Page 258 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Electron-Phonon
∂G
ε =
-------
–
∂σ (7.55a)
EHT
D = ( – ∇ G) σHT (7.55b)
E
B = ( – ∇ G) (7.55c)
H EσT
∂G
S = – ------- (7.55d)
∂T
EHσ
The subscripts indicate which parameters are to be kept constant during
differentiation. The Maxwell relation (B 7.2.15) enables us to obtain con-
stitutive equations from (7.55a)–(7.55d), and also to determine the sym-
metries inherent in the system of constitutive equations
( ∇ ε) = ( ∇ D) = d () (7.56a)
E σHT σ EHT HT
( ∇ B) = ( ∇ D) = n () (7.56b)
E σHT H EσT σT

d
defining the piezoelectric coefficients and the refractive index , and
n
so on. The relations rely on the fact that second derivatives of first order
relations are not dependent on the order of differentiation. We obtain six-
teen such constitutive relations, summarized by the following four equa-
tions

ε = s EHT :σ + d HT ⋅ E + d ET ⋅ H + α EH ∆T (7.57a)

D = d HT :σ + κ σHT ⋅ E + n σT ⋅ H + p σH ∆T (7.57b)
B = d :σ + n ⋅ E + µ ⋅ H + i ∆T (7.57c)
ET σT σET σE
∆S = α :σ + p ⋅ E + i ⋅ H + C ∆T (7.57d)
EH σH σE σEH

The symbols have the following meaning:
• s is the rank four elastic compliance tensor (the inverse of the
EHT
stiffness tensor derived in Chapter 2) measured at isothermal condi-
tions and at constant electromagnetic field.
• d and d are rank three piezoelectric tensors. It is unusual to
HT ET
have both in the same formulation. Typically we can apply a quasi-

Semiconductors for Micro and Nanosystem Technology 255