sergey l. bud’ko...second-order phase transitions –continuous in the first derivative –exhibit...

Novel Materials and Ground States

Thermal analysis –

heat capacity

590B F09

Sergey L. Bud’ko

Detour –

phase transitions of 2 ½

order

Heat capacity

(Сергей Леокадьевич Будько)


2 ½

order phase transition

First-order phase transitions–exhibit a discontinuity in the first derivative

of the free energy with a thermodynamic variable.

Second-order phase transitions–continuous in the first derivative–exhibit discontinuity in a second derivative

of the free energy.

Paul Ehrenfest:


2 ½

order phase transitionaka: electronic topological transition (ETT), Lifshitz

transition

also:


2 ½


electronic DOS

parameter:

thermodynamic potential

T=0, no scattering


2 ½


low temperature, no scattering

ALSO other control parameters

Li-Mg alloy


2 ½


LSCO high-Tc

SC

QCP in heavy fermions

New box –

the same good old taste


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


Heat capacity –

WHY?

* Useful knowledge for useful (structural) materials –

how easy to cool down

* 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)]


Lattice heat capacity –

Debye and Einstein

Einstein –

single frequency

Debye –

elastic continuum

(T << ΘD

)

(T >> ΘD

)


Lattice heat capacity –

Debye and Einstein

Debye

Einstein

One can also use realistic phonon DOS and calculate Cp

Einstein Debye

Blackman “realistic”


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


Heat capacity –

WHY?

•To learn something about superconductors

Cs

~ exp(-1.76Tc

/T) BCS


Heat capacity –

WHY?


BCS value (3.53)

phenomenological (1970s)α-model:2Δ0

/kB

Tc

–

fitting parameter;temperature dependence of SC gap –

BCS;can calculate thermodynamic properties.


Heat capacity –

WHY?

•To learn something about superconductors MgB2

Two-gap superconductor


Heat capacity –

WHY?


BCS superconductors with paramagnetic impurities

BCS

PM impurities La-Gd

jump in specific heat


Heat capacity –

WHY?

•To learn something about superconductorsIron-arsenides

ΔCp

~ Tc3

Smart person might be able to make sense out of this


Heat capacity –

WHY?•To learn something about crystal electric field (Schottky

anomaly)

maximum:


Heat capacity –

WHY?•To learn something about spin glasses

Scaling hypothesis


Heat capacity –

WHY?•To learn something about spin glasses

1/T

Tmax

~ 1.5 Tf


Heat capacity –

WHY?•To learn something about magnetic transitions

HoNi2

B2

C

Also can play entropy game and evaluate degeneracy of the ground

state

ΔSm

= R ln N


Heat capacity –

HOW?•relaxation (QD PPMS)

simple

more realistic

fit with two exponents


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


Heat capacity –

HOW?•ac modulation

τ1

–

sample to bath;

τ2

–

response time of the sample

if

τ2

<< 1/w << τ1


Heat capacity –


after Andreas Rydh


Heat capacity –

HOW?•ac modulation Commercial chips

with membrane

(this is a commercial thermal conductivity gauge)


Heat capacity –

HOW?•ac modulation Commercial chips

with membrane

you are invited to play with it…


Heat capacity –


after Andreas Rydh

differential cell


Heat capacity –


after Andreas Rydh

differential cell

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


Heat capacity –


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


Heat capacity –

HOW?•ac modulation –

under pressure


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 Statesafter Andreas Rydh


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…”

sergey l. bud’ko...second-order phase transitions –continuous in the first derivative –exhibit...

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