Physics
133/219:
Condensed
ma5er/materials
physics
laboratory
Lecture
1
(5/16/12)
M.
Brian
Maple
•  Project
oriented
course
−  Prepare
materials
and
inves3gate
physical
proper3es
in
connec3on
with
correlated
electron
phenomena
−  Real
research
–
some3mes
have
to
change
course,
objec3ve
−  Prepared
single
crystals
of
several
systems
containing
Ln
ions
−  Characteriza3on
by
x‐ray
diffrac3on
−  Electrical,
magne3c,
and
thermal
measurements
•  Lectures
and
discussion
to
provide
background
and
context
for
measurements
−  Prepara3on
for
research
paper
in
style
of
Physical
Review
LeEers
−  Miniconference
at
end
of
course
•  Correlated
electron
phenomena
arise
from
interac3ons
between
electrons
in
materials
−  Beyond
free
electron
theory
of
metal
–
noninterac3ng
electrons
(electron
gas)
−  Band
theory
of
solid
–
metals
and
insulators
–
scaEering
of
electrons
by
crystal
laJce
Physics
133/219:
Condensed
ma5er/materials
physics
laboratory
Examples:
•  Superconduc3vity
− Forma3on
of
electron
pairs
(Cooper
pairs)
− AErac3ve
interac3on
− Conven3onal
SC:
Electron
–
phonon
interac3on
(phonons
–
quan3zed
laJce
vibra3ons)
− Unconven3onal
SC:
Electron
–
magnon
interac3on
(magnons
–
quan3zed
magne3c
excita3ons)
•  Magne3c
order
− Ferromagne3c
order
− An3ferromagne3c
order
•  Kondo
effect
− ScaEering
of
electrons
by
paramagne3c
impuri3es
in
metal
− Minimum
in
electrical
resis3vity
(long
standing
puzzle
for
“pure”
noble
metals)
− Forma3on
of
nonmagne3c
ground
state
(many
body
singlet
ground
state)
Physics
133/219:
Condensed
ma5er/materials
physics
laboratory
Examples
(con3nued):
•  Heavy
fermion
metals
− Electrons
have
effec3ve
masses
of
102
‐103
3mes
mass
of
free
electron
− Anomalous
proper3es
− Unconven3onal
type
of
superconduc3vity
− Singlet‐spin,
d‐wave
(l
=
2);
triplet‐spin,
p‐wave
(l
=
1)
•  MoE
insulators
− Coulomb
repulsion
between
electrons
− Produces
insulator
where
band
theory
predicts
a
metal
•  High
temperature
copper
oxide
and
Fe‐based
superconductors
− Superconduc3ng
cri3cal
temperatures
up
to
130
K
at
atmospheric
pressure
and
160
K
under
high
pressure
(RT
=
300
K)
− Anomalous
proper3es
in
normal
state
− Superconduc3vity
emerges
from
insula3ng
or
magne3c
state
that
is
suppressed
by
chemical
subs3tu3on
or
applied
pressure
f‐electron
materials
4
Local
moment
paramagneGsm:
magneGzaGon
M
=
‐gJµBJ
J
=
L
+
S
(determined
from
Hund’s
rules)
gJ
=
1
+
[J(J+1)
+
S(S+1)
‐
L(L+1)]/2J(J+1)
Landé
g‐factor
µB
=
eћ/2mc
=
0.927
x
10‐20
erg/gauss
E
=
mJgJµBH;
mJ
=
J,
J‐1,
.
.
.
,
‐J
(2J+1
equally
spaced
levels)
M
=
NgJJµBBJ(x);
x
=
gJJµBH/kBT
BJ(x)
=
[(2J+1)/2J]ctnh[(2J+1)x/2J]
‐
(1/2J)ctnh(x/2J)
Brillouin
func3on
BJ(x)
gJJ
M(µB)
H/T
5
Local
moment
paramagneGsm:
magneGzaGon
x
<<
1
M
≈
[NgJ2J(J+1)µB2/3kBT]H
=
(Nµeff2/3kBT)H
=
χ(T)H
χ(T)
=
M/H
=
Nµeff2/3kBT
=
C/T
Curie
law
C
=
Nµeff2/3kB
Curie
constant
µeff
=
gJ[J(J+1)]1/2µB
Effec3ve
moment
Satura3on
moment
x
>>
1
M
≈
gJJµB
χ‐1
x
<<
1
Curie
law
slope:
C‐1
=
3kB/Nµeff2
T
6
Local
moment
paramagneGsm:
Hund’s
rules
S,
L,
J
for
lanthanide
ion
determined
from
Hund’s
rules
Hund’s
rules:
Lanthanide
ion
with
configura3on
4fn
4f
electron:
l
=
3,
s
=
1/2
S
=
maximum
value
Σ(sz)i
L
=
maximum
value
Σ(lz)i
(subject
to
Pauli
principle)
J
=
|L‐S|
4f
shell
less
than
half
filled
(n
<
7)
J
=
L+S
4f
shell
more
than
half
filled
(n
≥
7)
e.g.,
Ce3+
(4f1)
S
=
1/2,
L
=
3,
J
=
|L‐S|
=
5/2
Pr3+
(4f2)
S
=
1,
L
=
5,
J
=
|L‐S|
=
4
Gd3+
(4f7)
S
=
7/2,
L
=
0,
J
=
L+S
=
7/2
(so‐called
“S‐state”
ion)
Yb3+
(4f13)
S
=
1/2,
L
=
3,
J
=
L+S
=
7/2
(one
f‐“hole”)
2J+1
degeneracy
of
Hund’s
rule
ground
state
mul3plet
can
be
lired
by
crystalline
electric
field
(CEF)
⇒
CEF
ground
and
excited
states
7
Local
moment
magneGc
ordering
χ(T)
=
Nµeff2/3kB(T‐θ)
Curie‐Weiss
law
θ
–
Curie‐Weiss
temperature
Ferromagne3c
order:
θ
≈
θf
(Curie
temperature)
An3ferromagne3c
order:
θ
≈
‐TN
(Néel
temperature)
χ‐1
AFM
TN
θ
≈
‐TN
θ
≈
θf
FM
T
8
SuperconducGvity
–
zero
resistance
•  Zero
electrical
resistance
(R
=
0)
•  Tc
–
superconduc3ng
cri3cal
temperature
•  Superconduc3ng
ring
–
T
<
Tc
•  R
=
0
⇒
no
energy
dissipa3on
⇒
current
I
persists
forever!
•  Magne3c
field
H
∝
I
(SC’ing
electromagnet)
Discovery
of
superconducGvity
Heike
Kammerlingh
Onnes
• 1908:
liquified
helium
(b.p.:
4.2
K
≈
‐
453
oF)
• 1911:
measured
electrical
resistance
of
mercury
at
low
temperature
• 1913:
Nobel
Prize
in
physics
SuperconducGvity
–
Meissner
effect
•  Magne3c
flux
expulsion
(Meissner
effect)
Superconductor
expels
small
magne3c
field
H
from
its
interior
•  Magne3c
levita3on
Magnet
Superconduc3ng
disk
Fic33ous
(image)
magnet
MagneGc
levitaGon
with
a
high
Tc
cuprate
superconductor
SuperconducGvity
–
criGcal
magneGc
fields
Normal
core
Supercurrent
vortex
Type
I
Superconductor
• Elements
• Hc
–
Cri3cal
field
• Hc
~
102
Oe
• Too
small
for
applica3ons!
Type
II
Superconductor
• Alloys,
compounds
• Hc1
–
Lower
cri3cal
field
• Hc2
–
Upper
cri3cal
field
• Hc1
~
102
Oe,
Hc2
>
~
105
Oe
• Large
enough
for
applica3ons!
SuperconducGvity
–
criGcal
current
density
•  Cri3cal
current
density
H
A
Fluxoid
pinned
by
point
defect
I
Superconductor
F
•  Current
I
in
magne3c
field
H
where
Hc1
<
H
<
Hc2
L
⇒
Mo3on
of
fluxoids
(Lorentz
force
>
pinning
force)
⇒
Resistance,
since
fluxoids
have
“normal”
cores
⇒
Cri3cal
current
Ic
•  Increase
Ic
by
introducing
fluxoid
“pinning
centers”
•  Pinning
centers:
Defects
(grain
boundaries,
vacancies,
impuri3es)
•  Cri3cal
current
density
Jc
=
Ic/A
(A
=
cross‐sec3onal
area)
Origin
of
superconducGvity
•  Metal:
Periodic arrangement (lattice) of positively charge ions
“Gas” of mobile negatively charged conduction electrons
•  Normal state: R ≠ 0
o  Scattering of electrons by:
(1) thermal motion of ions
(2) impurities
(3) other electrons
⇒ dissipation of energy (finite R!)
•  Superconducting state: R = 0
o  Electrons with opposite
momentum p and spin s
”Cooper pairs”
are paired (p↑,-p↓)
o  Electron pairs move in concert through lattice
without scattering (R = 0!)
A5racGve
interacGon
(convenGonal
superconductors)
•  Formation of electron pairs
Attractive interaction between electrons, VA, stronger than repulsive
Coulomb interaction, VR ⇒ net attractive interaction V = VA – VR
•  Electron-lattice interaction
Electron (-) attracts ions (+)
⇒ lattice deformation
⇒region of excess (+) charge
⇒attracts second electron (-)
•  Tc depends on
(1)  Average vibrational frequency of lattice <ω> ∝ M-1/2
(2)  Strength of net attractive interaction V = VA – VR
(3)  Density of electrons (low in metallic oxides –
regarded as poor prospects for high Tc SC!)
Superconductivity in pure elemental solids