Magnetic susceptibility.

Basic principles and features on the PPMS.

Experimental methods:

• DC Magnetometry

• AC Magnetometry

• Torque Magnetometry

• Heat Capacity

1. DC Magnetometry. Basic principles.

DC magnetic measurements determine the equilibrium value of the magnetization in a

sample. The sample is magnetized by a constant magnetic field and the magnetic moment

of the sample is measured, producing a DC magnetization curve M(H). Inductive

measurements are performed by moving the sample relative to a set of pickup coils (one-

shot extraction, for example). In conventional inductive magnetometers one measure the

voltage induced by the moving magnetic moment of the sample in copper coils. The

Faraday’s Law is used in this case to extract useful information (see Appendix I).

DC extraction signal

2. AC Magnetometry. Basic principles.

In AC measurements, a small AC drive magnetic field is superimposed on the DC field,

causing a time-dependent moment in the sample. The field of the time-dependent

moment induces a current in the pickup coils , allowing measurements without sample

motion.

Sample Space AC Drive Coil (blue) Calibration Coils (dash) Thermometer

Detection Coils (red) AC Drive Compensation Coils Electrical Connections

Case 1: Low AC frequency

As long as the AC field is small, the induced AC moment is:

MAC = (dM/dH) HAC sin(ω t) HAC is the amplitude of the driving field,

ω is the driving frequency

χ = dM/dH is the slope of the M(H) curve Case 2: Higher frequencies

At high frequencies the AC moment of the sample doesn’t follow along the DC

magnetization curve due to the dynamic effects on the sample. That is why the AC

susceptibility is often known as the dynamic susceptibility. The magnetization of the

sample may lag behind the drive field. The AC method yields two quantities:

a) the magnitude of the susceptibility |χ|

b) the phase shift ϕ The magnetic susceptibility is defined as follows:

χ(q, ω)=χ' + χ'' or χ(q, ω)=|χ| exp(iϕ)

χ' - real part, or dynamical susceptibility

χ'' - imaginary part has the character of absorption coefficient ϕ = arctan(χ''/χ')

describes a phase lag of M(t) behind the external field H exp(iωt)

3. AC vs DC Magnetometry. Examples.

In the limit of low frequencies an AC measurement is rather similar to a DC extraction,

the real component χ' is just the slope of the magnetization curve M(H). χ'' or the

imaginary part is missing in the DC method.

Example #1

Ferromagnetic metal Pr0.75La0.25Ni measured on PPMS (E21/TUM).

Since the sample is conductive eddy(whirlpool) currents contribute to the dissipative

part χ''.(See Appendix II)

a) Pr0.75La0.25Ni measured by the DC method

0

1

2

3

1 10 100 1000T (K)

DC magn

0

100

200

300

0 100 200 300T (K)

1/M

(em

u-1)

experiment

Curie-Weiss

Mag

netiz

atio

n (e

mu)

K5.19,T

T⋅=Θ⋅⋅⋅

Θ−=χ

Tc=20.5K in pure PrNi

b) Pr0.75La0.25Ni measured by the AC method

0

1

2

3

1 10 100 1000T (K)

Mag

netiz

atio

n (e

mu)

DC magnM' MomentM''

0

20

40

60

80

100

1 10 100 1000

T (K)

Pha

se (d

egre

e), M

' (ar

b. u

nits

)

M' Phase

- In a ferromagnetic metal both χ' and the phase ϕ reveal well-defined maxima close

to the critical temperature Tc.

- χ'' or the dissipative part is huge in the phase transition region.

Example #2

AF insulator Ca2Y2Cu5O10 measured on PPMS (E21/TUM).

DC: H0 = 10 kOe

AC: f = 1 kHz, HAC = 10 OeThe Curie Law: χ(T)=C/T,

B

BA k

SSgNNC3

)1(22 +=

µ

C = 0.471 emu K/mole ⇒ S = 1/2 (g = 2.24)

0

0.002

0.004

0.006

0.008

0 100 200 300T (K)

χ (e

mu/

mol

e C

u)

experiment

Curie law

Since the sample isn’t conductive there

is no contribution due to eddy currents.

(em

u/m

ole

Cu)

DC = AC and χ'' ≈ 0 Re{X} AC measurements

0

0.002

0.004

0.006

0.008

0 20 40 60 80 100T (K)

χ

DC magnetization

Im{X} AC measurements

4. Torque Magnetometry

The torque magnetometer is based on the principle that a magnetic sample in a

(magnetic) field experiences a force. The torque exerted on a sample is proportional to

its magnetisation. Furthermore the torque is proportional to the derivative of the

anisotropy energy towards the angle of rotation.

The PPMS torque magnetometer incorporates a torque-level chip mounted on a

rotator platform for performing angular-dependent magnetic moment measurements.

It utilizes a piezoresistive technique to measure the torsion of the lever created by the

applied magnetic field on the sample moment:

τ = m × B

Some specifications of the PPMS Torque magnetometer:

- Moment sensitivity 1× 10-7 emu

- Maximum Torque 1× 10-5 Nm

- Maximum sample size 1.5 × 1.5 × 0.5 mm3

- Maximum sample weight 10 mg

4. Specific heat option of the PPMS

The PPMS (Quantum Design) heat capacity option measures the heat capacity at

constant pressure:

pp dT

dQC

=

It controls the heat added and removed from a sample while monitoring the resulting

change in temperature.

The picture below illustrates the thermal relaxation method employed by the PPMS

specific heat option.

The simplest model to analyse the thermal relaxation data assumes that the sample

and the sample platform are in good thermal contact with each other and are at the

same temperature. Within this model the temperature of the platform as a function of

time obeys the equation:

)()( tPTTKdTdQC bwtotal +−−==

where Ctotal is the total heat capacity of the sample and platform

Kw is the thermal conductance of the supporting wires

Tb is the temperature of the thermal bath

P(t) is the power applied by the heater

The solution of this equation is given by exponential functions with a characteristic

time-constant equal to Ctotal/Kw

PPMS. General description.

The Quantum Design PPMS (Physical Property Measurement System) represents an

open architecture, variable temperature-field system, designed to perform a variety of

automated measurements.

Sample environment controls include fields up to ± 9 Tesla and temperature range of

1.9 - 400 K.

Available PPMS applications:

- DC and AC Magnetometry

- Heat Capacity

- Electro-Transport

- Thermal Transport

Appendix I. Faraday's Law

Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.

Faraday's law is a fundamental relationship which comes from Maxwell's equations. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with magnetic field.

Appendix II. Eddy(Whirlpool) currents.

Frequency 10000 Hz

Appendix III. AC susceptibility of a weakly interacting spin system

Literature:

R.W.White “Quantum theory of magnetism” (1970) McGraw-Hill Inc.

Quantum Desing PPMS Brochure (2003)/ http://www.qdusa.com/products/ppms.html