Transport and thermodynamic properties of high temperature superconductors (HTS)-an experimentalist's view.
Overall aims:
Emphasize what can be achieved by `small-scale' experiments on new materials. New states of matter!
Importance of changing a control or tuning parameter in a systematic way, e.g. sample composition, pressure,
magnetic field.
Keep in mind access to shared facilities, e.g. high magnetic field, synchrotron radiation and neutron scattering
labs.
Powerpoint slides shown on April 16th.
Blackboard presentations were made on:
1. Revision of free electron, nearly free electron and tight binding theories of metals. 3D, quasi 2D and 1 D
Fermi surface. Counting states in k space. Qualitative description of Peierls transition.
2. Derivations of transport properties using the Boltzmann transport equation in the relaxation time
approximation. Based on the textbook “Principles of the Theory of Solids” J.M. Ziman, Cambridge
University Press (1964) Ch.7. Handwritten notes for this section to follow shortly.
Overview of HTS subjects to be covered
1. Systematic patterns in normal state transport properties, resistivity, thermoelectric power, Hall effect as a
function of hole doping and cation substitution
2. Thermodynamic properties, heat capacity and magnetic susceptibility/magnetisation in normal and s/c
states. Especially for hole-doped cuprates.
3. High magnetic field studies at international facilities especially Quantum Oscillation (QO) work.
But will try to make it relevant for students working on other problems / materials, e.g. by talking about
experimental methods
PLEASE INTERRUPT IF YOU THINK IT WILL BE HELPFUL!
Periodic table of elements showing superconducting transition temperatures
Courtesy of A. Carrington, University of Bristol, 2003
Superconducting elements and compounds
Elements (see table), “less good” metals superconducting.
Until quite recently superconductivity and magnetism thought to be incompatible.
Tc(K)
A15 compounds
Nb3Sn
18.2
Nb3Ge
20.3
Organics
(TMTSF)2PF6
1.2
k-(BEDT-TTF)2Cu(SCN)2
10.4
Cs3C60
33
Heavy Fermions
CeCu2.2Si2
0.9
CePd2Si2 (under pressure)
0.4
UBe13
0.9
UPt3
0.45
Magnetic materials UGe2 (under pressure)
0.4
Nearly ferro-magnetic Sr2RuO4
1.3
Cuprates
La2-xSrxCuO4
38
YBa2Cu3O7-d
93
HgBa2Ca2Cu3O8+d (under pressure)
130 (164)
Borides
HoNi2B2C
8
YNi2B2C
15
MgB2
39
Pnictides (P, As)
Nd(O1-xFx)FeAs
52
Also late last year Eremets
group H2S at extreme
pressure 190 K!
arXiv:1412.0460
H gO x
ELECTRONIC PHASE DIAGRAM
Temperature
BaO
Cu O 2
Ca
Cu O 2
Ca
Cu O 2
TN
AF Mott
insulator
BaO
Universal
pseudogap energy
H gO x
Tc
Glass
H g Ba 2Ca 2 Cu 3O 8+x
0
2+
Tc = 134K - 160K
d-wave
Superconductivity
0.05
Cu
correlated
metal
Pseudogap metal
0.16
2.05+
2.16+
Cu
Cu
Extremely weak s/c
Best
0.19
~0.3
Holes
/Cu
2.3+
Cu
s/c scales with Tc
Diagrams from Prof. W.Y. Liang
Cuprates (Bednorz and Muller, 1986)
Top view of CuO2 plane
Crystal structure of highest Tc
cuprate
Typical Cu-O co-ordination in cuprate
superconductors
6
p orbitals
PF6-
Jerome et al (1980)
CePd2Si2
Julian et al (1999)
Saxena et al. (2000)
Cuprates (Bednorz and Muller, 1986)
Parent compounds anti-ferromagnetic Mott insulators e.g. La2CuO4,
Count electrons 2La3+  6+, 4O2-  8-, Cu2+, so Cu (3d10 4s1)  3d9.
one hole in d shell, one hole per unit cell, but not conducting.
 Correlated electrons, Mott-Hubbard insulator. Antiferromagnetic,
Neel temperature 300 K.
Dope lightly with holes, e.g. replace 8% of La3+ atoms with Sr2+
get 40 K superconductor! La2-xSrxCuO4 , 1+x holes per CuO2 unit. Now
think that all hole doped cuprates show similar behaviour as function of
number of added holes, p (p = x for LSCO).
YBa2Cu3O6+x , add O2- , add holes to CuO2 layers until have YBa2Cu3O7
Layered compounds, wide range of anisotropy, CuO2 layers source
of superconductivity
8
Two Bpopular theories: 1.The “t-J” model - starting from the Mott-Hubbard side.
J S .S
i
j
t
+
+
Holes moving in an a-f background attract each other because minimise
magnetic perturbation.
Analytical theory difficult because double occupancy is excluded. Numerical
calculations at high T account for many of the strange properties of the cuprates.
See e.g. “Numerical studies of the t-J model”, J. Jaklic and P. Prelovsek, Adv. Phys.
(2000), 1, 1-85 starting with “t-J” Hamiltonian:
+
Does give strong p dependences of electrical conductivity and entropy that are observed experimentally. Almost no adjustable
parameters. Origin of PG?
9
2. The spin-fluctuation model or nearly anti-ferromagnetic Fermi liquid model, NAFL (Pines et al., Moriya et al.). Start from LDA
Fermi surface containing 1+p holes per Cu. Introduce antiferromagnetic correlations in self-consistent way.
Enhanced magnetic response at Q = (+/-p/a, +/-p/a), corresponds to
proximity to a-f ordered state. Long-lived a-f spin excitations of wave-vector
Q can be formed by correlated occupancy of k and (k+Q) states. These are
spin fluctuations, collective excitations of Fermi liquid. Analogy : mixing of
k and k+G states in nearly free electron theory of metals gives modulations
in charge density with period of lattice parameter.
Early (1993) model used by us to calculate Hall
coefficient and thermoelectric power. lQ and lc are
electron mfp’s in spanned and unspanned
regions. x is range of a-f correlations,
Similar picture works well for superfluid He3 (ferromagnetic fluctuations).
Supported by NMR, neutron scattering and ARPES data, predicted d-wave
pairing. Pairing given by virtual exchange of spin fluctuations. Are they
strong enough to produce the high Tc s found in the cuprates? And strong
p-dependences of conductivity and entropy?
10
Four terminal resistance measurements. R0 is
resistance at 273.16 K. As increase purity of metal
resistance ratio, R273.16/R4.2 increases.
Controversy between Kelvin, Dewar and Matthiessen
about R(T  0).
Kamerlingh Onnes (Leiden 1911).
Present limit for small s/c Pb ring with trapped field
(Quinn and Ittner, J. Appl. Phys, 33,748, 1962) r < 3.6 10-23
 cm (no decay in magnetic moment over a time of expt.,
so L/R > 7 hrs). Nowadays use s/c coils in persistent mode
to provide high magnetic fields ( e.g. 18 T) without power
dissipation.
From G. Gruner Rev. Mod.
Phys. 60, 1129 (1988)
From P. Monceau
et al. PRL, 37,
602 (1976)
From G. Gruner
Rev. Mod. Phys. 60, 1129 (1988)
From D. Jerome and
H.J. Schulz, Adv. Phy.
31, 299-490 (1982)
From G. Gruner
Rev. Mod. Phys. 60,
1129 (1988)
Interacting Fermions in 1 D
MgB2 Fermi Surface with dHvA orbits
J. Kortus et al, PRL 86, 4656 (2001)
From JRC, Handbook of Superconducting Materials,
D.A. Cardwell and D.S. Ginley, Eds, IoP (2002) p. 1461
Another pair of contacts
opposite
Diagram from A. Carrington’s PhD thesis 1993.
For crystal, thickness t = 20 microns, 0.2 mm between voltage contacts and 0.2 mm wide and ρa = 50 e-6 ohm-cms (OII YBCO
just above Tc), Ra only 25 milliohms. One reason why people using pulsed magnetic fields often measure ρc . Resistance of
contacts – 1 ohm considered good, therefore separate I and V contacts – 4 terminal method - essential
RH = 1/ne, (n is carrier density).
Estimate Hall voltage
VH= RH I B/t
Take RH = 2 e-9 m3/C , e.g. Carrington et al, Phys Rev B 48 13051, 1993, then for I = 1 ma, t = 20 microns and B = 10T get VH = 1e6 V. In practice need to see how large I can be without getting too much heating. Also check that voltage noise does not increase
with I. If it does then no point in increasing I further. Stability of contacts important for measuring RH,
e.g. Heat treated silver paint (Dupont 6838) often better than room temperature cured Dupont 4929.
Sketch set up
matching transformers.
Twisted pairs especially high field.
Johnson noise.
From JRC, Handbook of
Superconducting Materials,
D.A. Cardwell and D.S.
Ginley, Eds, IoP (2002) p.
1461
Thermoelectric power, thermopower, Seebeck coefficient (S)
From JRC, Handbook of
Superconducting Materials, D.A.
Cardwell and D.S. Ginley, Eds,
IoP (2002) p. 1461