Ballistic Thermal Transport by Phonons at Low Temperatures in Low-Dimensional Quantum Structures - Nanoscale Energy Transport and Harvesting - page 168

( λ + 2 μ ) D nm

N I

+ ( B SVm − A I SVm ) k SVm m W I − ( A I Pm + B I Pn )( k I Pm ) 2

m = 0

B I Pm ) m

2

N I

m = 0 λ

W I W I

m

( A I Pm + ( A I SVm − B SVm ) k SVm

+ − +

D nm

= ( λ + 2 μ ) + ( − A I SVn + B I SVn ) k I SVn n W II − ( A I Pn + B I Pn )( k I Pn ) 2

− ( A I Pn + B I Pn ) n W II

2

+ ( A I SVn − B I SVn ) k I SVn n W II + λ

, (4.78)

2

( A I Pm − B I Pm ) k I Pm

+ ( A I SVm + B SVm )( − ) m W I

N I

m = 0 μ F nm

m W I

−

F nm ( A I Pm − B I Pm ) k I Pm B SVm )( k SVm ) 2

N I

m = 0 μ

W I

m

( A I SVm +

+ − +

2

( A I Pn − B I Pn ) k I Pn

+ ( A I SVn + B I SVn )( − ) n W II

n W II

= μ −

( A I Pn − B I Pn ) k I Pn

+ ( A I SVn − B I SVn )( k I SVn ) 2

n W II

+ μ −

(4.79)

= W I
0

= W I
0

I n ( y ) dy . Note
thathere D mn isthesameasthatpresentedinEqs.4.39-4.43forthe
SH wave. Similarly, we can also get the explicit form of F mn in terms
of Eq.4.65

I m ( y ) η I n ( y ) dy and F mn m ( y ) χ

η χ

with D mn

F mn = 0, if m = 0or n = 0; (4.80)

W I
W II , if m

0, m

n
W II ;

F mn =

=

0, n

=

W I =

(4.81)

sin m

− π

1
√ W I W II

1 n W I

W II

F mn =

π +

m W I + n W II

sin m π −

n π W I
W II

1
m W I +

,if m = 0, n = 0, m

n
W II .
(4.82)

+

W I =

n W II