ELEG 667–016; MSEG-667-016 - Solid State Nanoelectronics – Fall 2005
Solution
Homework #3 - due Thursday, 22 September 2005, in class
1. Ferroelectric criterion for atoms: Consider a system of two neutral atoms separated by a
fixed distance a, each atom having a polarizability α. Find the relation between a and α for such
a system to be ferroelectric. Hint: The dipolar field is strongest along the axis of the dipole.
2. Saturation polarization at Curie point: In a first-order transition, the equilibrium condition:
(T-Θ)/C − |g4|Ps2 + g6Ps4 = 0
with T set equal Tc gives one equation for the polarization Ps(Tc) at the transition temperature. A
further condition at the Curie point is that F(Ps,Tc) = F(0,Tc).
(a) Combining these two conditions, show that Ps2(Tc) = 3|g4| / 4g6.
(b) Using this result, show that Tc = Θ + 3 C g42 / 16g6.
Solutions:
1. For field along z axis due to p1; Ez(at p2) = 2p1/a3. p2 = αEz = 2αp1/a3. This has solution p1 = p2
≠ 0 if 33
1
2a; a
2
α= α= .
2. (a). One condition is, from the minimization,
()
24
C0 4s 6s
TT gP gP 0.γ−− + =
The other condition is
()
246
c0s 4s 6s
111
TTP gP gP 0.
246
γ− − + =
Thus
24 2 4
4s 6s 4s 6s
11
gP gP gP gP;
23
−+=− +
22
4
6s 4 s
6
g
21 3
gP g ; P .
32 4g
==
(b) From the first line of part (a),
()
22
2
444
c0
666
ggg
39 3
TT .
4g 16g 16g
γ− = − =