specific heat in thermodynamics · vd ds d r r van hove singularity extremal points of phonon...
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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
Introduction to thermal and transport techniques
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 ∈+= ),()(
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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(
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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 =
Introduction to thermal and transport techniques
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:
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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.
Introduction to thermal and transport techniques
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.
Introduction to thermal and transport techniques
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
−−=
Introduction to thermal and transport techniques
Quantum Design PPMS
Introduction to thermal and transport techniques
PPMS Heat Capacity Option Hardware
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
+=
= ∫∞
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
Heavy fermion superconductor PrOs4Sb12
Phys. Rev. B 73, 104503 (2006)
Introduction to thermal and transport techniques
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
=
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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.
Introduction to thermal and transport techniques
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)
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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)
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
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
Introduction to thermal and transport techniques
Thermal Expansion by Capacitive Dilatometry 2Rev. Sci. Instr. 77, 123907 (2006)
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
Introduction to thermal and transport techniques
Thermal Expansion by Capacitive Dilatometry 3Rev. Sci. Instr. 77, 123907 (2006)
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
Introduction to thermal and transport techniques
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)
Introduction to thermal and transport techniques
Thermal Expansion in CeCoIn5
Phys. Rev. Lett. 100, 136401 (2008)