Page 257 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
This fundamental relation includes the effects of mechanical deforma-
tion, electric and magnetic fields, as well as thermal phenomena. In
σ
(7.51) the intensive variables: the stress tensor , the electric field vector
E , the magnetic field vector H and the scalar temperature , are inde-
T
ε
pendent, and the extensive variables: the strain tensor , the dielectric
displacement vector D , the magnetic induction vector B and the scalar
S
entropy , are dependent. By this property we can write that
∂ε ∂ε ∂ε ∂ε
dε = ------ dσ + ------- dE + -------- dH + ------- dT (7.52a)
∂σ ∂E ∂H ∂T
EHT σHT EσT EHσ
∂D ∂D ∂D ∂D
dD = ------- dσ + ------- dE + -------- dH + ------- dT (7.52b)
∂σ ∂E ∂H ∂T
EHT σHT EσT EHσ
∂B ∂B ∂B ∂B
dB = ------- dσ + ------- dE + -------- dH + ------- dT (7.52c)
∂σ ∂E ∂H ∂T
EHT σHT EσT EHσ
∂S ∂S ∂S ∂S
dS = ------ dσ + ------- dE + -------- dH + ------- dT (7.52d)
∂σ ∂E ∂H ∂T
EHT σHT EσT EHσ
With this starting point, we use a Legendre transformation (see Box 7.2)
to convert the fundamental relation from an internal energy formalism to
a Gibbs free energy formalism. In doing so, we switch the roles of pres-
sure and volume, of entropy and temperature, of electrical displacement
and electric field, and of magnetic induction and magnetic field, to obtain
the full and differential form of the Gibbs free energy

⋅
⋅
–
G = U – σ:ε – ED – HB TS (7.53a)
⋅
⋅
dG = – ε:dσ – D dE – B dH – SdT (7.53b)
But, by the first order homogeneous property of fundamental relations
(i.e., (B 7.2.4)), we can also write that
∂G ∂G ∂G ∂G
dG = ------- :dσ + ------- ⋅ dE + -------- ⋅ dH + ------- dT (7.54)
∂σ ∂E ∂H ∂T
EHT σHT EσT EHσ
so that the following four associations may be made

254 Semiconductors for Micro and Nanosystem Technology