Low temperature specific heat of solids

James C. Ho

For some 200 articles on low temperature specific heat and superconductivity, I have many coauthors including:

Members of the U.S. National Academy of Sciences

Ching-Wu Chu, Theodore Geballe, John Hulm,

Bernd Matthias, Maw-Kuen Wu

Members of the U.S. National Academy of Engineering

Karl Gschneidner, John Hulm, Robert Jaffee

Fellow of British Royal Society

Sir John Enderby

Members of the Academia Sinica (Taiwan)

Ching-Wu Chu, Maw-Kuen Wu

Member of Chinese Academy of Science

Weiyan Guan

Low temperatures

Specific heat and entropy in chemical thermodynamics

Low temperature specific heat in solid state physics

Phonon (lattice)

Conduction electrons

Nuclear magnetic moments

Critical phenomena

Superconductivity

Magnetic transitions

Some applications to materials science

Magnetic clusters

Electron localization

Non-uniform superconductors

Nano-size effect

Low temperatures

How low is low?

How low can we go?

How low do we need to go?

17th-century depiction of Leiden University Library

Walther Nernst (Nobel prize in Chemistry, 1920) developed in 1906-1912 the 3rd law of thermodynamics (zero entropy at zero degree)

Specific heat and entropy in chemical thermodynamics

Entropy change determination relies on specific heat measurements:

ΔS = δQ/T = ∫(C/T) dT______________________________________________

U = Q – W dU = δQ – PdV

(∂U/∂T)V = (∂Q/∂T)V ≡ CV

H = U + PV dU = δQ +VdP

(∂H/∂T)p = (∂Q/∂T)p ≡ Cp

Cp – CV = VT(α2/βT)

α: Coefficient of linear thermal expansion

βT: Isothermal compressibility

Early models of specific heat of solids

The Dulong-Petit law (1819): CV = 3Nk

(harmonic oscillators)

_______________________

The Einstein model (1907)

(quantized harmonic oscillators)

When Nernst learned of Einstein's paper, he was so excited that he traveled all the way from Berlin to Zurich to meet with Einstein.

Heike Kamerlingh Onnes (Leiden University, 1853-1926) first liquefied helium in1908 and discovered superconductivity in 1911.

1913 Nobel Prize in Physics

Einstein and Kamerlingh Onnes

Specific heat and entropy in chemical thermodynamics

Low temperature specific heat in solid state physics

Phonon (lattice)

Conduction electrons

Nuclear magnetic moments

Critical phenomena

Superconductivity

Magnetic transitions

Some applications to materials science

Magnetic clusters

Electron localization

Non-uniform superconductors

Nano-size effect

Einstein model (individual atomic vibrations)Debye model (phonon modes)

Peter Debye (1884-1966) 1936 Nobel prize in chemistry 1937-1939, President, Deutsche

Physikalische Gesellschaft 1940-1950, Cornell University

Debye model of phonon (lattice) specific heat at low temperatures

at T << ϴD, Cph = Nk(12π4/5)(T/ϴD)3 ≡ βT3

SiC, ϴD = 990 K (Fe-477, Al-433, Cu-347, Au-162, In-112, Pb-105)

Free electron model of electronic specific heat Ce = (π2/3)D(EF)k2T ≡ γT at T << TF ,

For metallic materials at low temperatures, C = Ce + Cph = γT + βT3 or C/T = γ+ βT2

Schottky anomaly in specific heat for multi-energy level systems(from a homework problem in C. Kittel: Introduction to Solid State Physics)

Nuclear specific heat

In magnetically ordered materials, interactions between

s-electrons (non-zero density at nucleus) and the nuclear moment μ behave as an extremely large effective field He. It causes nuclear energy splitting, and in turn yields a Schottky-type specific heat.

At temperatures T ˃˃ μHe/k (often near or less than 1 K),  

CN = Nk[(I+1)/3I](μHe/kT)2

– [(I+1)(2I2+2I+1)/30I3](μHe/kT)4 + ….

= A/T2 – A’/T4 + …. 

.

Specific heat of manganese

C = 0.055T3 + 9.20T + 0.264/T2

with I = 5/2 and μ = 3.4532μN, He = 65 kOe

Effective hyperfine field at the nuclei of Pt dissolved in Fe

C = 8.25x10-2T + 1.40x10-3/T2 – 1.31x10-6/T4

Specific heat and entropy in chemical thermodynamics

Low temperature specific heat in solid state physics

Phonon (lattice)

Conduction electrons

Nuclear magnetic moments

Critical phenomena

Superconductivity

Magnetic transitions

Some applications to materials science

Magnetic clusters

Electron localization

Non-uniform superconductors

Nano-size effect

Superconducting transitionBCS theory: Ces/γT at Tc ≈ 1.43; Ces/γTc = a exp(-bTc/T)

BCS theory: Tc ≈ ϴD exp[-1/D(EF)V]V: electron-phonon interaction parameter

Superconducting transitions in Ti-Mo alloys (V = constant?)

An antiferromagnetic transition at 3.8 K in bulk CeAl2

S = Rln2 = 5.76 J/mol K

Two sets of erbium ions in erbium sesquioxide -- Er3+-I and Er3+-II in 1:3 ratio

Entropy associated with the observed anomaly, ΔS = ∫(Cm/T)dT = 4.14 J/mol K,a value about 72% of Rln2 = 5.76 J/mol K for all Er3+ ions (ground state doublet),

indicating that only Er3+-II undergoes the magnetic transition at 3.3 K.

Specific heat of GdBa2Cu3O6+δ revealed a magnetic ordering of Gd3+ at 2.2 K, independent of whether superconducting transition occurs above 90 K (δ = 0.7) or not (δ = 0.5).

J.C. Ho, P.H. Hor, R.L. Meng, C.W. Chu and C.Y. Huang, Solid State Commun. (1987)J.C. Ho, C. Y. Huang, P.H. Hor, R.L. Meng and C.W. Chu, Mod. Phys. Lett. (1988)

President Ronald Reagan gave the keynote address at the Conference on “Superconductivity: Challenge for the future”, Washington, D.C., July 1987.

Specific heat and entropy in chemical thermodynamics

Low temperature specific heat in solid state physics

Phonon (lattice)

Conduction electrons

Nuclear magnetic moments

Critical phenomena

Superconductivity

Magnetic transitions

Some applications to materials science

Magnetic clusters

Electron localization

Non-uniform superconductors

Nano-size effect

Physical Review (1893)

Physical Review B -- Solid State Physics (1970)

Physical Review B – Condensed Matter (1978)

Physical Review B -- Condensed Matter and Materials Physics

Magnetic clusters (classical harmonic oscillators)Eδ = N(kT), δ = Nk

C = δ + γT + βT3 or (C – δ)/T = γ + βT2

Nickel-base MAR-M200 superalloy: N ≈ 3x1020/mol ≈ 0.5x10-3NA

Professor Sir John Enderby

Kt, CBE, PhD (London), DSc (Lough), FInstP, FRS

Vice President of Royal Society, 1999-2004

President of Institute of Physics, 2004-2006

Ordering effect on the electronic structure and subsequently the mechanical behavior (stronger but brittle) of titanium-aluminum alloys

Light and strong Al-Li alloys

Large electronegativity difference: 1.47 for Al and 0.97 for Li Intermetallic compound AlLi has its m.p (700oC) extruding into the liquidus.

Detection of superconducting transitions

Transport property measurements (qualitative):

Electrical resistivity

(impurity forming a continuous network?)

Thermodynamic property measurements (quantitative):

Magnetic susceptibility

(shielding effect?)

Specific heat (qualitative)

fraction of superconducting component in sample

(BCS theory: (Ces/γT)Tc ≈ 2.43)

Superconducting transition in an non-uniform sample

Specific heat and entropy in chemical thermodynamics

Low temperature specific heat in solid state physics

Phonon (lattice)

Conduction electrons

Nuclear magnetic moments

Critical phenomena

Superconductivity

Magnetic transitions

Some applications to materials science

Magnetic clusters

Electron localization

Non-uniform superconductors

Nano-size effect

An antiferromagnetic transition at 3.8 K in bulk CeAl2 is completely suppressed in 8 nm particles, yielding to a heavy

Fermion behavior.

The dashed line represent the sum of phonon and crystal-field contributions (open circles), based on LaAl2.

In conclusion, measurements of low temperature specific heat, as a thermodynamic quantity, provide some fundamental information about a solid, as well as a relatively simple and effective evaluation technique in materials science.