Novel Materials and Ground States

Thermal analysis – heat capacity

590B F18

Sergey L. Bud’ko

Novel Materials and Ground States

Heat capacity

Cp = (dQ/dT)p = T(∂S/∂T)p Q – heat, S - entropy

Cv = (dQ/dT)v = T(∂S/∂T)v

Cp – Cv = TVβ2/kT β – volume thermal expansion, kT – isothermal volume compressibility

For solids Cp ~ Cv

Usually we measure Cp

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

simple

sample + platform

thermal conductance of wires

T of the bath

power of the heater

P0

0

solution: exponent with t = Ctotal/Kw

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS) more realistic

fit with two exponents

addenda

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

Addenda - simple model Sample – simple model

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

Sample – 2-tau model

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

- Good for flat relatively thin samples with reasonable thermal conductivity- Liquids: need thin-walled sealed container made out of material with

high thermal conductivity and low heat capacity. – personally did not try (yet?)

- Powders: - press a pellet (if has structural integrity); - mix with some metallic powder and press a pellet; - seal powder in aluminum DSC container (can be used for air-

sensitive powder samples as well, if assembled in a glove box)

Remember addenda!

Novel Materials and Ground States

Heat capacity – HOW?

•relaxation (QD PPMS)

calibrated heater and thermometer

sophisticated temperature control and fitting software

need to measure addenda (platform + grease) every time

need to calibrate heater and thermometer in magnetic field

need to shape your sample

vertical 3He platform may oscillate in magnetic field

remember about torque

assembly is fragile

measurements take long time

not good for 1st order phase transitions

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation

τ1 – sample to

bath;

τ2 – response

time of the sample

if τ2 << 1/w << τ1

Heat capacity – HOW?

•ac modulation

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation

after Andreas Rydh

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation Commercial chips with membrane

(this is a commercial thermal conductivity gauge)

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation Commercial chips with membrane

you are invited to play with it…

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation

after Andreas Rydh

differential cell

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation

after Andreas Rydh

differential cell

need to have (at least primitive) thin film technology and lithography

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation

fast

scalable, good for small sample

accurate (relative measurements)

easy to put on rotator

can use e.g. in pressure cells

hard to get absolute values

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation – under pressure

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation – under pressure

Bridgman cell-heater and thermometer are at ambient pressure

for DAC – laser heating with thermocouple as sensor

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation – under pressure

Novel Materials and Ground States

Heat capacity – HOW?

•ac modulation – under pressure

need to choose frequency

Novel Materials and Ground Statesafter Andreas Rydh

Novel Materials and Ground States

Heat capacity – WHY?

* Useful knowledge for useful (structural) materials – how easy to cool down

Novel Materials and Ground States

Heat capacity – WHY?

* To learn something about electrons and phonons

T -> 0: Cp = γT + βT3 (no magnetism)

γ ~ N(0)(1 + λ) - electronic density of states

λ - measure of e-e (and e-ph) interactions, m*/m0 = (1 + λ)

γ > 400 mJ/mol K2 – “heavy fermions” – an enormous, separate research field that entertains number of us

ΘD = (1944/β)1/3 - Debye temperature (use right units)

tells you something about stiffness of the lattice, helps to understand BCS superconductors [Tc ~ ΘD exp(-1/N(0)V)]

Novel Materials and Ground States

Heat capacity – WHY?

Novel Materials and Ground States

Lattice heat capacity – Debye and Einstein

Einstein – single frequency

Debye – elastic continuum

(T << ΘD)

(T >> ΘD)

Novel Materials and Ground States

Lattice heat capacity – Debye and Einstein

Debye

Einstein

One can also use realistic phonon DOS and calculate Cp

Einstein Debye

Blackman “realistic”

Novel Materials and Ground States

Lattice heat capacity –Einstein mode

Peak associated with the Einstein mode @ ~ QE/20

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about magnons

Ferro (ferri) magnets

isotropic

anisotropic

Antiferromagnets

isotropic

anisotropic

And need to remember contributions from electrons and phonons: Cp = γT + βT3

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors

Cs ~ exp(-1.76Tc/T) BCS

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors

RMP 62, 1027 (1990)

Strong coupling

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors

BCS value (3.53)

phenomenological (1970s)α-model:2Δ0/kBTc – fitting parameter;temperature dependence of SC gap – BCS;can calculate thermodynamic properties.

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductorsMgB2

Two-gap superconductor

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductorsMgB2

Two-gap superconductor

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors

four gaps

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors

BCS superconductors with paramagnetic impurities

BCS

PM impurities La-Gd

jump in specific heat

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductors Kondo effect in SC

jump in specific heat

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about superconductorsIron-arsenides

ΔCp ~ Tc3

Smart person might be able to make sense out of this

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about phase transitions below superconducting Tc

Tc

TWFM?

TN

Novel Materials and Ground States

Heat capacity

Bulk, thermodynamic property

In case of “not so single phase” samples results:

-do not suffer from “M&M effect” in case of superconductors

-do not suffer from impurity SC or FM signal in magnetization (although for FM need to remember about entropy)

normal

Novel Materials and Ground States

Resistance

SC

normal bulk, superconducting shell

VI Will get R=0 below Tc of the shell

normal

Novel Materials and Ground States

Low field ZFC susceptibility

SC

normal bulk, superconducting shell

Will get complete flux expulsion below Tc of the shell

normal

Novel Materials and Ground States

Heat capacity

SC

normal bulk, superconducting shell

Will get complete jump of Cp at Tc of the shell proportional to the fraction of the SC phase, e.g. very small

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about crystal electric field (Schottky anomaly)

maximum:

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about spin glasses

Scaling hypothesis

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about spin glasses

1/T

Tmax ~ 1.5 Tf

Novel Materials and Ground States

Heat capacity – WHY?

•To learn something about magnetic transitions

HoNi2B2C

Also can play entropy game and evaluate degeneracy of the ground state

ΔSm = R ln N

Novel Materials and Ground States

Heat capacity – beware of fool’s gold

low temperature C = AT in insulators

Novel Materials and Ground States

Heat capacity – beware of fool’s gold

low temperature C = AT in insulatorsglasses - distribution of two level systems

Novel Materials and Ground States

Heat capacity – beware of fool’s gold

“fake” heavy fermions (apparently large Sommerfeld coefficient):

- (really) low temperature magnetic ordering

- low temperature spin glass behavior

One measurement technique, whatever sophisticated, is not enough

Novel Materials and Ground States

Reading:

Quantum Design Manual and refs therein

E.S.R. Gopal, Specific heats at low temperatures.

A.Tari, Specific heat of matter at low temperatures.

T.H.K. Barron and G.K. White, Heat capacity and thermal expansion at low temperatures.

Review of Scientific Instruments – search on “heat capacity…”