Thermal analysis – heat capacity
590B S14
Sergey L. Bud’ko
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Detour – phase transitions of 2 ½ order
Heat capacity
2 ½ order phase transition
First-order phase transitions–exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable.
Paul Ehrenfest:
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thermodynamic variable.
Second-order phase transitions–continuous in the first derivative–exhibit discontinuity in a second derivative of the free energy.
Fermi surface - reminder
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2 ½ order phase transition
aka: electronic topological transition (ETT), Lifshitz transition
also:
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2 ½ order phase transition
electronic DOS
parameter:
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parameter:
thermodynamic potential
T=0, no scattering
2 ½ order phase transition
Li-Mg alloy
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low temperature, no scattering
ALSO other control parameters
2 ½ order phase transition
QCP in heavy fermions
New box – the same good old taste
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LSCO high-Tc SC
QCP in heavy fermions
2 ½ order phase transition
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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
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For solids Cp ~ Cv
Usually we measure Cp
Heat capacity – WHY?
* Useful knowledge for useful (structural) materials – how easy to cool down
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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
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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)]
Heat capacity – WHY?
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Lattice heat capacity – Debye and Einstein
Einstein – single frequency
Debye – elastic continuum
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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
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Blackman “realistic”
Heat capacity – WHY?
•To learn something about magnons
Ferro (ferri) magnets
isotropic
anisotropic
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Antiferromagnets
isotropic
anisotropic
And need to remember contributions from electrons and phonons: Cp = γT + βT
3
Heat capacity – WHY?
•To learn something about superconductors
Cs ~ exp(-1.76Tc/T) BCS
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Cs ~ exp(-1.76Tc/T) BCS
Heat capacity – WHY?
•To learn something about superconductors
BCS value (3.53)
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(3.53)
phenomenological (1970s)α-model:2∆0/kBTc – fitting parameter;temperature dependence of SC gap – BCS;can calculate thermodynamic properties.
Heat capacity – WHY?
•To learn something about superconductorsMgB2
Two-gap superconductor
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Heat capacity – WHY?
•To learn something about superconductorsMgB2
Two-gap superconductor
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Heat capacity – WHY?
•To learn something about superconductors
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four gaps
Heat capacity – WHY?
•To learn something about superconductors
BCS superconductors with paramagnetic impurities
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BCS
PM impurities La-Gd
jump in specific heat
Heat capacity – WHY?
•To learn something about superconductors Kondo effect in SC
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jump in specific heat
Heat capacity – WHY?
•To learn something about superconductorsIron-arsenides
∆Cp ~ Tc3
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∆Cp ~ Tc
Smart person might be able to make sense out of this
Heat capacity – WHY?
•To learn something about crystal electric field (Schottky anomaly)
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maximum:
Heat capacity – WHY?
•To learn something about spin glasses
Scaling hypothesis
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Heat capacity – WHY?
•To learn something about spin glasses
1/T
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Tmax ~ 1.5 Tf
Heat capacity – WHY?
•To learn something about magnetic transitions
HoNi2B2C
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Also can play entropy game and evaluate degeneracy of the ground state
∆Sm = R ln N
Heat capacity – beware of fool’s gold
low temperature C = AT in insulators
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Heat capacity – beware of fool’s gold
low temperature C = AT in insulatorsglasses - distribution of two level systems
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Heat capacity – beware of fool’s gold
“fake” heavy fermions (apparently large Sommerfeld coefficient):
- (really) low temperature magnetic ordering
- low temperature spin glass behavior
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One measurement technique, whatever sophisticated, is not enough
Heat capacity – HOW?
•relaxation (QD PPMS)
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Heat capacity – HOW?
•relaxation (QD PPMS)
simple
sample + platform
thermal conductance of wires
T of the bath
power of the heater
P0
0
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solution: exponent with ττττ = Ctotal/Kw
Heat capacity – HOW?
•relaxation (QD PPMS) more realistic
fit with two exponents
addenda
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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
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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
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τ2 – response
time of the sample
if τ2222<< 1/w << << 1/w << << 1/w << << 1/w << τ
1111
Heat capacity – HOW?
•ac modulation
Heat capacity – HOW?
•ac modulation
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after Andreas Rydh
Heat capacity – HOW?
•ac modulation Commercial chips with membrane
(this is a commercial
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(this is a commercial thermal conductivity gauge)
Heat capacity – HOW?
•ac modulation Commercial chips with membrane
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you are invited to play with it…
Heat capacity – HOW?
•ac modulationdifferential cell
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after Andreas Rydh
Heat capacity – HOW?
•ac modulation differential cell
need to have (at least primitive) thin film
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after Andreas Rydh
thin film technology and lithography
Heat capacity – HOW?
•ac modulation
fast
scalable, good for small sample
accurate (relative measurements)
easy to put on rotator
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easy to put on rotator
can use e.g. in pressure cells
hard to get absolute values
Heat capacity – HOW?
•ac modulation – under pressure
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Heat capacity – HOW?
•ac modulation – under pressure
Bridgman cell-heater and thermometer are at ambient pressure
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pressure
for DAC – laser heating with thermocouple as sensor
Heat capacity – HOW?
•ac modulation – under pressure
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Heat capacity – HOW?
•ac modulation – under pressure
need to choose frequency
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after 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…”
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