11 ε 33
À
Á
A
D
n
33 ε 33 I
, a 2 ¼
qμ
n
11 An 0
D
n
33 ε 33 I
þ
D
n
11 ε 11 A
2
D
n
33 ε 33 I
2
:
ð4:44Þ
Substituting Eqs. (4.39), (4.40), (4.41), and (4.42) into the boundary conditions in
Eqs. (4.37) and (4.38), we obtain eight linear algebraic equations for C 1 through C 8 .
These equations are solved on a computer.
As an example, consider a beam whose geometric parameters are that L ¼ 600 nm
and a ¼ 25 nm. n 0 ¼ N
þ
D ¼ 10
23 m
À3 . D
n
11 ¼ D
n
33 is used in the calculation because
of insufficient material data [2]. For these parameters, λ 1 through λ 4 are all real and so
are Eqs. (4.39), (4.40), (4.41), and (4.42). In addition, λ 1 ¼ À λ 2 and λ 3 ¼ À λ 4 .
Numerical results show that when f y < 0.2 nN, Δn is at least an order of magnitude
smaller than n 0 , and the linearized theory is valid.
Δn ¼ x 2 n
(1) shows that the behavior of the electron concentration perturbation is
determined by n
(1) . In Fig. 4.3, n
(1) is plotted for different values of f ¼ f y . (a) shows
that n
(1) is nearly constant except near the left end where it is large and varies rapidly.
For the relevant applications, its behavior away from the fixed end is of interest. To
show the behavior of n
(1) away from the fixed end more clearly, it is plotted in
(b) without a small region near the fixed end. It can be seen that a larger end force
corresponds to a larger n
(1) as expected. From Eqs. (4.33) and (4.34), it can be seen
that this electron distribution is caused by the shear force Q or the related shear stress
T 4 through e 24 ¼ e 15 and the bending moment M as well as D
1
ð Þ
3 through e 33 . The
behavior of ϕ
(1) is found to be similar to that of n
(1) and therefore is not shown here.
4.3 Static Bending of a Cantilever
97
1
ð Þ
¼
1
qI
½ ε 33 Iλ
2
1 À ε 11 A
À
Á
C 1 sinh λ 1 x 3 þ C 2 cosh λ 1 x 3
ð
Þ
þ ε 33 Iλ
2
3 À ε 11 A
À
Á
C 3 sinh λ 3 x 3 þ C 4 cosh λ 3 x 3
ð
Þ
À
n 0 μ
n
11 I e 33 c 44 À e 15 c 33
ð
Þ
qn 0 μ n
11 I þ D
n
11 ε 11 A
À
Á
c 44
C 5 ,
ð4:41Þ
v ¼
e 33
c 33
À
e 15
c 44
1
λ 1
C 1 cosh λ 1 x 3 þ C 2 sinh λ 1 x 3
ð
Þ
þ
1
λ 3
C 3 cosh λ 3 x 3 þ C 4 sinh λ 3 x 3
ð
Þ
!
À
C 5
6
x
3
3 À
C 6
2
x 3
2
þ
c 33 I
c 44 A
À
D
n
11 A e 33 c 44 À e 15 c 33
ð
Þ
2
D
n
33 ε 33 Ic 33 c
2
44 a 2
!
C 5 À C 7
"
#
x 3 þ C 8 ,
ð4:42Þ
where C 1 through C 8 are eight arbitrary constants. λ 1 through λ 4 are the four roots of
the following equation:
λ
4
À a 1 λ
2
þ a 2 ¼ 0,
ð4:43Þ
and
a 1 ¼
qμ
n
33 n 0
D
n
33 ε 33
þ
D
n
33 ε 11 þ D
n
11 ε 33
À
Á
A
D
n
33 ε 33 I
, a 2 ¼
qμ
n
11 An 0
D
n
33 ε 33 I
þ
D
n
11 ε 11 A
2
D
n
33 ε 33 I
2
:
ð4:44Þ
Substituting Eqs. (4.39), (4.40), (4.41), and (4.42) into the boundary conditions in
Eqs. (4.37) and (4.38), we obtain eight linear algebraic equations for C 1 through C 8 .
These equations are solved on a computer.
As an example, consider a beam whose geometric parameters are that L ¼ 600 nm
and a ¼ 25 nm. n 0 ¼ N
þ
D ¼ 10
23 m
À3 . D
n
11 ¼ D
n
33 is used in the calculation because
of insufficient material data [2]. For these parameters, λ 1 through λ 4 are all real and so
are Eqs. (4.39), (4.40), (4.41), and (4.42). In addition, λ 1 ¼ À λ 2 and λ 3 ¼ À λ 4 .
Numerical results show that when f y < 0.2 nN, Δn is at least an order of magnitude
smaller than n 0 , and the linearized theory is valid.
Δn ¼ x 2 n
(1) shows that the behavior of the electron concentration perturbation is
determined by n
(1) . In Fig. 4.3, n
(1) is plotted for different values of f ¼ f y . (a) shows
that n
(1) is nearly constant except near the left end where it is large and varies rapidly.
For the relevant applications, its behavior away from the fixed end is of interest. To
show the behavior of n
(1) away from the fixed end more clearly, it is plotted in
(b) without a small region near the fixed end. It can be seen that a larger end force
corresponds to a larger n
(1) as expected. From Eqs. (4.33) and (4.34), it can be seen
that this electron distribution is caused by the shear force Q or the related shear stress
T 4 through e 24 ¼ e 15 and the bending moment M as well as D
1
ð Þ
3 through e 33 . The
behavior of ϕ
(1) is found to be similar to that of n
(1) and therefore is not shown here.
4.3 Static Bending of a Cantilever
97