uniform magnetic susceptibilities€¦ · uniform magnetic susceptibilities typical behaviors and...
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SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 1 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Uniform magnetic
susceptibilities
Typical behaviors and
measurement techniques
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 2 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Various behaviors of M(H)
Applied field H
magnetization M
H
Msaturation
Mremanent
H
M
Paramagnetism ferromagnetism
type II superconductor
M
Applied field H
magnetization M
Diamagnetism
Linear
response
Linear
response
Non linear
response
Non linear
response
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 3 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Uniform susceptibilities in some simple
solids
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 4 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
where DE0 is the energy perturbation due to the effect of B
the free energy
-> the effect of B in the Hamiltonian (DE0) needs to be
calculated at the 2nd order in perturbation to get finally M in
first order of B.
B
E
V
N
B
E
V
NM
D
0
0
B
M B
E
BV
N
D
00
Calculation of the uniform magnetic susceptibility
Uniform susceptibility
F
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 5 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Calculation for insulators (with isolated and localised spins)
SBgVm
rAep
B
Z
ielectrons
i
el
ii.
2
)(
1
2
H
2/rBA
Z
ielectrons
i
i
i
i
Z
ielectrons el
iirB
m
erBp
m
e
m
p
m
rAep
1
222
1
2
82.
22
)(
LBprBrBp
i
ii
i
ii
.)(.).(
elBme /
Z
i
iB
rBm
eSgLB
1
22
8).(
0HH
gauge
->
In the 2nd term, one recognizes the orbital momentum
and the Bohr magneton
Hamiltonian of an atom in a
magnetic field
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 6 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
D
nn nn
n
EE
nnnnE
' '
2
0
0
'HH
HH
)(0,,2222
2
iiiii yxBxyBrB
Taking B//Oz and keeping only the quadratic contributions in B:
Z
i
iB
rBm
eSgLB
1
22
8).(
0HH
nn nn
BZ
i
iB
EE
nSgLBn
nrBm
ennSgLnB
' '
2
1
22 ').(
8)(
nn nn
BZ
i
iiB
EE
nSgLBn
nyxnBm
enSgLnB
' '
2
1
222
2 ').(
)(8
)(
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 7 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Larmor diamagnetism
D
nn nn
BZ
i
iiBn
EE
nSgLBn
nyxnBm
enSgLnBE
' '
2
1
222
2 ').(
)(8
)(
2
1
22
3
20)(0 rZyx
Z
i
ii
D
22
2
000
3
2
8rZB
m
e
BBV
N
B
E
BV
N
dia
2
2
0
6rZ
m
e
V
N
dia
Contribution of this term to the magnetic susceptibility:
We use: where r is the ionic radius
We assume that temperature is low enough so that only the ground
state contributes significatively
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 8 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
- negative, weak, T independent
- Exists in all solids
- important contribution for
insulators with full shells (L=S=0)
Larmor diamagnetism
T
2
2
0
6rZ
m
e
V
N
dia
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 9 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Strongly diamagnetic materials can levitate in strong
magnetic field gradient
graphite
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 10 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
graphite frog
High Tc
superconductors
(cuprates)
Strongly diamagnetic materials can levitate in strong
magnetic field gradient
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 11 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Van Vleck Paramagnetism
D
nn nn
BZ
i
iiBn
EE
nSgLBn
nyxnBm
enSgLnBE
' '
2
1
222
2 ').(
)(8
)(
We keep only the effect of the mixing of the exited states into the ground state
D
0 0
2
000
0).(
n n
B
EE
SgLBn
BBV
N
B
E
BV
N
0 0
2
)(0
n n
B
EE
nSgL
a
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 12 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
- positive, weak, T independent
- Detectable in solids with only full shells (J=0 in ground
state)
Van Vleck paramagnetism
T
0 0
2
)(0
n n
B
EE
nSgL
a
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 13 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
D
nn nn
BZ
i
iiBn
EE
nSgLBn
nyxnBm
enSgLnBE
' '
2
1
222
2 ').(
)(8
)(
Contribution for atoms with partially filled shells (J<>0)
All the states with different Jz (J=L+S) must be taken into account
because temperature is comparable with the difference in energy
of these different states.
Statistical physics calculation (see Kittel or Ashcroft)
)(Tk
JHgJBg
V
NM
B
B
JB
)2
coth(2
1)
2
12coth(
2
12)(
J
x
Jx
J
J
J
JxB
J
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 14 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
)(Tk
JHgJBg
V
NM
B
B
JB
TkJHgxBB
/ Is not small
1/ TkJHgyBB
=>
Linear response ok
HTk
JJg
V
NM
B
B
3
)1()(2
T
C
Curie
B
B
k
JJg
V
NC
3
)1()(2
0
Curie constant
Curie Paramagnetism
Uniform susceptibility in insulators
is not well defined
in linear response
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 15 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
- positive, varies as 1/T, dominates all other terms at low T
- exists only in solids with partially filled shells (unpaired
spins)
Curie Paramagnetism
T
T
C
Curie
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 16 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Uniform susceptibility in a solid with partially filled shells
Larmor
diamagnetism
Temperature
Van Vleck
paramagnetism
Curie
paramagnetism Total
susceptibility
CurieVanVleckdia
Uniform susceptibility in insulators
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 17 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Metals
Different calculation involving spin and orbital momentum of
the conduction electrons (see tutorial).
Two contributions :
-Spin susceptibility «Pauli »
-Orbital susceptibility « Landau »
(Larmor and Van Vleck contributions are still there)
)(2
0 FBPauliEn
PauliLandau
3
1
Uniform susceptibility in non correlated metals
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 18 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Uniform susceptibility in a « standard » metal
Temperature
Van Vleck para.
Pauli para.
Global
susceptibility
Landau dia.
Larmor dia.
LandauPauliVanVleckdia
Uniform susceptibility in non correlated metals
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 19 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Temperature dependence of Pauli susceptibility in some
transition metals
Uniform susceptibility in non correlated metals
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 20 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Susceptibilities in some simple solids
metals
closed shell insulators
Ferromagnetic metals
Uniform susceptibility in simple solids
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 21 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
More « exotic » behaviors of
magnetic uniform
susceptibilities
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 22 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Shimizu et al., PRB (1995)
csteT
D exp
Gap in the magnetic
excitations
Spin 1 chain with antiferromagnetic interactions
(no long range order)
Y2BaNiO5
Ni O
Uniform susceptibilities in exotic systems
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 23 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
High TC superconducting cuprates
as a function of doping for T>TC
YBa2Cu3O6+x
Cu
O
superconductivity dopage x
Antiferro Supraconducteur
T
Anti
Ferro
~ Pauli like, T-independent
TC
Strong T dependence: pseudogap
Johnson et al
Uniform susceptibilities in exotic systems
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 24 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
0 100 200 300 4000.00
0.05
0.10
x=0.44
x=1
T (K)
2
3K
c (%
)
Cobaltates NaxCoO2
Very good ionic conductors (batteries)
Non conventional superconductors
Large thermoelectric power
Lang et al., PRB
0 0.25 0.50 0.75 1
x
supra
antiferro
magnétisme
x=0.66
x=0.58
x=0.3
x=0.5
x=0.62
x=0.44
Uniform susceptibilities in exotic systems
Co 3d74s2
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 25 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Magnetometries :
different techniques to
measure uniform
susceptibilies
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 26 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Measurement based on a force
Faraday balance
Sample is in a magnetic field gradient
designed so that:
cstez
Hz
z
HMHMF
z
zz
00.
One measures the force on the
balance
from which one gets Mz the
magnetization along z
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 27 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Faraday balance
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 28 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
M
Inductive techniques
Extraction of the sample:
Sample is moved inside a pick-up coil:
dt
dtU
)(
Variation of flux when sample is
inserted:
U(t)
bobine
MtU D dt)(
echantillon
MAAHAMHaaemptycoilmplecoilwithsa 000
)(
Ha
Leading to a voltage at the coil:
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 29 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
M U(t)
coil
sample
t
U
Area under the curve is
proportional to the magnetization
MtU D dt)(
Experimental measurement of uniform susceptibility
Inductive techniques
Extraction of the sample:
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 30 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Inductive techniques
vibration : VSM vibrating Sample Magnetometer
Sample is moved periodically
with a frequency n
The periodic voltage is measured with
a lock-in detection
t
U
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 31 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
VSM : vibration frequencies about 100Hz
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 32 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
SQUID magnetometer
SQUID : Superconducting Quantum Interference Device
supra
insulator
supra
squid DC squid AC (rf)
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 33 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
AC SQUID (rf):
sin)sin(0120
IIIs
Josephson junction: Cooper pairs tunnel through
the insulator. A dc current IS appears due to the
phase jump accross the junction Is
Flux quantization in the ring:
enhnring
2/0
2/00
nring
With one junction:
Without junction:
One applies an external flux (L : ring inductance) :
0
0
2sin
22sin
sin
0
0
0
ring
ring
LI
nLI
LI
LI
ext
ext
ext
sextring
supra
isolant
supra
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 34 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
0
2sin0
tLI
et
example : 00
25.1 LI
The flux in the ring varies
periodically with the
external flux, with the
period 0
By measuring the number of 0 passing through the squid ring, one can
measure very precisely the external flux, therefore the magnetization
of the sample with an accuracy (much better than) 0 : very good
sensitivity because 0 =2.10-15 Wb is small.
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 35 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Ac squid setup : inductive detection
M
Ha
Mutual
inductance
Mutual inductance
The magnetized sample
induces a flux in the
pickup coil
the flux is
transfered to
the squid by
inductive
coupling
Measured in a
resonant circuit
used at
resonance : flux
change induces L
change and voltage
change
Rf current
4.2K below the
superconducting TC
U
t
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 36 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
SQUID
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 37 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
DC squid DC : alternative setup
Experimental measurement of uniform susceptibility
SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 38 - M2 ICFP- Electronic properties of solids (Fabrice Bert)
Advantages and drawbacks of SQUID
Strong sensitivity + no need of (fast) extraction
BUT
Max magnetic field 7T (commercial) and
expensive
Experimental measurement of uniform susceptibility