Download Lecture 23

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10PM-2:00 PM)
Lecture 24, room 502 Nielsen
Chapter 13: Dielectrics and ferroelectrics
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
Polarization
The polarization P is defined as the dipole moment per
unit volume. The total dipole moment is defined as
p = ∑ qn rn , where rn is the position vector of the charge.
3( pir )r − r 2 p
E (r ) =
r5
Dai/PHYS 342/555 Spring 2006
Chapter 13-2
If Px , Py , Pz are the components of the plarization P referred
to the principle axes of an ellipsoid, then
E1x = − N x Px ; E1 y = − N y Py ; E1z = − N z Pz ;
A uniform applied field will induce uniform polarization. We
introduce the dielectric susceptibility χ
P = χ E ; Therefore
E = E0 + E1 = E0 − NP
P = χ ( E0 − NP )
Dai/PHYS 342/555 Spring 2006
Chapter 13-3
The macroscopic electric field in a sphere is
E = E0 + E1 = E0 − 4π P / 3;
The local field at an atom is the sum of the electric field
E0 from external sources and of the field from the dipoles.
E = E0 + E1 + E2 + E3 .
3( pi iri )ri − ri 2 pi
E1 + E2 + E3 = ∑
5
r
i
i
E0 = field produced by fixed charges external to the body;
E1 = depolarization field, from a surface charge density nˆ i P on
the outer surface of the specimen;
E2 = Lorentz cavity field;
E3 = Field of atoms inside cavity.
Dai/PHYS 342/555 Spring 2006
Chapter 13-4
Dai/PHYS 342/555 Spring 2006
Chapter 13-5
Lorentz Field, E2
The electric field at the center of the spherical cavity of
radius a is
π
4π
E2 = ∫ (a −2 )(2π a sin θ )(adθ )( P cos θ )(cos θ ) =
P.
0
3
Dai/PHYS 342/555 Spring 2006
Chapter 13-6
Dielectric constant and polarizability
The dielectric constant ε of an isotropic medium relative
to vacuum is defined as
E + 4π P
ε≡
=1+4πχ
E
The polarizability α of an atom is defined in terms of the local
electric field at the atom:
p = α Elocal
The polarization of a crystal is then
P = ∑ N j pj =
j
∑N α E
j
j
local
( j ).
j
Dai/PHYS 342/555 Spring 2006
Chapter 13-7
If the local field is given by the Lorentz relation, then
4π
4π
P = ∑ N jα j ( E +
P )= ( ∑ N jα j ) ( E +
P).
3
3
j
N jα j
P
∑
χ= =
.
E 1 − 4π
N jα j
∑
3
ε = 1 + 4πχ
ε − 1 4π
=
N jα j
∑
ε +2 3
Dai/PHYS 342/555 Spring 2006
Chapter 13-8
The total polarizability can be separated into three parts:
1. electronic: arises from the displacement of the electron shell
relative to a nucleus.
2. ionic: comes from the displacement of a charged ion with
respect to other ions.
3. dipolar: from molecules with a permanent electric dipole
moment that can change orientation in an applied electric field.
Dai/PHYS 342/555 Spring 2006
Chapter 13-9
Classical theory of electronic polarizability
Equation of motion in the local electric field Eloc sinωt
d 2x
m 2 + mω02 x = −eEloc sinωt
dt
m(−ω 2 + ω02 ) x0 = −eEloc .
α = p / Eloc = −ex / Eloc
e2
=
m(ω02 − ω 2 )
Dai/PHYS 342/555 Spring 2006
Chapter 13-10
Structural phase transitions
The stable structure at a temperature T is determined by the
minimum of the free energy F = U − TS .
Ferroelectric crystals
A ferroelectric state is a state where the center of positive charge
of the crystal does not coincide with the center of negative charge.
Ferroelectricity disappears above a certain temperature, where the
crystal is in the paraelectric state.
Dai/PHYS 342/555 Spring 2006
Chapter 13-11
Classification of ferroelectric crystals
Ferroelectric crystals can be classified into two main
groups: order-disorder and displacive transition.
Dai/PHYS 342/555 Spring 2006
Chapter 13-12
Dai/PHYS 342/555 Spring 2006
Chapter 13-13
Displacive transitions
polarization catastrophe:
8π
N jα j
1+
∑
3
ε=
4π
1−
∑ N jα j
3
Dai/PHYS 342/555 Spring 2006
Chapter 13-14
Antiferroelectricity
Dai/PHYS 342/555 Spring 2006
Chapter 13-15
Ferroelectric domains
Dai/PHYS 342/555 Spring 2006
Chapter 13-16
Piezoelectricity
Dai/PHYS 342/555 Spring 2006
Chapter 13-17
Consider a semiclassical model of the ground state of the hydrogen
atom in an electric field normal to the plane of the orbit. Show that
for this model α = aH3 , where aH is the radius of the unperturbed
orbit.
Dai/PHYS 342/555 Spring 2006
Chapter 13-18
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.
Dai/PHYS 342/555 Spring 2006
Chapter 13-19