Specific Heat in Thermodynamics · Vd dS d r r van Hove singularity extremal points of phonon dispersion curves in BZ Phonon dispersion relation by ... s t i c ( L A) M o d e T r

Introduction to thermal and transport techniques

Specific Heat in Thermodynamics

⎟⎠⎞

⎜⎝⎛

ΔΔ

=→Δ T

QCT 0lim

dTdq

mCc ==

Heat capacity: Specific Heat:PP dT

dqc =V

V dTdqc =

Per mass or usually “per mole”: CCPP and CCVV

VV dT

dUCPdVdUdWdUdQ ⎟⎠⎞

⎜⎝⎛=⇒+=+=

VV

PP T

STCTSTCTdSdQ ⎟

⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=⇒= ;

reversible system in equilibrium with its environment

VVV T

FTTSTCPdVSdTTSUddF ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−=⎟⎠⎞

⎜⎝⎛

∂∂

=⇒−−=−= 2

2

)(

ZTkF B ln−= ∑−

=i

TkEi

BeZFree energy:

neglect the volume expansion whensolid is heated, dW ~ 0

• Powerful method in materials characterization

• Modern time: high level of automatization (PPMS)

• Notice that what we calculate is CV and what we measure is CP.

∫=1

01 d)(

T

TC TTS

sum over allpossible states (i)with energies Ei

PPP T

HdTdQCdQTdSdH ⎟

⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛=⇒==

For isobaric conditions

Note lower boundaryOn integration


Simplest Example: 1D Monoatomic LatticeChain of N identical atoms with M:

( ) ( ) NnuuKuuKuM nnnn ∈−−−= −+ ,11

..

)()( tnaqijn

jjeAtu ω−=Search for harmonic solution:

Periodic boundary condition un(t)=un+N(t) → exp(iqjNa)=1

NjNa

jq j ∈= ,2π

)2(2 −+= − aiqaiqjjj

jj eeKAAMω

2sin2)cos1(22 aq

MKaq

MK j

jjj =⇒−= ωω

1

2

3

( )( ) 0

cusp,max,min

ω∇ =q q

( ) const.ω∇ =q q

Extremal values: )2

(; Nja

q j =±=π

First Brilluen zone: ⎟⎠⎞

⎜⎝⎛−=

aaq j

ππ ......

Number of possible q values = N (number of unit cells in the system)

4For low q → ω(q) ~ |q|cs(cs ~ speed of sound)

5

velocity of excitations in the chain (group velocity): cg = ∂ω/∂q = cscos(qa/2)sgn(q)cg=0 at the zone boundary |q| = π/a = 2π/λ (Bragg law)→ at the zone boundary lattice effects are strongest, atoms oscillate out of phase for |q|=π/a

6

What is phonon density of states in 1D chain?

Nntunatx nn ∈+= ),()(


Simple example: Diatomic chain in 1D

nn-1 n+1

u1(na): displacement of atom n,1

n

u2(na): displacement of atom n,2Spring K Spring G

Coupled equations of motion:

[ ] [ ]))1(()( 212111 anu(na)uGnau(na)uK(na)..uM −−−−−=

[ ] [ ]))1(()( 121222 anu(na)uGnau(na)uK(na)..

uM −−−−−=

( ) qaKGGKMM

GKkeffeff

cos21 222 ++±+

=ω

See textbook, also Aschroft/Mermin “Solid State Physics”

Optical branch. Atoms vibrateagainst each other with theircenter of mass fixed

Acoustic branch. Atoms and theircenter of mass move together as inacoustical vibration

Long wavelength modes can interact withelectromagnetic radiation (oppositely chargedions can be excited by E of the light wave

ω~ck (sound waves)

There are N values of q, N={1…N}:

Nn

aq π2

=

For each q, there are 2 solutions,total of 2N normal modes - phonons


General Case and Phonon Density of StatesIf there is s ions per unit cell and N cells, there will be 3Ns degrees of freedom and 3s normal modes for each phonon. The lowest 3 branches are acoustic. Remaining 3(s-1) branches are optical.

Each mode has its own polarization vector:

paralel to - longitudinal mode

perpendicular to - transverse mode

Phonon spectrum of real materials (diamond, s=2, 6 normal modes)include also interaction beyond nearest neighbors, electron – phonon coupling, anharmonicity…

qr

qr

qrmixed excitations areof course possible

3)2())(()(

πωωδωρ

μμ

kVdqr

∑∫ −=

Define phonon DOS: number ofstates per energy interval:

ωωρ d)( total number of modes ininfinitesimal range ω+dωper total V

sqVdN 31;)2( 3 == ∑∫

μπ

sIf then expand:ωωμ =)( 0qr

qqqqq rrrrr

∂∂

−+= μμμ

ωωω )()()( 00

group velocity )( 0qv rμ

)()(

qddSdqdqqd

qq r

r

μ

ωμ ω

ωωω∇

=⇒∇= ⊥ ∑ ∫= ∇

=μ ωω μ

ω

μωπ

ωωωρ)(

3 )()2()(

q q qdSVdd

rr

van Hove singularity

extremal points of phonondispersion curves in BZ

Phonon dispersion relation by INS:λ, E match


0 0.2 0.4 0.6 0.8 1.00.20.40

2

4

6

8

(111) Direction (100) DirectionΓ XL Ka/π

LA

TATA

LA

LO

TO

LO

TO

Freq

uenc

y (1

0 H

z)12

Longitudinal and transverse modesFr

eque

ncy,

ω

Wave vector, K0 π/a

Longitudinal Acoustic (LA) M

ode

Transverse

Acoustic (T

A) Mode

Group Velocity:dKdvg

ω=

GaAs


Linear oscillator in quantum representation

Consider harmonic oscillator: V=(1/2)kx2 = (1/2)mω2x2 → [ ]22222

)(21

21

2xmp

mxm

mpH ωω +=+=

Introduce raising and lowering operators: † ( ) ( )xmipm

axmipm

a ωω

ωω

+=+−=hh 21,

21

So that [a,a†]=1 (easy to show since [xi,pj]=iħδij) → aa†=1+a†a

11122111 +=→+=+====⇒= +

∗++ ncnaaccccaaca nnnnnnnnnnnnnn ψψψψψψψψ

† † †

11 ++= nn na ψψ†

ndddddnaada nnnnnnnnnnnn =⇒====⇒= −∗

−−2

111 ψψψψψψ†

1−= nn na ψψ

Now we write Hamiltonian in terms of raising and lowering operators:

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=−++=

++−=

mmxm

mppxxpimxmp

mmxmipxmipaa

2221)()(

21

2))(( 222

22 h

hhh

ωωω

ωωωω

ωω†

H

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ +=

21

21 aaaaH ωω hh

† †

energy of linear harmonic oscillatorof frequency ω is quantized in ħω


Einstein Model of Lattice Specific HeatCollection of uncoupled quantum oscillators, each vibrating with the same frequency ωE

Number of oscillators is equal to number of degrees of freedom in the system

Average energy:

ωωωβ

ωβ

ωω

ωω

ωβωβωβ

ωβ

ωβ

ωβ

ωβ

hhhhh

hh

h

hh

h

h

h

h

h

⎟⎠⎞

⎜⎝⎛ +=

−+=

−∂∂

−=⎟⎠

⎞⎜⎝

⎛∂∂

−=+=⎟⎠⎞

⎜⎝⎛ +

= −−

∞−

∞−

∞−

∞−

∞−

∑∑

∑

∑

∑21

1211ln

2ln

2221

)( nee

ee

en

e

enTE

n

n

n

n

n

n

n

n

n

n

<n> = average quantum number for oscillator = [exp(βħω)-1]-1 (Bose-Einstein distribution)

( )2/

/2

1)/(3)

21(33

−=⎟

⎠⎞

⎜⎝⎛ +

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=Tk

TkBEBA

EAV

AV

VBE

BE

eeTkkNnN

TEN

TTUC

ω

ωωωh

hhh

Specific heat in Einstein approximation:ΘE – Einstein temperature

High temperature limit (T>> ΘE) – exponents are replaced with expansions and we get Dulong – Petit result:

RT

RC EV 3.....

12113

2

≈⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ Θ

−=

Low temperature limit (T<<ΘE) – predicts that specific heat should decrease to zero exponentially:

( ) )/(2/3 TEV

EeTRC Θ−Θ=

Real materials: atoms are coupled, vibrate collectively at many different ω

∑ −−=n

n xx 1)1(


Debye model of Lattice Specific Heat 1For thermodynamic properties optical modes are irrelevant (low T).

We keep acoustic modes and replace them with purely linear mode with the same initial dispersion

Enter discrete nature of the solid: total number of vibrational modes is normalized to 3NA

maximum (cutoff) frequency ωD∃

The number 3NA is large (10-24), therefore weconsider vibrational levels as continuous andwrite number of modes in ω, ω+dω:

( ) ANdD

30

=∫ ωωρω

Relation between ω and wave vectorq is defined by Debuye approximationfor sound waves:

λπω cqc ==

2

Periodic boundary conditions (L is the dimension of representative cube of continuum):

( )[ ] [ ] ,...4,2,0,,)()()(

LLqqqee zyx

qLzqLyqLxizqyqxqi zyxzyx ππ±±=⇒= +++++++

Volume in reciprocal space for each wavevector is (2πL)3 → number of allowed values of per unit volume of space is (L/2π)3 = V/8π3qr qr

Number of allowed values is large, so q ~ continuous variable → number of modes with q or less is V/8π3 · volume of sphere with R = q:qr

( ) 23

33

434

8ωπ

ωωρπ

π cV

ddNqVN ==⇒=

( ) 23

12 ωπωρc

V=Real solid, elastic waves for each q have

longitudinal component and 2 transverse:

Cutoff frequency for vibrational spectrum 3/1

43

⎟⎠⎞

⎜⎝⎛=

VNc A

D πω

Total number of oscillatorsin interval ω, ω+dω:

ωωω

ωωπ dNdc

V

D

A 23

23

912=

~sound velocity


Debye model of Lattice Specific Heat 2Specific heat in Debuye model (x=ħω/kBT, ΘD=ħωD/kB):

( ) ( )∫∫∫Θ

−⎟⎟⎠

⎞⎜⎜⎝

⎛Θ

=−⎟⎟

⎠

⎞⎜⎜⎝

⎛=+

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=T

x

x

DD

ATk

Tk

BB

VV

DD

B

BD

edxexTRdN

ee

Tkkdn

TTUC

/

02

432

032/

/

0 199

)1()

21( ωω

ωωωωωρ

ω

ω

ωω

h

hhh

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ Θ

−= ...20113

2

TRC D

V

High temperatures: T>>ΘD we replace exponential withseries function and obtain Dulong Petit result

Low temperatures T<<ΘD , upper limit of integral is ∞ and we get

33D

33

4 θ R2345

12 TTTRCD

V−==⎟⎟

⎠

⎞⎜⎜⎝

⎛Θ

= βπ

4π4/15

Usually good for T < (ΘD/10 - ΘD/50). For T > Debuye region:

.......)( 73

52

31

63

42

21 +++=⇒+++= TTTCV βββωαωαωαωρ

Physical significance: Debuye temperature is a measure of the stiffnes of the crystal: above ΘD all modes are gettingexcited, and below ΘD modes begin to be “frozen out”, marking rapid reduction in CV with decreasing temperature.

Shortcoming of the model: θD = θD(T) but at high T all vibrational modes are excited so θD=const. (classical result)

Note that it holds for 1 atom in unit cell = 3 phonon branches

3D βR234)atoms (θ NN =


Debye model can be applied succesfully to many materials with more than one atom in the unit cell.

T 2(K2)

0 100 200 300 400 500

Cp/

T (J

/mol

.K2 )

0.0

0.1

0.2

0.3

0.4

UCoGa5

Application of Debye Model

Sometimes high temperaturerange can also have CP ~ T3

which is unphysical.

At T > θD the model is not valid. Optical phonon mode contributionis not negligible. Thermal expansion may not be negligible

Debye temperatures of solids:


Specific Heat Dominated by ~ T3 Phonon Contribution

10 410 310 210 110 1

10 2

10 3

10 4

10 5

10 6

10 7

Temperature, T (K)

Spec

ific

Hea

t, C

(J/

m -

K)3

C ∝ T 3

C = 3ηkB = 4.7 ×106 Jm3 −K

θD =1860 K

Diamond

ClassicalRegime

Each atom has a thermal energy of 3kBT


Electronic specific heat

( ) ∫∫ ∫∞∞ ∞

+=

∂∂

−=∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

=0

2

22

0 0 )1()()()(

xx

x

BFFV

V edxexTkd

Tfndnf

TTUC εεεεεεεε

Gas obeying Fermi-Dirac statistics. Fermi Dirac distribution function: 11)( /)( +

= − TkBFef εεε

At T=0 all energy levels with ε<εF (~ TF=εF/kB) are occupied, rest are vacant. At T>0 electrons with ε~kBT of εF have sufficient thermal energy to become excited to vacant levels.

At T<TF, fraction of electrons with ε~kBT is of the order of T/TF, contributing to U ~ NA(T/TF)kBT and to C ~ 2RT/TF

Since TF ~ (104 – 105)K → Ce ~ 10-2R which is ~ 1% of lattice contribution (think Dulong Petit).

Internal energy of a system of N electrons is: where ∫∞

=0

)()( εεεε dnfU ∫∞

=0

)()( εεε dnfN

Total density of states for both spin directions is n(ε), so that n(ε)dε is the number of energy levels in interval ε, ε+dε

At very low temperatures T<<TF, we can write ∫∞

∂∂

=∂∂

=0

)(0 εεεε dnTf

TN

FF

Nonzero only in the vicinity of εF.

We introduce x = (ε-εF)/kBT and x0 = -TF/T

For x0 = -∞~ π2/3

TTknCBFe γεπ

== 22

)(3 Linear T dependence of electronic specific heat


Free electron model

2/3

22

23

⎟⎠⎞

⎜⎝⎛=h

επ

mVN

Simplest but rather useful approximation. Electrons move as free particles with dispersion:

[ ]22222

22 zyx kkkm

km

++==hhε

The lowest energy state is obtained by placing pairs of electrons in states with smallest ,occupying all states within sphere of radius kF:

kr

Fermi energy and radius of Fermi sphere depend only on electron density N/V

Number of states N with ε < εF is: ,density of states:

3/2223/12

3

3 32

3283

4⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛=⇒=•⎟⎟

⎠

⎞⎜⎜⎝

⎛V

NmV

NkNVkFF

F πεππ

π h

23/2

3/1200

22

3)()(

3 Bm

AF kVzNmTTknCeB

⎟⎠⎞

⎜⎝⎛=⇒==

πγγεπh

Ce/T in free electron model

Periodic boundary conditions: ki = 2πn/L (i=x,y,z), (n = 0, ±1, ±2….) → there is one allowedvalue of k for each cell of (2π/L)3 = 8π3/V → number of allowed wave vectors per unit V of

space is V/8π3

kr

2/12/3

22

22

)( επε

ε ⎟⎠⎞

⎜⎝⎛==h

mVddNn

Density of states at εF: 3/12

2

3)( ⎟

⎟⎠

⎞⎜⎜⎝

⎛=

ππε m

VzNmn AF

h

Where z=N/NA is the conduction electron/atom ratio and Vm is the molar volume


Thermal Relaxation Calorimetry 1Sample of unknown heat capacity Cx is attached to a sample platform with a thermal grease (e.g., apiezone N grease).

The platform consists of a thin sapphire or silicon disc, which has high thermal conductivity. A thin-film heater is evaporated onto the bottom of the platform, and the platform temperature Tp is determined from a bare temperature sensor attached to it.

Wires thermally link the platform to a copper heat sink held at a constant temperature T0. They create a thermal link between the bath and platform with a thermal conductance K1. They also provideelectrical connections to the temperature sensor and heater.

Power P is applied to the platform via the thin-film heater, and a system of differential equations is solvedCryogenics 43, 369 (2003)

Heat flow diagram for a standard relaxation calorimeter

Based on a measurement of thermal response of a sample calorimeter assembly to a change in heating conditins.


Thermal Relaxation Calorimetry: Single τ

Cryogenics 43, 369 (2003)

Power P is applied to the heater, the platform sample assembly warms to a temperature T0 + ΔT =T0 +P/K1. If the thermal connection sample - platform is very strong (K2 >> K1, Tx ≈ Tp), we get:

)(0

)()(

2

012

pxx

x

pxpP

a

TTKdt

dTC

TTKTTKdt

dTCP

−+=

−+−+=

Thermal conductance ofsample – platform thermal link

Combined addenda HC(platform, T sensor, heater,and grease)

( ) )( 01 TTKdt

dTCCP pP

xa −++=

Power P is discontinued and the platform/sample assembly will cool to the bath temperature T0:

1/

0 /)(;)( KCCTeTtT xat

p +=Δ+= − ττ

For small ΔT (ΔT/T << 1), we can ignore T dependence of Ca, Cx and K1 and get Cx via τ

In this method (using single τ) a steady state at constant P and Tp > T0 followed by relaxationto T0 can be used to determine both K1 and Cx. K1 is determined by measuring the temperature change ΔT that results when power P is applied. , Addenda heat capacity Ca can be determined from a decay measurement with no sample attached to the platform.


Thermal Relaxation Calorimetry: Double τ

This method is applied when K2 >> K1 due to poor sample - platform thermal link. Therefore Tx ≠ Tp. Thermal decay of Tp is described using two exponentials:

21 //0)( ττ tt

p BeAeTtT −− ++=

Time constant τ2 is usually much shorter than the other. Thus, there are two relaxation times:1. Shorter relaxation process (τ2) between the sample and platform 2. Longer gradual process (τ1) due to thermal relaxation between the platform/sample and the

heat-sink temperature bath.

By measuring decay curves it is possible to determine τ1, τ2, K2, and Cx given known values for Ca and K1. Series of decays cycles (10–100) can be averaged at each temperature to obtain data scatter of less than 1%.

Drawbacks of the method:1. Measurements can be time consuming and are becoming impractical for τ1 ~ 100s2. It is assumed that sample Cx does not vary much during T0 + ΔT, which may not be satisfied

near the phase transition. Better results are obtained using single τ method near the phase transition since near the phase transition Cx >> Ca and we get:

dtdTTTKTCx /

)( 01

−−=


Quantum Design PPMS


PPMS Heat Capacity Option Hardware


PPMS Heat Capacity Puck

sample greaseplatform (3x3 mm)

heater thermometerwires

vacuum T baseT ba

se

After applying heat pulse P(t), thermal response of a system is:

))(()()(total bw TtTKtP

dttdTC −−=

Ctotal = (sample+platform+grease). Heater power P(t) is varied, and T vs time t is measured, minimizing the difference between measured temperatures and the model. Here, Tb is the base temperature, Kw is the heat coupling through the wires.

Base temperature

Heat coupling through wires, thermal conductance

We choose heatpulse as: ⎩

⎨⎧

>≤≤

=)(0

)0()(

0

00

ttttP

tP


Time (sec)0 2 4 6 8 10 12 14 16 18 20

Tem

pera

ture

(K)

33.57

33.71

33.86

34.00

34.14

34.29

34.43

34.57 Heat pulse applied

Sample, platform T

Thermal coupling sample – platform through grease

( ) ( )

( ))()()(

)()()()()(

sample

platform

tTtTKdt

tdTC

tTtTKTtTKtPdt

tdTC

psgs

psgbpwp

−−=

−+−−=

Sample is not 100% thermally attached to the platform, so we have to solve two coupled differential equations. The T(t) then usually looks like the blue curve.

Quantum Design Two τ Method


Let’s consider first ideal case of 100% coupling: ))(()()(total bw TtTKtP

dttdTC −−=

( )( )⎩

⎨⎧

>+−=≤≤+−=

= −−−

−

)(/1)()0(/1)(

)(0total

/)(/0off

0total/

0on00 ttTCeePtT

ttTCePtTtT

bttt

bt

ττ

τ

ττ

⎩⎨⎧

>≤≤

=)(0

)0()(

0

00

ttttP

tP),()(

)0(

0off0on

on

tTtTTT b

==

For applied heat pulse: and boundary conditions:

We obtain T(t):wKC /total=τ

Where:

P0, t0 are known.

All uknowns are obtained by minimizing the differencebetween measured temperatures Ti and those obtainedfrom the model at the same time ti.

( )∑ −i

ii TtT 2)(

measuredmodel

In the real case of non-ideal coupling, more unknowns are fitted at the same time. The process is more extensive numerically. We fit only to the expression for Tp since the thermometer in the system is on the platform, no thermometer is attached to the sample and we assume T(sample) = T (platform) = T(wires).

Data Fitting Procedure


Schottky Anomaly 1Consider system with discrete energy levels. When temperature is comparable tolevel separation Δ, specific heat has a broad peak (Schottky anomaly).

Consider a general case of a system with multiple non-degenerate levels ε1…..εn.. Average thermal energy is:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−Δ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−Δ

==⇒Δ

=

∑

∑

∑

∑

∑

∑

=

=

=

=

=

Δ−

=

Δ−

2

0

0

0

0

2

2

0

)/(

0

)/(

exp

exp

exp

exp

n

i B

i

n

i B

ii

n

i B

i

n

i B

ii

n

i

Tk

n

i

Tki

A

Tk

Tk

Tk

TkTR

dTdUCschottky

e

eNU

Bi

Bi

Now reduce this to a two level system with energies ε0=0 and ε1=Δwith degeneracies g0 and g1:

[ ]2)/(10

)/(2

1

0)/(

10

)/(1

)/(1 Tk

Tk

BschottkyTk

TkA

B

B

B

B

egge

TkggR

dTdUC

eggegNU

Δ

Δ

Δ−

Δ−

+⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ==⇒

+Δ

=

TΔ=Δ/kBLevel separationin K

Low temperature limit (T<<TΔ):

)/(2

1

0 TTschottky e

TT

ggRC Δ−Δ ⎟

⎠⎞

⎜⎝⎛=

High temperature limit (T>>TΔ):

2

210

10

)(⎟⎠⎞

⎜⎝⎛

+= Δ

TT

ggggRCschottky

Entropy:

( ))/(1ln 01

0

ggRS

dTT

CS

schottky

schottkyschottky

+=

= ∫∞


Schottky Anomaly 2Peak magnitude depends on ratio of g’s. It increases with degeneracy difference. Usually large when compared with other contributions, and dominant if occurs at low temperatures:

Cschottky ~ R, other contributions ~ 10-2R

In everyday life presents a problem since separation of other contributions is non - trivial

Nuclear Schottky anomaly – when interaction removes degeneracies of nuclear levels – could beproduced by external magnetic field, by hyperfine magnetic field from conduction electrons or byCEF gradients.

ΔE In the simplest case of two-level system, Cschottky shows as an anomaly withmaximum at ~ 0.4·ΔE. For a more complex level system, the Schottkyspecific heat is smooth function withoutany clear anomalies, but it can be fit withexponential and ~ T-2 terms


Given the Hund’s rule, groundstate multiplet J we would expect Rln(2J + 1) entropy associated with the magnetic state. This is what we find for Gd: (S = 7/2, L = 0 and J = 7/2) therefore S ~ R ln(8). For other rare earths the spin-orbit coupling gives rise to crystalline electric field (CEF) splitting.

Δ = 18 K

e.g.: Ce (J =5/2) in cubic point symmetry Δ

This can be seen in the Cp as a Schottky anomaly. This is clearly shown at the right in PrAgSb2. CP is modeled as a two level system.

Schottky Anomaly Example


Heavy fermion superconductor PrOs4Sb12

Phys. Rev. B 73, 104503 (2006)


Specific Heat in Disordered Solids

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛= ΔΔ

TTh

TTkC Bschottky

22

sec

Disordered solids ~ collection of two level systems characterized by ±ε:

Think of Schottky for two single level system with energy configurations ±ε:

∫∫∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

0

20

222

sec2sec)(2 hxdxxTPkdTk

hTk

PkC BBB

BV εεεε

Assume P(ε)=const=P0 and x=ε/kBT

Low T upper limit ∞

π2/12

Specific heat of disordered two level system has linear T contribution to heat capacity:

TkPC BV2

0

2

6π

=


At low temperatures both electronic and vibrational excitations contribute:

Cp = aT + bT3

Clean separation of a and b is usually done by plotting C/T versus T2.

As we have seen a is proportional to N(EF) and b is proportional to the Debye temperature.

Paramagnetic Metals at Low Temperatures


T2 (K2)0 20 40 60 80 100

Cp/T

(mJ/

mol

.K2 )

0

10

20

30

40

50

60

LuNiAl

Au

Cu

f.e.

exp

e γγ

m*m

=

β

γ

2p βTγT/C +=

γ (mJmol-1K-2):

exp. 1.6 1.4 2.1 4.6 15.2 0.7 0.6 0.6 0.7 1.3 0.6

Free e 0.8 1.1 1.7 0.6 0.6 0.5 0.8 0.6 0.6 0.9 1.0Li Na K Fe Mn Cu Zn Ag Au Al Ga

We can introduce an effective mass m* to account for this difference, since n(EF) is proportional to the carrier mass me.

Electronic specific heat coefficient γ can be estimated from the low-temperature specific-heat data, where the lattice part reduces to ~T3 dependence.

In real metals, the γ value is often differentfrom that obtained by free electron model.

Effective mass


γ = kB2T ·(mkF/3ħ2)

The electronic specific heat can be used as a measure of the electron effective mass, based on the free electron result of:

Heavy Fermions are compounds with exceptionally high values of γ for

T < TK, the Kondo temperature.

They are defined as compounds with γ > 100 mJ/mol-K2, about 100 times the value of g for Cu. Large γ means large electron mass, ergo, heavy fermion.

Heavy Fermions


Specific heat can also be used to locate and characterize phase transitions. We can suppress superconducting Tc with an applied magnetic field, so the Cp feature can be more clearly seen via comparison to the non-superconducting (in high applied magnetic field) compound.

Phase Transitions

Local moment ordering can be seen even more clearly (larger entropy). Shown here are a series of transitions in antiferromagnetic DyAgGe.


First and Second Order Phase Transitions

T (K)20.0 25.0 30.0 35.0 40.0

Cp /

T (J

/mol

.K2 )

0

2

4

6

8

UO2

Not ideal δ function but sharp anomaly.Clear identification of TC.

T (K)0 4 8 12 16 20

Cp/T

(J/m

ol.K

2 )0.0

0.5

1.0

1.5

2.0

2.5

HoNiAl

T2

T1

TC can be determined e.g. by idealization of thespecific heat jump under the constraint of entropyconservation. Circles are real experimental points,red line is the idealization drawn in the way thatthe yellow and green areas are equal to satisfy theentropy conservation.

Think that P

P TSTC ⎟

⎠⎞

⎜⎝⎛

∂∂

=

So if S has discontinuity, Cp will have sharper(like ∂ρ/∂T at magnetic transition)


Basic scaling in the critical region (near critical temperature that corresponds to order – disorder transition):

for T > TC and T < TC

α is the critical exponent and )c

c

TTTt −

=

Magnetic specific heat in Heisenberg magnetic systems scales with α = α’ ≈0.01 (Fe, Ni, EuO…)

Specific Heat and Critical Region

',

α−≈ tC Vpα−≈ tC Vp,

Phys. Rev. Lett. 89, 137002 (2002)

Sometimes nature of transition may changein magnetic field: Magnetic field inducedfirst order transition in heavy fermionsuperconductor CeCoIn5 detected byspecific heat measurement


Specific Heat of Spin Waves 1

For T > 0 thermal excitations create spin waves in magnetically ordered materials that propagate due to exchange coupling between neighboring spins. Heisenberg hamiltonian:

Consider spin oscillations along linear ferromagnet. Ground state: all spins are aligned along z with excitation:

Exchange field Be will induce a torque on spin :

)()( , lqatiy

yl

lqatix

xl eSeS ++ == ωω ξξ

∑−=ji

jiSSJH,

2ˆ rr Sum over all spins i and all j of their NN’sEach spin has associated magnetic moment

elllllll BSSJSSSJrrrrr

h

rrrμμ

γ−=+−=+− +−+− )(2)(2 1111

Effective magnetic(exchange) field

lSr

)(2)(11 +− +×=⇒×= lll

lel

l SSSJdtSdBS

dtSd rrr

h

rrr

h

rh γ

With approximation of small spin deviation (good at low T) : SSSSS yl

xl

zl <<≈ ,,

0),2(2)(),2(2)(1111 =−−−=−−= +−+− dt

dSSSSJSdtSdSSSJS

dtSd z

lxl

xl

xl

yly

lyl

yl

xl

hh

We look for wave solutions (a = lattice parameter):

And get new sets of equations: ( ) ( ) 0cos14,0cos14=+−=−+− yxyx iqaJSqaJSi ωξξξωξ

hh

ll Sr

hr γμ =

Gyromagnetic ratio


Specific Heat of Spin Waves 2

Nontrivial solutions if determinant is zero, we get spin wave dispersion relation:

Now in 3D in the small q limit this becomes:

2/3

2⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

JSTkkNc

dTdUC B

BAfV

M

( ) ( )hhh

222 22

4cos14 qJSaqaJSqaJS=≈−=ω At low T where low frequencies (long

wavelength dominate so (1-cosqa)=(1/2)(qa)2

2/33 TTCP δβ +=

∫∫∑∞

−∞

−=

−=

−=

0

42/52/3

20

)/(

2

2)/( 1)(

2121 2xTkq

qTkq

edxxkBTbV

edqqV

eU

BqB πω

πω

ωω hh

hh

Specific heat is then:

h

222 qJSaf •=αω

αf -constant that depends on crystal structure

Works well in long wavelength limit formetallic ferromagnets too.

Antiferromagnets: h

qSaJa

2'2•=αω Solutions doubly degenerate, Two

spin wave modes with ω for each q (see Rev. Mod. Phys. 30, 1 (1950))

We can write:

Sum over allowed values of q (1st BZ) Number of modes with q,q+dq is n(q)dq = (dN/dq)dq=(V/2π2)q2dq(Recall Debuye model)

(long wavelength, small frequencies)(x2 = q2b/kBTand b=2αfJSa2)

For antiferromagnets:

In insulators: In metals:2/33 TTTCP δβγ ++=

3

'2⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

SJTkkNc

dTdUC B

BAaV

M

(see Rev. Mod. Phys. 30, 1 (1950))Hard to resolve lattice vs magnons,in metallic AFM linear term as well


S = ∫(Cp/T)dT

We can determine how much entropy (change) is associated with a given state. For magnetic systems we need to use the magnetic Cp. In practice, this is done by subtracting off the Cp(T) data from a non-magnetic analogue (e.g. LuAgGe from TmAgGe).

For these local moment systems S ~ R ln D

Entropy Associated with Magnetic Order


Specific Heat of BCS Superconductors

Attraction of electron pairs by virtual phonon exchange, leading to T – dependent gap 2Δ at theFermi level. At T = 0: 2Δ(0)=3.52kBTC

Other complications may involve : strong coupling, gap anisotropy, presence of two distinct energy scales (two gaps).

Number of broken pairs as the temperature is increased is proportional to exp[-2Δ(0)/kBT], so contribution to electronic specific heat in superconducting state is:

)/( TbT

C

sel Cae

TC −=γ

a = 8.5, b = 1.44 for 2.5 < TC/T < 6a = 26, b = 1.62 for 7 < TC/T < 12

Near TC there is abrupt discontinuity since gap vanishes at TC. No latent heat is released (secondorder phase transition):

43.1)(=

−

C

Csel

TTTCC

γγ

Schrieffer, J. R. Theory of Superconductivity, Benjamin, New York 1965


Spin fluctuations (critically – damped spin waves) in nearly ferromagnetic systems whenexchange interaction is not strong enough to produce ordered state, for T << Tsf:

Some Other Contributions to Specific Heat

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

sfsfe T

TTTTC ln

3

αγ (Phys. Rev. Lett 17, 750 (1966))

BCS superconductors with anisotropic gap have reduced value of specific heat anomaly atsuperconducting transition (see Ann. Phys. 40, 268 (1966)):

( )241426.1 aTC

C

−=Δγ

Phys. Rev. B 75, 064517 (2007)


Superconducting and Magnetic Entropy

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1 100

1

2

3

4

SM

AG (R

ln2)

T(K)

X=

[C-C

latt]/T

(J/m

ol K

2 )

T (K)

x = 1 x = 0.9 x = 0.7 x = 0.6 x = 0.4 x = 0.2 x = 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1

10

1

10HEAVY FERMION

SUPERCONDUCTOR

TN(C) TN(χ) TCOH (ρ) TC(ρ) TC(C) TM (C)

T (K

)

X

MAGNETIC ORDER

LMM

Phys. Rev. B 77, 165129 (2008)

Nd1-xCexCoIn5

Heat capacity and magnetic entropy of electronicsystem that evolves from local moment magnetismto heavy fermion supeconductivity via disorderedheavy fermion magnetic states


Thermal Expansion in Thermodynamics

BTVCC m

Vp

2β=−

β (volumetric expansion)α (linear expansion)Vm (molar volume)B (isothermal compressibility)

At T = 0 CP = CV (entropy S = 0): PP

P TH

dTdQC ⎟

⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛=

At T ≠ 0 CP is always greater than CV (heating with P = const for dT cost energy to dowork expanding against external pressure; if V = const, there is no work done):

General relation between CP and CV at T ≠ 0 is

αβ 331 3

3 =⎟⎠⎞

⎜⎝⎛

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=PP T

LLT

LL

In practice, CP – CV ~ 1% of CP at θD/3 and 0.1% of CP at θD/6 and is maximum (~ 10%) at Tm (melting temperature).

TPV

VB ⎟

⎠⎞

⎜⎝⎛

∂∂

−=1


Thermal Expansion of Anharmonic Crystals

Real materials have non-parabolic U(r) so there is thermal expansion and CV increases aboveDulong – Petit at T>θD

VVTT T

PBT

PPV

V⎟⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

−=11β

TVFP ⎟

⎠⎞

⎜⎝⎛

∂∂

−=VV T

UTST ⎟

⎠⎞

⎜⎝⎛

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

( ) )()(1)1(

)()(21

,,)(

,qn

Tq

VBeqs

VqU

VP s

sqsT

sqqs

sqseq

rh

hh

h ∂∂

⎥⎦⎤

⎢⎣⎡

∂∂

−=⇒⎥⎦

⎤⎢⎣

⎡−∂

∂−+⎥

⎦

⎤⎢⎣

⎡+

∂∂

−= ∑∑∑ ωβωω ωβ

γs,q ~ 2

ωβ

ωωωh

hhh −−

+=+⎟⎠⎞

⎜⎝⎛ +=

eUnU eq 122

1

Harmonic = 0

BVCV

Tγβ =Grüneisen

Grüneisen approximation: Δω/ω ~ γΔV/VGrüneisen parameter is a measure of anharmonicity

Cu

Harmonic crystals do not expand when heated and do not shrink when cooled since average interatomic spacing does not increase with T.

PT T

VV

⎟⎠⎞

⎜⎝⎛

∂∂

=1β

)(ln))((ln

, Vqs

qs ∂∂

=ωγ

this We get

)()(,

qnT

qC sqs

sVr

h∂∂

= ∑ ω


Thermal Expansion by Capacitive Dilatometry 1Rev. Sci. Instr. 77, 123907 (2006)

In a capacitive dilatometer the dilation L of a sample of length L manifests as a change in the gap D between a pair of capacitor plates. For an ideal parallel-plate capacitive dilatometer in vacuum the relationship between the measured capacitance C and D is

DAC 0ε

= ε0 = 8.854·1019 pF/mA = area of capacitor plates

Thermal expansivity

Thermal expansiondT

LddTdL

L

LLTL

)(ln1)0(

)]0()([

==

−=

α

ε



Commercial capacitance bridges have ~ 10-7 pF resolution at 1kHz → 3·10-3Å for dilatometer operation at 18 pF.

Part of calibration processs is to find an appropriate functional relationship between the capacitor gap D and the measured capacitance C. This is done using a sample platform e long enough to adjust the capacitor gap from its largest zero-force to its smallest shorted value.

A protractor (with an appropriately sized hole in its center)is attached to the main flange o. The dilatometer is invertedand the sample platform e is then rotated and tightened in small steps; after each step the angular position of the sampleplatform (read off the protractor) and the capacitance C are measured. The capacitor gap is:

( )θθε

θθ −=⇒−= mm Ac

CcD

0

11

1)(

Thread pitch, # threads/mm → #μm/mm

Angle of dilatometer shorting

Deviation from linear behaviorsince plates are not perfectlyparallel. Can be estimated as:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2

max1

0 1C

Cc

AD εDistance from straight to tilted plate

C when they short



Data acquisition process in temperature or magnetic field consists of two steps:

1. Measurement of reference sample (Cu)2. Measurement of unknown sample.

[ ] ⎥⎦⎤

⎢⎣⎡ −

++−==L

DDDDdTd

LdTdL

Lcs

Cusc 111 αα

Cell effect, measurement with Cu standard installed

When the reference Cu or Sample are mounted

Since distance between platesis T,H dependent but is independentof length or nature of the samplein dilatometer

For long samples with thermal expansion close to Cu [Ds-Dc]/L ≈ 0, therefore (since dL = -dD):

CuCucellsamplecell dT

dLLdT

dLLdT

dLL

αα +⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛==

++

111

Derivatives are evaluated as:

Dilatometer is surrounded by Cu can to provide electrical and thermal insulation.

⎥⎦

⎤⎢⎣

⎡−−

+−−

=⎟⎠⎞

⎜⎝⎛

−

−

+

+

1

1

1

1

21

ii

ii

ii

ii

i TTDD

TTDD

dTdD

Resolution ~ 0.03 – 0.11ÅDepends on cryogenicsystem used and cell


Thermal Expansion by DiffractionBased on the measurement of unit cell by X-ray, neutron methods (will be discussed later in the course).

High resolution data are required. Resolution comparable or smallerthan for capacitive dilatometry. Advantage: bond length, angles, atomic position information is available

Phys. Rev. B 72, 045103 (2005)

0 100 200 300 400 5000

5

10

15

20

25

0 100 200 300 400 5000

10

20

30

40

50

α(1

/K)*

10-6

T(K)

Unit cell volume

c-axis b-axis a-axis

α (1

/K)*

10-6

T(K)


Thermal Expansion in CeCoIn5

Phys. Rev. Lett. 100, 136401 (2008)

Documents

Specific Heat in Thermodynamics · Vd dS d r r van Hove singularity extremal points of phonon dispersion curves in BZ Phonon dispersion relation by ... s t i c ( L A) M o d e T r