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


• 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


• 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


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TT1-Chap3 - 7


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


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


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


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


• 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


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


• 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


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:


note that

𝑑𝑔𝑛 = −𝑠𝑛𝑑𝑇

𝑔𝑛 = − 0

𝑇

𝑠𝑛𝑑𝑇 ∝ − 𝑇2

since 𝑠𝑛 ∝ 𝑇 for electron gas

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TT1-Chap3 - 17


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


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


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


< 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


• 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


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

Chapter 2 I/SL_Chapter 2_2014.pdfwe fill the coil with magnetic medium change of magnetization M = m...

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Transcript of Chapter 2 I/SL_Chapter 2_2014.pdfwe fill the coil with magnetic medium change of magnetization M = m...