Uniform magnetic susceptibilities€¦ · Uniform magnetic susceptibilities Typical behaviors and measurement techniques . SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 2 - M2 ICFP-

SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 1 - M2 ICFP- Electronic properties of solids (Fabrice Bert)

Uniform magnetic

susceptibilities

Typical behaviors and

measurement techniques


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


Uniform susceptibilities in some simple

solids


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


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


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

)(



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



- 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



Strongly diamagnetic materials can levitate in strong

magnetic field gradient

graphite


levitating frog.flv

meissner.mov


graphite frog

High Tc

superconductors

(cuprates)

Strongly diamagnetic materials can levitate in strong

magnetic field gradient


levitating frog.flv

meissner.mov


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



- 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



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



)(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


is not well defined

in linear response


- 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 a solid with partially filled shells

Larmor

diamagnetism

Temperature

Van Vleck

paramagnetism

Curie

paramagnetism Total

susceptibility

CurieVanVleckdia



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


Uniform susceptibility in a « standard » metal

Temperature

Van Vleck para.

Pauli para.

Global

susceptibility

Landau dia.

Larmor dia.

LandauPauliVanVleckdia



Temperature dependence of Pauli susceptibility in some

transition metals



Susceptibilities in some simple solids

metals

closed shell insulators

Ferromagnetic metals

Uniform susceptibility in simple solids


More « exotic » behaviors of

magnetic uniform

susceptibilities


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


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



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


Co 3d74s2


Magnetometries :

different techniques to

measure uniform

susceptibilies


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


Faraday balance



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:



M U(t)

coil

sample

t

U

Area under the curve is

proportional to the magnetization

MtU D dt)(



Extraction of the sample:



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



VSM : vibration frequencies about 100Hz



SQUID magnetometer

SQUID : Superconducting Quantum Interference Device

supra

insulator

supra

squid DC squid AC (rf)



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



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.



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



SQUID


DC squid DC : alternative setup



Advantages and drawbacks of SQUID

Strong sensitivity + no need of (fast) extraction

BUT

Max magnetic field 7T (commercial) and

expensive


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Uniform magnetic susceptibilities€¦ · Uniform magnetic susceptibilities Typical behaviors and measurement techniques . SUSCEPTIBILITE UNIFORME & MAGNETOMETRIES page 2 - M2 ICFP-