Physics
133/219:
Condensed
ma5er/materials
physics
laboratory
Lecture
2
(5/30/12)
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
ConvenGonal
and
unconvenGonal
superconducGvity
ConvenGonal
superconductors
•  Conven3onal
superconductors
−  Metallic
elements,
alloys,
compounds
•  Superconduc3ng
state
− Electron
pairs
(“Cooper
pairs”):
(k↑,‐k↓)
L
=
0
(s‐wave),
S
=
0
(spin
singlet)
− Macroscopic
quantum
state:
ρ
=
0,
Φ
expulsion
(Meissner
effect)
− Nearly
isotropic
(nodeless)
energy
gap
Δ(k)
≈ Δ over
Fermi
surface
− SC’ing
proper3es:
“ac3vated”
behavior:
e.g.,
Ce(T)
≈
exp(‐Δ/T)
− Pairing
mechanism:
Electron‐phonon
interac3on
− SC’ing
cri3cal
temperature:
Tc
≈ θDexp(‐1/N(EF)V)
kF
Δ
UnconvenGonal
superconductors
•  Unconven3onal
superconductors
–  Cuprate,
Fe‐based,
heavy
fermion,
organic
–
•  Superconduc3ng
state
–  Electrons
paired
in
states
with
L
>
0
(L
=
1,
S
=
1),
p‐wave,
spin
triplet
(L
=
2,
S
=
0),
d‐wave,
spin
singlet
–  Anisotropic
(nodal)
energy
gap
Δ(k)
vanishes
at
points
or
on
lines
on
Fermi
surface
+
Δ(k)
kF
–
–  SCing
proper3es
have
“power
law”
T‐dependences
e.g.,
Ce(T)
~
Tn
(n
=
2,
line
nodes;
n
=
3,
point
nodes)
–  Mul3ple
SC’ing
phases
(different
OP
symmetries)
(e.g.,
UPt3,
UBe13,
PrOs4Sb12)
–  Time
reversal
symmetry
breaking
(e.g.,
PrOs4Sb12)
–  Coexistence
of
SC
&
magne3c
order
(e.g.,
Fe‐based,
heavy
fermion
SC’s)
–  Pairing
mechanism:
Magne3c
dipole
fluctua3ons
(e.g.,
cuprate,
Fe‐based,
heavy
fermion
SC’s)
Electric
quadrupole
fluctua3ons?
(PrOs4Sb12?)
+
SuperconducGng
–
magneGc
interacGons
8
SuperconducGng‐magneGc
interacGons
Superconductor
containing
ions
with
spin
S
and
magne3c
moment
µ
ElectromagneGc
interacGon
(e/mc)(p•A)
m(r)
j(r)
µ =
gµBS
J

Exchange
interacGon
Hex
=
‐2JS•s
s
k
Both
are
“pair
breaking”
interacGons
Conven3onal
SC
–
(k↑,
‐k↓)
(Cooper
pairs)
Raise
energy
of
one
electron
of
Cooper
pair,
lower
energy
of
other
⇒
suppresses
SC!
(NOTE:
µ
=
gJµBJ
&
Hex=‐2J(gJ‐1)J•s
for
R
ions)
k
‐k
‐k
9
Kondo
effect
•  Nonmagne3c
host
metal
containing
small
concentra3on
of
paramagne3c
impuri3es
(spin
S)
•  Paramagne3c
impuri3es
(par3ally‐filled
d‐
or
f‐electron
shells)
3d
transi3on
metal
(Cr,
V,
Mn,
Fe,
Co,
Ni);
e.g.,
Cu1‐xFex
4f
rare
earth
(Ce,
Pr,
Yb);
e.g.,
La1‐xCexAl2
5f
ac3nide
(U);
e.g.,
Th1‐xUx
•  Conduc3on
electrons
(spin
s)
of
host
metal
scaEered
by
paramagne3c
impurity
ions
(spin
S)
via
exchange
interac3on
Vkf
–
Hybridiza3on
matrix
element
Covalent
mixing
(Friedel
‘58,
Anderson
‘61)
εf
=
(EF
–
Ef)
—
f‐electron
binding
energy
Kondo
effect
•  Physical
picture
Many‐body
singlet
forms
as
T
decreases
below
TK
AFM
screening
of
S
by
σ’s
of
conduc3on
electrons
(S↑
,σ
↓)
•  Thermodynamic
energy
scale
TK
~
TF
exp(‐1/N(EF)|J|)
•  T‐dependent
Kondo
(Abrikosov‐Suhl)
resonance
e.g.,
Ce3+
Kondo
effect
NOTE:
impurity
ion
magne3c
for
T
>>
TK,
nonmagne3c
for
T
<<
TK
Wilson‐Sommerfeld
ra3o:
RW
=
(δχ/χ)/(δCV/CV)
=
2
for
S
=
½
Free
electron
gas:
RW
=
1
Kondo
effect
in
Th1‐xUx:
electrical
resisGvity
• Th1‐xUx:
conven3onal
Kondo
effect
(Fermi
liquid
‐
low
T)
•  Δγ
≈
270
mJ/mol
U‐K2
•  Δρ(T)
=
ρo[1‐(T/TK)2];
TK
≈
100
K
M.
B.
Maple
et
al.
(70)
13
Kondo
effect
in
Th1‐xUx:
magneGc
suscepGbility
and
thermopower
Th1‐xUx:
•  χ(T)
=
C/(T‐θ)
•  θ
≈
‐
3
TK
• C
=
Nµeff2/3kB
• Peak
in
thermoelectric
power
⇒
TK
≈
100
K
M.
B.
Maple
et
al.
(70)
Kondo
effect
in
superconductors
•  Superconduc3ng
state
Compe33on
between:
(1)
Singlet
spin
paired
(k↑,‐k↓)
SCing
state
(ESC
~
kBTc);
(2)
Kondo
many
body
singlet
state
(EK
~
kBTK)
–  TK
<<
Tco:
Reentrant
Tc(x)
curve!
–  TK
>>
Tco:
Exponen3al‐like
depression
of
Tc
with
x
–  TK
≈
Tco:
Maximum
in
ini3al
rate
of
depression
of
Tc
Theory:
Müller‐Hartmann,
Zi5artz
(70‐71);
Zuckermann
(68):
Ludwig,
Zuckermann
(71)
15
Kondo
effect
in
La1‐xCexAI2:
reentrant
Tc
vs
x
curve
•  Kondo
effect:
TK
<<
Tco
“reentrant
SC”
•  Riblet,
Winzer
(71)
(U.
Köln)
•  Maple,
FerGg,
Mota,
DeLong,
Wohlleben,
Fitzgerald
(72)
(UCSD)
Maple
(68)
16
Kondo
effect
in
Th1‐xUx:
exponenGal
Tc
vs
x
curve
Comparison
of
Tc
vs
x
curves
of
Th1‐xUx,
Al1‐xMnx,
Th1‐xCex
Kondo
effect:
TK
>>
Tco
Maple,
Huber,
Coles,
Lawson
(70);
17
Huber,
Maple
(70)
Pressure
dependence
of
Tc
in
the
La1‐xCex
system
•  Maximum
in
ΔTc
at
~15
kbar
•  |J|
&
TK
increase
with
P
⇒
TK/Tco
increases
with
P
from
<<
1
to
>>
1
through
TK/Tco
≈
1
at
~15
kbar
•  Consistent
with
measurements
of
P‐dependence
of
TK
•  P‐induced
demagne3za3on
of
Ce
impuri3es
in
La
•  Ce
impurity
analogue
of
γ‐∝
phase
transi3on
in
Ce
metal
K.
S.
Kim,
M.
B.
Maple
(71);
J.
S.
Schilling
(79)
M.
B.
Maple,
J.
Wifg,
K.
S.
Kim
(69)
FerromagneGc
(local
moment)
superconductors
Example:
ErRh4B4
(Similar
behavior
–
HoMo6S8)
•
Experiments:
e.g.,
• FerGg,
Johnston,
DeLong,
McCallum,
Maple,
Ma5hias
(77)
• Moncton,
McWhan,
Schmidt,
Shirane,
Thomlinson,
Maple,
MacKay,
Woolf,
Fisk,
Johnston
(80)
(neutron
sca5ering)
• Sinha,
Crabtree,
Hinks,
Mook
(81)
(neutron
sca5ering)
Tc2
θC
FM
SC
Tc1
SCing
&
FM
regions
SC
&
sinusoidally‐modulated
magne3c
state
(λ
~
100
Å)
• FM
destroys
SC
below
Tc2
<
θC
(Curie
temperature)
• Inhomogeneous
coexistence
of
SC
and
FM
between
Tc2
and
θC
• Homogeneous
coexistence
of
SC
and
new
sinusoidally
modulated
magne3c
state
with
wavelength
λ ~
100
Å
between
Tc2
and
θC
• Screening
of
exchange
or
EM
interac3on
at
long
λ
by
supercurrent
19
MagneGc
ordering
via
RKKY
interacGon
RKKY
interac3on
–
mediated
by
conduc3on
electrons
Hex
=
‐2JS•s
⇒
s(r)
~
cos(2kFr)/(2kFr)3
(kFr
>>
1)
S(r)
µ
=
gµBS
⇒
FM,
AFM,
complex
magne3c
structures
HRKKY
=
‐
(4g2J2/g2µB2V)∑χ(q)exp(iq∙r)Ji∙Jj
∝
J2
q
Kondo effect vs RKKY interaction
•  Screening
of
the
local
moment:
Kondo
effect
•  Indirect
interac3on
between
isolated
magne3c
moments:
RKKY
interac3on
•  Compe3ng
interac3ons
•  “Doniach
phase
diagram”
•  TN
–
Néel
temperature
staggered
magne3za3on
•  Decreases
with
increase
of
|J
N(EF)|
•  QCP
–
quantum
cri3cal
point
SuperconducGvity
near
pressure‐induced
AFM
QCP
AFM
QCP:
Pc
≈
28
kbar
ρ(T)
≈
ρo +
AT1.2
Tc
≤
T
≤
40
K
Tc(max)
≈
0.4
K
Similar
behavior
for
CeIn3
under
P
Julian,
Lonzarich
et
al.
(98)
Suggests
AFM
spin
fluctua3ons
responsible
for
NFL
behavior
in
ρ(T)
and
SC’ing
electron
pairing
SuperconducGvity
within
the
ferromagneGc
state
in
UGe2
•  First
P‐induced
FM‐SC
Saxena
et
al.
(00)
High
purity
crystal
(l
>>
ξ)
⇒
microscopic
coexistence
of
triplet‐spin
SC
&
FM?
•  I3nerant
electron
FM
θC
=
53
K
(P
=
0)
•  γ
≈
35
mJ/mol
K2
m*
≈
20
me
Onuki
et
al.
(93)
•  θC
→
0
K
at
Pc
≈
16
kbar
Oomi
et
al.
(98)
•  Experiments
on
polycrystalline
UGe2
(l
≈
ξ)
Inhomogeneous
state:
coexistence
of
singlet‐
spin
SC
regions
&
FM
regions?
Bauer,
Zapf,
Ho,
Maple
(01)
H–T
phase
diagram
of
PrOs4Sb12
•  Heavy
fermon
behavior
(m*
~
50
me)
•  Nonmagne3c
groundstate
•  Unconven3onal
superconduc3vity
(TRSB,
gap
nodes)
QCP
T.
Yanagasawa
(06)
AFQ
order
HFOP
QCP
Ho
et
al.,
PRB
(03)
•  Related
to
crossover
of
CEF
energy
levels
•  Iden3fied
with
an3ferro‐
quadrupolar
order:
neutron
diffrac3on
Kohgi
et
al.,
JPSJ
(03)
•  Anisotropic
phase
boundary:
M(H,T)
Tayama
et
al.,
JPSJ
(03)
•  SC
in
vicinity
of
an3ferro‐
quadrupolar
QCP!
Generalized
T
–
x
phase
diagram
for
hole‐doped
cuprates
Aler
D.
M.
Broun,
Nature
Physics
4,
178
(2008)
T
–
x
phase
diagrams
of
Fe
pnicGde
systems
H.
Luetkens
et
al.,Nature
Materials
8,
305
(2009)
J.
Zhao
et
al.,
Nature
Materials
7
(2008)
phase
separa3on?
H.
Chen
et
al.,
Europhys.
Le5.
85,
17006
(2009)
S.
Nandi
et
al.,
PRL
104,
057006
(2010)
Heavy
fermion
compounds
Electrical resistivity of selected heavy fermion compounds
After Fisk, Ott, Rice & Smith 86
28
Magnetic susceptibility of selected heavy fermion compounds
After Fisk, Ott, Rice & Smith 86
29
C/T
=
γ
vs
T2
for
CeCu2Si2,
UBe13,
and
UPt3
Ce(T)
=
γ(T)T
γ(0)
≈
1
J/mol‐K2
30
Heavy
fermion
superconductors
Tc (K)
CeCoIn5
* CeCu Si
2 2
2.3
0.49
CeIrIn5
0.4
U6Fe
3.7
* UPd2Al3
2.0
* URu2Si2
1.5
* UNi2Al3
1.0
UBe13
0.85
* UPt
3
0.55
* URhGe
0.4
PuCoGa5
18
PrOs4Sb12
1.8
* Magnetic order
31
Heavy
fermion
superconductors
Tc (K)
CeCoIn5
* CeCu2Si2
2.3
0.49
CeIrIn5
0.4
U6Fe
3.7
* UPd2Al3
2.0
* URu2Si2
1.5
* UNi2Al3
1.0
UBe13
Superconducting under pressure:
0.85
* UPt
3
0.55
* URhGe
0.4
PuCoGa5
18
PrOs4Sb12
1.8
Tc (K)
P (kbar)
* CeRhIn5
2.2
21
* Ce2RhIn8
2
23
* CeCu2Ge2
~2
165
* CePd2Si2
0.43
28
* CeRh2Si2
0.26
11
CeNi2Ge2
0.23
23
* CeIn3
0.17
25
* UGe2
0.7
10
* Magnetic order
32
C/T
≡
γ
vs
T2
for
UBe13
γ(0)
≈
1
J/mol‐K2
⇒
m*
~
102
me
ΔC
≈
γTc
Origin:
Kondo
effect,
valence
fluctua3ons,
narrow
bands?
33
H = 0:
Superconductivity
H = 50 kOe:
NFL behavior
C Petrovic et al., J. Phys.: Condens. Matter 13 (2001) L337
34
Upper critical field Hc2(T) of UBe13
M. B. Maple et al.35
(85) Hc2(T)
for
convenGonal
superconductor
Hc2
T
36
MulGple
superconducGng
phases
in
UPt3
Two
dis3nct
SCing
transi3ons
(sensi3ve
to
H
&
P)
Hasselbach,
Taillefer,
Flouquet
89
Coupling
between
mul3component
SCing
OP
&
AFM
OP
AFM:
TN
≈
5
K
µ
≈
0.02
µB/U
(basal
plane)
Aeppli
et
al.
88
37
H
–
T
phase
diagram
of
UPt3
Ultrasonic
velocity
measurements
Adenwalla,
Ke5erson,
Yip,
Lin,
Levy,
Sarma
92
B‐phase:
odd‐parity,
spin‐triplet
SCing
state
Sauls
‘94
38
MulGple
superconducGng
phases
in
U1‐xThxBe13
H.
R.
O5
et
al.
PRB
31
‘85
R.
H.
Heffner
et
al.
PRL
65
‘90
39
MulGple
superconducGng
phases
in
U1‐xThxBe13
R.
H.
Heffner
et
al.
PRL
65
‘90
S.
E.
Lambert
et
al.
PRL
57
‘86
40
High
temperature
superconducGvity
in
PuCoGa5
Sarrao
et
al.
‘02
41