Page 228 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

From Global Balance to Local Non-Equilibrium

Figure 6.9. Two–dimensional tri- k
L
angular element with nodal indi- j
d j d
ces i, j, k. The three vectors L , i L i
i
L and L represent the edges
j k
connecting nodes j to k, k to i and i
d
to j, respectively. The cross sec- k
tions for the fluxes are d , d and
i i
d . i L k j
i
The discrete form of (6.90a) is found by assuming that ψ varies linearly
along the edges connecting two nodes and thus is a linear function on the
i j
two dimensional triangular element, e.g., defined by the nodes , and k
in Figure 6.9. Therefore, the corresponding flux D is constant. Let us
k
integrate on the part of the box associated with note that lies in the ele-
i j
k
ment defined by , and . This gives us two contributions from the sur-
face integral form the flux displacement vector

∫ Ds S = D l d + D l d j (6.91)
d
i i i
j j
S k
where S is the surface of the part of the box around node lying in the
k
k
triangle ij k,,( ) . Here we introduced the unit vector in the respective
⁄
⁄
directions l = L L and l = L L , that are associated with the box
i i i j j j
surface elements d and d . In terms of the electrostatic potential (6.91)
i j
reads
k Tε ε d i d j d L + d L j
j
∫ Ds S = ------------------- ---- u –( k u ) + ----- u –( k u ) = ρ -------------------------- (6.92)
i i
B
0 r
d
k
j
i
2
q
L
L
S k i j
Semiconductors for Micro and Nanosystem Technology 225