Chapter 2 I/SL_Chapter 2_2014.pdfwe fill the coil with magnetic medium change of magnetization M = m...
Transcript of Chapter 2 I/SL_Chapter 2_2014.pdfwe fill the coil with magnetic medium change of magnetization M = m...
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TT1-Chap3 - 1
Chapter 2
2. Thermodynamic Properties of Superconductors
2.1 Basic Aspects of Thermodynamics
2.1.1 Thermodynamic Potentials
2.2 Type-I Superconductor in an External Field
2.2.1 Free Enthalpy
2.2.2 Entropy
2.2.3 Specific Heat
2.3 Type-II Superconductor in an External Field
2.3.1 Free Enthalpy
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TT1-Chap3 - 2
2. Thermodynamic Properties of Superconductors
• already in 1924:W.H. Keesom tries to describe the superconducting state using basic laws of
thermodynamicsMeißner effect not know that time: unclear whether superconducting state
is a thermodynamic phase
• after 1933: after discovery of Meißner effect it is evident that superconducting state
is real thermodynamic phase this fact is used in development of GLAG theory
• questions: what is the suitable thermodynamic potential to describe superconductors ? what can we learn on superconductors from basic thermodynamic
properties (e.g. specific heat) ?
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TT1-Chap3 - 3
2.1 Thermodynamic Potentials
• thermodynamic potentials: used for the description of equilibrium states
equilibrium state: determined by extremal value of the potential
• thermodynamics: describes systems with large particle number by small quantity of variables: T, V, N, …
• extensive variables: V, S, N, … depend on system size (amount of substance)
• intensive variables: T, p, n, … do not depend on system size
• example: S, V and N are the natural variables of a systems, all other fixed by external constraints internal energy U(S,V,N) yields full information on the system
dU = TdS – pdV + µdN adiabatic-isochore processes are characterized by minimum of U
• other potentials: - Helmholtz free energy F(T,V,N)- enthalpy H(S,p,N)- free enthalpy or Gibb energy G(T,p,N)
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TT1-Chap3 - 4
2.1 Thermodynamic Potentials
• question:which is the suitable potential to represent the system under consideration ? find the set of independent variables
• discussion of magnetic and electronic systems: additional variables such as polarization P and magnetization M
• discussion of systems with N = const.
A: Internal Energy
• differential of the internal energy
infinitesimal heat flow into the
system
infinitesimal mechanical work
done by the system
infinitesimal electromagnetic
work done by the system
m = magnetic momentM = m/V = magnetization
since magnetization is mostly parallel or anti-parallel to B
statevariable
generalizedforce
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TT1-Chap3 - 5
2.1 Thermodynamic Potentials
• important question: how to calculate the em work ? do we have to use mdB or Bdm in the expression of dU ?
• Answer: depends on experimental situation
closed metallic ring with infinite conductivity: flux density in ring stays constant
open metallic ring with current source attached: current through ring (magnetic field)
stays constant
interaction energy of dipole with fieldhas to be taken into account
no interaction energy of dipole and field
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TT1-Chap3 - 6
2.1 Thermodynamic Potentials
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TT1-Chap3 - 7
2.1 Thermodynamic Potentials
work required tomagnetize sample in a coil
• Example: current I through coil of length L, cross-section F, and winding number n = N/L Hext = n I
we fill the coil with magnetic medium change of magnetization M = m /F L results in flux change µ0F DMtemporal change of flux through coil induced voltage V = - nL µ0F dM/dtcurrent source has to perform work -I V dt = µ0nIFLDM = µ0Hext DM
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TT1-Chap3 - 8
2.1 Thermodynamic Potentials
expressions for internal energy
two different expressions depend on definition of the considered systems
Scheme II: interaction energy is not included it is assigned to external circuit which performs work on the system and
is not part of the considered system
Scheme I: interaction energy is included into the considered systems
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TT1-Chap3 - 9
2.1 Thermodynamic Potentials
B: Helmholtz Free Energy F
• definition:
• differential:
• F is potential for the natural variables T, V, 𝐵 = 𝜇0 𝐻𝑒𝑥𝑡 +𝑀 isothermal-isochore processes at B = const. are characterized by minimum of F problem: in experiments it is difficult to keep V and m (or 𝑩) constant
use free enthalpy or Gibbs energy
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TT1-Chap3 - 10
2.1 Thermodynamic Potentials
C: Free Enthalpy G
• definition:
• differential:
• G is potential for the natural variables T, p, Hext
isothermal-isochore processes at Hext = const. are characterized by minimum of G this is appropriate for most experimental situations
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TT1-Chap3 - 11
2.1 Thermodynamic Potentials
• if the suitable thermodynamic potential is known derivation of various thermodynamic quantities by partial differentiation
• specific heat
for p, Bext = const.
dU = dQrev = TdS
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TT1-Chap3 - 12
2.1 Thermodynamic Potentials
superconductor in external field: free energy
• normal state
• superconducting state
with we obtain for
problem: free energies of normal and superconducting state do not coincide atphase boundary
origin of discrepancy: interaction energy of magnetic moment of SC with Hext
with and we obtain
total volume
volume outside SC
volume of SC
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TT1-Chap3 - 13
2.1 Thermodynamic Potentials
• where is the energy coming from ?
it is provided by the current source maintaining 𝐻𝑒𝑥𝑡 = 𝑐𝑜𝑛𝑠𝑡 by doing workagainst the back electromotive force induced as the flux is entering the sample
• at 𝐻𝑒𝑥𝑡 = 𝐻𝑐𝑡ℎ superconductivity collapses flux density 𝐵𝑒𝑥𝑡 = 𝜇0𝐻𝑒𝑥𝑡 = 𝐵𝑐𝑡ℎ enters the volume 𝑉𝑠 of the superconductor
• work done by current source 𝑊 = 𝑎
𝑏
𝑈 𝐼𝑐𝑜𝑖𝑙 𝑑𝑡 = 𝑎
𝑏
−𝑁 Φ 𝐼𝑐𝑜𝑖𝑙 𝑑𝑡 = 𝑎′
𝑏′
−𝑁𝐼𝑐𝑜𝑖𝑙 𝑑Φ
𝑊 = 𝑁𝐼𝑐𝑜𝑖𝑙 0
𝐵𝑐𝑡ℎ
𝐴 𝑑𝐵 = 𝑁 𝐼𝑐𝑜𝑖𝑙𝐴 𝐵𝑐𝑡ℎ
with 𝐵 = 𝜇0𝐼𝑐𝑜𝑖𝑙𝑁/𝐿 and 𝑉𝑠 = 𝐴 𝐿
𝑊 = 𝑁𝐵𝑐𝑡ℎ𝐿
𝜇0𝑁𝐴 𝐵𝑐𝑡ℎ = 𝑉𝑠
𝐵𝑐𝑡ℎ2
𝜇0
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TT1-Chap3 - 14
with and we obtain
2.1 Thermodynamic Potentials
superconductor in external field: free enthalpy
• normal state
• superconducting state
with we obtain for
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TT1-Chap3 - 15
2.2 Type-I Superconductor in Bext
• perfect diamagnetism
• we assume p, T = const.
free enthalpy in S-state:
@ Bext = Bcth
same free enthalpy density gs and gn
of S- and N-state
condensation energy
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TT1-Chap3 - 16
0.0 0.2 0.4 0.6 0.8 1.0
free
en
thal
py
den
sity
T / Tc
gn(0,T)
gs(0,T)
Bcth2/2µ0
empirically found T dependence:
2.2 Type-I Superconductor in Bext
note that
𝑑𝑔𝑛 = −𝑠𝑛𝑑𝑇
𝑔𝑛 = − 0
𝑇
𝑠𝑛𝑑𝑇 ∝ − 𝑇2
since 𝑠𝑛 ∝ 𝑇 for electron gas
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TT1-Chap3 - 17
2.2 Type-I Superconductor in Bext
Entropy density
• with and 𝑠𝑠 = 𝑆𝑠/𝑉, 𝑠𝑛 = 𝑆𝑛/𝑉
• how does sn (T) look like?
we use and Cp proportional T for electron gas (phonons can be
neglected at low T)
Sn proportional T
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TT1-Chap3 - 18
2.2 Type-I Superconductor in Bext
Sn = a ∙ T
• 𝑇 → 𝑇𝑐:𝐵𝑐𝑡ℎ → 0, Δ𝑠 → 0
Δ𝑄/𝑉 = 𝑇𝑐Δ𝑠 → 0no latent heat, 2nd order phase transition
• 𝑇 → 0:𝜕𝐵𝑐𝑡ℎ/𝑑𝑇 → 0
Δ𝑠 → 03. law of thermodynamics (Nernst theorem)
• 0 < 𝑇 < 𝑇𝑐:𝜕𝐵𝑐𝑡ℎ/𝑑𝑇 < 0 Δ𝑠 > 0 lower entropy of S-state
state with less disorder Δ𝑄/𝑉 = 𝑇 Δ𝑠 > 0
finite latent heat, 1st order phase transition
0.0 0.2 0.4 0.6 0.8 1.0
entr
op
y d
ensi
ty, D
Q/V
T / Tc
sn(T)
ss(T)
DQ(T)/V
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TT1-Chap3 - 19
for : Δ𝐶𝑇=𝑇𝑐
𝑉= −
8
𝑇𝑐
𝐵𝑐𝑡ℎ2 0
2𝜇0
2.2 Type-I Superconductor in Bext
Specific Heat
• with we obtain
• T = Tc: Bcth = 0 but 𝜕𝐵cth/ 𝜕𝑇 ≠ 0 Δ𝐶 < 0
since 𝜕Bcth/ 𝜕T is discontinuous at T = Tc, we obtain jump of specific heat
Rutgersformula
specific heat jump can be measured directly and calculated from measured Bcth
usually good agreement
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TT1-Chap3 - 20
2.2 Type-I Superconductor in Bext
< 0 for all T decreases with T
Ds changes sign
0.0 0.4 0.8 1.2 1.6 2.0
0
1
2
3
m
ola
r sp
ecif
ic h
eat
(mJ/
mo
l K)
T (K)
DCCs
Cn
Tc
Al
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TT1-Chap3 - 22
2.2 Type-I Superconductor in Bext
• determination of Sommerfeld coefficient 𝜸:
at low T we can neglect 𝐶𝑠 𝛾 =
𝐶𝑛
𝑉𝑇=
𝑐𝑛
𝑇≃
𝑐
𝑇
• with
we can determine 𝛾 =𝜋2
3𝑘𝐵2 𝐷 𝐸𝐹
𝑉𝑇 by measuring Bcth(0) and Tc
- 11
𝛾 ≃4
𝑇𝑐2
𝐵𝑐𝑡ℎ2 0
2𝜇0
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TT1-Chap3 - 23
2.2 Type-I Superconductor in Bext
volume change at S/N transition:
• we use
volume approaches zero for T Tc since Bcth 0
T < Tc : Vs > Vn, i.e. superconductor has larger volumetypical relative length change ≈ 10-7 to 10-8
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TT1-Chap3 - 24
2.3 Type-II Superconductor in Bext
• everything stays the same for Bext < Bc1
• for Bext > Bc1 more complicated behavior