single crystal growth and basic characterization of ... · single crystal growth and basic...

Single crystal growth and basic characterization of intermetallic compounds

CIFAR Summer School 2015

Eundeok MunDepartment of Physics

Simon Fraser University

Beautiful single crystals! Then What?

We know how to make a noodle soup. Enjoy!

Is it delicious?

Is it interest? Magnetic, electronic, mechanical, thermal property, and so on…

If so, continue to make it, share the recipe with friends, and discuss how good it is.

Collaborations - open to community and try to understand the nature of the material.

If not, make another one!

Characterizations

Structural ! Electronic !

Magnetic ! Thermal !

Thermodynamic and transport measurements “bulk”

Quantum mechanics, electrodynamics, statistical mechanics, and solid state physics

R = V/II

V

Q/T

Determination of crystal structure

Identify crystal structure: powder and single crystal x-ray

Synchrotron and elastic neutron Micro and surface structure: SEM, TEM, etc

Determine crystallographic orientations: Laue back reflection

3-fold4-fold

Analyze elemental compositions: EDX, WDS, EPMA, etcBa Fe1-x Cox As2 : nominal and actual xNote: site-deficiency is a nature of some materials

Electronic properties - resistivity Let’s categorize materials like metal, semiconductor, superconductor, and insulator

We can get a lot of information from the temperature dependence of resistivity.

Lakeshore.com

Metal

Semiconductor

thermometer

Superconductor

phase transition

anisotropy

Resistivity / Resistance

In a simple wayOhm’s law I = (1/R) V

J = E

1 ne

2m

AlR

IV

w

tl A

Electrical contact: silver/gold paste, soldering

Electronic properties - resistivity

n: carrier densitym: effective mass: relaxation time

Categorize materials from (T)

For simple metal• Temperature dependence• Residual resistivity• Residual resistivity ratio• Spin disorder scattering• Fermi surface

Electronic properties – Hall, TEP Carrier density - electron and hole

carrier type (electron, hole) – Hall coefficient and Thermoelectric power

S

N H

Resistivity- even (H) = xx(H) ~ H2

Hall resistivity- odd H(H) = xy ~ H

Hall coefficientRH = xy / H

Thermoelectric power (thermocouple)S = - V / T (V/K)S is positive when the direction of electric current is the same as the direction of thermal current. Order of ~V/K

ThighTlowHeat flow

electron

holeI

V+ + + + +

+

- - - - - -

Electronic properties - resistivity Let’s categorize materials like metal, semiconductor, superconductor, and insulator

Metal: Cu, Ag, Au…

Semiconductor: Si, Ge, InSb…

Superconductor: Pb, Sn, In…

Insulator: wood, plastic, glass…

This is a simple picture.

In materials -AntiferromagneticFerromagneticSuperconductorMetal-to-InsulatorCDWSDWMeta-magneticTopologically non-trivialNematic phase

There are more…Temperature (K)

Res

istiv

ity (o

hm c

m)

metal

superconductor

semiconductor

insulator

Residual resistivity and RRR

Temperature (K)

Res

istiv

ity (o

hm c

m)

NOTE: the measured electrical resistivity will contain an impurity scattering term, 0 which appears additively. This is often associated with chemical impurities as well as a variety of structural defects.

~T 2 ~T 5

~T

Metal at low temperaturese-e scattering - Fermi liquid

= 0 + AT2

0

Residual resistivity 0

Residual resistivity ratio RRRR(300K) / R(2K) – geometric

factors cancel outgrain boundary, imperfection, dislocation, impurity

Magnetic materials: spin disorder scattering

Temperature (K)R

esis

tivity

0

sd

TM

grain boundary, imperfection, dislocation, impurity

spin disorder scattering

J. MMM 205 (1999) 27-52

AFM orderTN = 10.9 K sd

0

Residual resistivity and RRR

0 grain boundary, imperfection, dislocation, impurity

Extremely pure & fully ordered

0 ~ 0.04 cm

RRR ~ 1000

Highly disordered: Ni site-deficiencyLuNi0.81Ge3

J. MMM 205 (1999) 27-52

0 ~ 22 cm

RRR ~ 30 ~ 1 cm

RRR ~ 17

In general, high RRR means low 0.

Residual resistivity and RRRIf the grown material is fully ordered, expected low 0, high RRR, and

Quantum oscillationsdHvA

Very large MRat low temperatures

Quantum oscillationsSdH

MR: 500000% at 2 K

Simple metal(semimetal)MR ~ H2

Kohler rule

(0)

F H(0)

n

MR ~ H1.8

Quantum oscillations

Oscillation frequency: related to area of extremal orbit, periodic in 1/H, p=1/F~1/A

Oscillation amplitude:increases as magnetic field increasesdecreases as temperature increasesL-K formula: effective (band, renormalize) mass

Dingle temperature, g-factor, and so on…

Fermi surface of Cu

Landau gauge:

H H k^

A Hyi^

dHvA-oscillations in magnetization

Harmonic oscillatorLandau level

Fermi surface of high Tc cupratesQuantum oscillations SdH

Fourier Transform Angle dependence – mapping Fermi surface

Multi-band system: PtSn4

ARPESARPES+ +Complex andDifficult…

Nature 511, 61 (2014), samples come from UBC

CDW, SDW, magnetic superzone gap

K. Myers, et al., JMMM 205, 27 (1999)

M. Maple, et al., PRL 56, 185 (1986)

R. Elliot, Magnetic properties of rare earth metals (1972)E. Fawcett, Spin-density-wave

antiferromagnetism in chromium, RMP 60, 209 (1988)A. Arrott, Antiferromagnetism in Metals Vol. IIB, p.295

CDW (and SDW) transitions, where nested parts of the Fermi surface become gapped below TCDW. This leads to an decrease in conductivity due to a decrease SF. For a partial gapping of the F.S. the sample remains metallic and ultimately returns to (T) with positive slope.

Resistivity is very sensitive to changes in the Fermi Surface:

Magnetism

Iconduction electron

valence electron

nuclear

Magnetic susceptibility & Magnetization

Larmor diamagnetism (Langevin result)Pauli paramagnetism of conduction electronsLandau diamagnetism of free electronsCurie/Langevin paramagnetism (~ 1/T)

Magnetic ordering: antiferromagnetic/Ferromagnetic

NMR, Neutron scattering, ESR, etc.

Atom or molecule: Na, NO, …?Free atoms and ions with a partially filled shell?Exchange interactions?

Hund Rules

The Hund rules as applied to electrons in a given shell of an atom affirm that electrons will occupy orbitals in such a way that the ground state is characterized by the following:1. The maximum value of the total spin S allowed by the exclusion principle2. The maximum value of the orbital angular momentum L consistent with this value of S3. The value of the total angular momentum J is equal to |L-S| when the shell is less than half full and to L+S

when the shell is more than half full. When the shell is just half full, the application of the first rule gives L = 0, so that J = S

Fe2+ 3d6

Lz = -2 -1 0 1 2

L = -2+ -1+0+1+2+ -2 = 2S = 1/2 * 4 = 2J = L + S = 4

2S1LJ

5D4

Kittel Introduction to solid state physics

Ce3+ 4f 1

3 2 1 0 -1 -2 -3

L = 3S = 1/2J = L - S = 5/2

2F5/2

Gd3+ 4f 7

L = 0S = 7/2J = S = 7/2

8S7/2

3 2 1 0 -1 -2 -3

Hund Rules



3 2 1 0 -1 -2 -3

L = S = J =

Exercise

2S1LJ

Dy3+ 4f 9

Hund Rules



Exercise

Hund Rules



Dy3+ 4f 9

3 2 1 0 -1 -2 -3

L = 5S = 5/2J = L+S = 15/2

6H15/2

3d and 4f electron basic level


d-electronsExperiments close to S

f-electronsExperiments close to J

Itinerant and localized

Magnetic properties of rare earth metals – R. J. Elliot

(CEF) > (L-S)

Crystalline Electric Field (CEF) spin-orbit coupling

(CEF) < (L-S)

3d-electrons 4f-electrons

Note for d-electrons – most of the case considering t2g and eg energy level splitting due to the octahedron environment. But, in general we have to consider actual local point symmetry.

4f-electron magnetism

By controlling S, L, J as well as point symmetry (CEF) we can get a wide variety of magnetic properties: eff, sat and anisotropy.

Effective moment and saturated magnetizationMagnetic susceptibility


Effective moment

In the low field region, the magnetization of a paramagnetic system is linear in H. For such low field,

dMdH

(T ) M (T )H

Magnetic moment vs fieldBrillouin function - paramagnet

(T ) M (T )H

Np2B

2

3kBT CT

p = g[J(J+1)]2


Fit with Curie-Weiss Law: the temperature dependence of the magnetic susceptibility in the paramagnetic state is given by

(T ) CT P

0

Plot inverse magnetic susceptibility

C Np2B

2

3kB

Determine p from the slop and compare to the free ion value

Gd3+ free ion value p = 7.94

from the slop 7.9 B/Gd3+

J. MMM 205 (1999) 27-52


Fit with Curie-Weiss Law: the temperature dependence of the magnetic susceptibility in the paramagnetic state is given by

(T ) CT P

0

0: basically temperature independent term that is the sum of core diamagnetism, Pauli paramagnetism, and Landau diamagnetism

Weiss temperature p

negative

Temperature

1/

J. MMM 205 (1999) 27-52

Effective moment and saturated magnetizationMagnetization isotherms - spin state of magnetic ions

Multiferroicity with coexisting isotropic and anisotropic spins in Ca3Co2−xMnxO6

PRB 89, 060404(R) (2014)

Mn4+ : 3d3 S = 3/2

Co2+ : 3d7 S = 3/2 or 1/2 ?

No evidence of low-spin to high-spin transition

Neutron – seeing ordered moment should be complementary

Anisotropy in magnetic susceptibility

J. MMM 205 (1999) 27-52

The anisotropy can be clearly seen in 1/

CEF anisotropy

NO CEF anisotropy Gd3+ or Eu2+

(s-state, L = 0)

For rare-earth with finite L, there will be CEF splitting that can give rise to significant anisotropy, depending on local point symmetry.

Exchange interaction - anisotropy

MagnetismExchange interaction J Magnetic ordering and quasi-particle

• Direct• Indirect

• RKKY• Kondo

Specific heatElectron and phonon contribution

Specific heat of a metal at low temperatures

c T T 3

Plot C/T vs T2 – extract and

: n(EF) density of state: Debye temperature

~0.7 mJ/mol K2

Specific heat of Cu matter ad methods at low temperature

Specific heat of CeAl3PRL 35, 1779 (1975)

~1620 mJ/mol K2

Heavy fermion- large !- large effective mass m*- 1000 times bigger than Cu

Note: estimate value - carefully consider all possible contributions in specific heat, especially magnetic and Schottky contributions

Specific heatElectron and phonon contribution

Heavy fermion superconductor

Specific heat jump at the Tc- related to the

The Cp jump - not related to any other contributions like electronic schottky, nuclear, magnetic fluctuations, and all strange stuffs…

PRL 43, 1892 (1979)

Specific heat of CeCu2Si2

Heavy fermion – Kondo effect

The electronic specific heat - measure of the electron effective mass

Phys. Rev. Lett. 94, 057201 (2005)

Relationship among thermodynamic and transport

Specific heatResistivitySusceptibilityThermal conductivityThermoelectric powerThermal expansion…

Fermi liquid: KW, Wilson number, and more…

Please make summary note just for yourself…

Anything deviates from conventional behavior?

EntropySpecific heat is intimately related to entropy

Tell us how much entropy is associated with a given state. For magnetic systems we need to use the magnetic specific heat.

Remove phonon contributions: often do this by subtracting a non-magnetic analogue.

S Cm

TdT

Cm = Cp(R) –Cp(Y, La, Lu)

Gd-compound, no CEF J(=S, L=0) = 7/2

Entropy associated with magnetic orderingRln(8)

Energy level splittingCEF effect – Schottky anomaly

C UT

kB( /T )2e/T

(1 e/T )2

Consider a two-level system

Nuclear: 1 ~ 100 mK splitting, often observed below 1K

Estimate effective mass Subtracting nuclear contribution by simply use ~ 1/T2

~1/T2

Energy level splittingCEF effect – Schottky anomaly

Given the Hund’s rule, groundstate multiplet J we would expect R ln(2J + 1) entropy associated with the magnetic state. For rare earths the spin-orbit coupling give rise to crystalline electric field (CEF) splitting.

Ce (J = 5/2) in cubic symmetry

2J +1 = 6 doublet

quartet

C UT

kBg1

g2

( /T )2e/T

1 g1

g2

e/T

22

PrAgSb2: no magnetic ordering at low temperatures – can be considered as a two level system

Specific heat at low temperatures and neutron at high temperatures

Phase transitions

We have now seen how (T), Cp(T), and (T) can be used to determine magnetic ordering temperatures (as well as many other useful parameters).

M.E. Fisher, Philos. Mag. 7 (1962) 1731. M.E. Fisher, J.S. Langer, Phys. Rev. Lett. 7 (1968) 665.M.P. Kawatra, et al., Phys. Rev. Lett. 23 (1969) 83. D.J.W. Geldart, T.G. Richard, Phys. Rev. B 12 (1975) 5175.

Cm

Phase transitions

We have now seen how (T), Cp(T), and (T) can be used to determine magnetic ordering temperatures (as well as many other useful parameters).

M.E. Fisher, Philos. Mag. 7 (1962) 1731. M.E. Fisher, J.S. Langer, Phys. Rev. Lett. 7 (1968) 665.M.P. Kawatra, et al., Phys. Rev. Lett. 23 (1969) 83. D.J.W. Geldart, T.G. Richard, Phys. Rev. B 12 (1975) 5175.

Bulk transition

Specific heat: good measure of bulk phase transition - superconductors

Poster: QMSS-18 Hyunsoo Kim (University of Maryland)H. Kim, et al., Philosophical Magazine, 2015Crystal growth and annealing study of fragile, non-bulk superconductivity in YFe2Ge2

Phase transitions

Jump in the specific heat!!!!

Guess from periodic table-tuning systems

Density of stateExchange interaction

Thank you!

Questions…

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