Preparation and Characterization Preparation and Characterization of of RealReal MaterialsMaterials

A CCMS Summer Lecture SeriesA CCMS Summer Lecture SeriesJuly 21 July 21 –– August 1, 2008August 1, 2008

Acknowledgements: Parts of these lectures Acknowledgements: Parts of these lectures were give at Seoul National University the were give at Seoul National University the week of July 7, 2008. Work at University of week of July 7, 2008. Work at University of Florida performed under the auspices of the Florida performed under the auspices of the US Dept. of Energy, contract no. DEUS Dept. of Energy, contract no. DE--FG02FG02--86ER45268. I have attempted to give proper 86ER45268. I have attempted to give proper citations on each slide citations on each slide –– for where these are for where these are lacking, please consult the author lacking, please consult the author –– G. R. G. R. Stewart.Stewart.

Preparation and Characterization Preparation and Characterization of of RealReal MaterialsMaterials

A CCMS Summer Lecture SeriesA CCMS Summer Lecture SeriesJuly 21 July 21 –– August 1, 2008August 1, 2008

00--d (small d (small –– a few 100 a few 100 ÅÅ -- particles, particles, molecular magnets)molecular magnets)11--d (carbon d (carbon nanotubesnanotubes/polymers/below a /polymers/below a dimensionality transition)dimensionality transition)22--d (films, 3d (films, 3--d systems with strong d systems with strong anisotropy)anisotropy)33--d (the rest)d (the rest)

Examples of 1Examples of 1--d Materials Prep. d Materials Prep. Univ. of FloridaUniv. of Florida

1D coordination polymer:[Me4N]3{[Mn(5-MeO-salen)][Nb6Cl12(CN)6]}Chem. Mater. 2007, 19, 2238-2246Zhou, et al. (Kevin Little, Dan Pajerowski, Mark Meisel –Physics U of F; Dan Talham – Chemistry U of F)

ClN

Mn

Nb

O

C

Examples of 1Examples of 1--d Materials d Materials Prep./Characterized Univ. of FloridaPrep./Characterized Univ. of Florida

(CH(CH33))22NHNH22CuClCuCl33, referred to as MCCL, a , referred to as MCCL, a system of onesystem of one--dimensional chains of copper dimensional chains of copper spins (spins (Meisel/TalhamMeisel/Talham))lowlow--dimensional (quasi 1dimensional (quasi 1--d) charged) charge--transfer transfer salts salts (EDO(EDO--TTF)TTF)22XX ((XX=PF=PF66 and AsFand AsF66))(Tanner group)(Tanner group)Quasi 1Quasi 1--d organic d organic superconductorssuperconductors (Hill (Hill group), group), antiferromagnetsantiferromagnets ((BaBa33MnMn22OO88--Andraka/Takano)Andraka/Takano)

Examples 1Examples 1--d Materials Prep. U of Fd Materials Prep. U of F

Single-Walled Carbon Nanotubes (Lecture July 25 by A. Rinzler)Zhihong Chen, Xu Du, Mao-Hua Du, C. Daniel Rancken, Hai-Ping Cheng, andAndrew G. Rinzler (Physics – U of F)NANO LETTERS 3, 1245-1249 (2003)

22--dd

Thin films of Fe Thin films of Fe –– the the ““Quantum DanceQuantum Dance””measurements on ultrathin ( measurements on ultrathin ( < < 100 100 ÅÅ thick) Fe films (thick) Fe films (HebardHebard group)group)

In 2In 2--d, when the thickness falls d, when the thickness falls to a certain value (e. g. to a certain value (e. g. 50 nm), the dc electrical conductivity has a peak as a function 50 nm), the dc electrical conductivity has a peak as a function of of thickness due to higher carrier mobility. Higher mobility is thickness due to higher carrier mobility. Higher mobility is expected in one dimensional confined paths where expected in one dimensional confined paths where transverse transverse scatteringscattering is reduced thus improving the mean free path of the is reduced thus improving the mean free path of the carriers.carriers.Quasi 2Quasi 2--d hid hi--TTcc superconductors like superconductors like BiSCOBiSCO

22--d character in hid character in hi--TTcc

Y Ba2 Cu3 O7

Cu Oxide planes

33--dd

Much work is still in 3Much work is still in 3--d; even a thin film is d; even a thin film is 33--dimensional as long as the thickness > dimensional as long as the thickness > mean free path/coherence length. Contrary mean free path/coherence length. Contrary wise, a 3wise, a 3--d material made small d material made small →→ 00--ddAlso, many Also, many ““quasi quasi –– 2d2d”” materials have materials have anisotropies that are < 10.anisotropies that are < 10.holehole--doped doped perovskiteperovskite manganitesmanganitesRERE11--xxCa/SrCa/SrxxMnMnOO33 ((BiswasBiswas) in film form) in film form

33He and He and 44He, solid HHe, solid H22

33He in He in aerogelaerogel: effects on the : effects on the superfluidsuperfluid (p(p--wave) condensate of disorder/restricted wave) condensate of disorder/restricted dimension (dimension (YoonseokYoonseok Lee group)Lee group)Solid HSolid H22 –– NMR in Sullivan groupNMR in Sullivan group

First Cautionary RemarkFirst Cautionary Remark

Compositional purity and subCompositional purity and sub--lattice disorder lattice disorder are very often critical.are very often critical.Examples: Examples:

--High vacuum needed to prevent oxidation of the High vacuum needed to prevent oxidation of the thin Fe film.thin Fe film.--In flux growth of single crystals, the flux is the In flux growth of single crystals, the flux is the obvious candidate for obvious candidate for impuritesimpurites snssns ((‘‘ssee ee nnext ext sslidelide’’))

Superconducting Transition Width in Superconducting Transition Width in fluxflux--growngrownvsvs arcarc--meltedmelted UBeUBe1313

C (J

/mol

-K)

1.0

2.0

T (K) 1.0

Flux grown Arc melted

Tc Data from H. R. Ott, et al.

The moral of the story The moral of the story –– just because itjust because it’’s a s a ““single crystalsingle crystal”” doesndoesn’’t mean itt mean it’’s perfect!s perfect!

Beware Hazards of SamplesBeware Hazards of SamplesSuperconducting HiSuperconducting Hi--TTcc TlTl compound (compound (““superconducting rat superconducting rat poisonpoison””))Asbestos like hazard in carbon Asbestos like hazard in carbon nanotubesnanotubes: : ““researchers researchers observed that observed that longerlonger nanotubesnanotubes behaved like asbestos, behaved like asbestos, provoking inflammation andprovoking inflammation and lesions in mice.lesions in mice.”” Associated Associated Press May 20 reporting on article in Nature. Press May 20 reporting on article in Nature. Nanotechnology Nanotechnology nanotubesnanotubes are already used in some are already used in some products, like tennis rackets, golf clubs or bicycle frames, products, like tennis rackets, golf clubs or bicycle frames, researchers say that the fibers appear to pose little risk researchers say that the fibers appear to pose little risk toto consumers. consumers. The new The new ‘‘highhigh’’ TTcc ArsenicArsenic--containing compoundscontaining compoundsOr, you can work really hard Or, you can work really hard –– e. g. OsOe. g. OsO44 is very toxic but is very toxic but only forms in the bulk above 670 K.only forms in the bulk above 670 K.

I. Always check for I. Always check for hazardoushazardousproperties of the materials/processes properties of the materials/processes being used for a sample being used for a sample ‘‘newnew’’ to you.to you.

II. Just because some element is II. Just because some element is harmless in its bulk form does harmless in its bulk form does not not

mean that in, e. g., the powdered form mean that in, e. g., the powdered form that it continues to be harmless. that it continues to be harmless. (simple example, flammability.)(simple example, flammability.)

III. Mixtures of ~ harmless components can III. Mixtures of ~ harmless components can be very dangerous (Se + acid be very dangerous (Se + acid →→ HH22Se)Se)

4 =4 = Very short exposure could cause death or major residual Very short exposure could cause death or major residual injury (Hinjury (H22Se exposure limit: 0.3 Se exposure limit: 0.3 ppmppm over an 8 hour period over an 8 hour period

vsvs 140 140 ppmppm for HCN.)for HCN.)

4 = will burn readily. Flash point below 234 = will burn readily. Flash point below 23°°C C

Safety advisory shield for H2Se

blue: health red: flammability yellow: reactivity

4 is the highest (worst risk) number

Addition of alcohols to highly Addition of alcohols to highly concentrated hydrogen peroxide concentrated hydrogen peroxide

forms powerful explosives.forms powerful explosives.

(Brief) History of Sample Prep(Brief) History of Sample PrepGold is found as a free element (i. e. not a compound) in natureGold is found as a free element (i. e. not a compound) in natureand is associated with quartz, pyrite and other minerals. Gold hand is associated with quartz, pyrite and other minerals. Gold has as been known since ~ 6000 BC.been known since ~ 6000 BC.To extract gold from pyrite compounds, they are sometimes To extract gold from pyrite compounds, they are sometimes treated in a proprietary chlorine process. The roasted pyrites atreated in a proprietary chlorine process. The roasted pyrites are re moistened with water and impregnated with chlorine gas. Gold moistened with water and impregnated with chlorine gas. Gold chloride is then leached out with water, and the gold is precipichloride is then leached out with water, and the gold is precipitated tated by reduction with ferrous sulfate or solid charcoal. Bromine watby reduction with ferrous sulfate or solid charcoal. Bromine water er is sometimes used instead of chlorine. The gold obtained is thenis sometimes used instead of chlorine. The gold obtained is thenprocessed into bullion. The gold bullion is then refined. If theprocessed into bullion. The gold bullion is then refined. If the gold gold contains copper, it is removed by oxidizing it with borax and contains copper, it is removed by oxidizing it with borax and nitric acid.nitric acid. (Also cyanide leaching, mercury amalgamation)(Also cyanide leaching, mercury amalgamation)

((http://www.newton.dep.anl.gov/askasci/gen01/gen01415.htmhttp://www.newton.dep.anl.gov/askasci/gen01/gen01415.htm) )

* Ag is etched out of * Ag is etched out of ‘‘granulatedgranulated’’ Au Au –– achieved by putting molten achieved by putting molten gold into cold water gold into cold water –– via HNOvia HNO3 3 (starting in the 1200(starting in the 1200’’s)s)

Removal of noble metal Pt from Au (discovered by the Removal of noble metal Pt from Au (discovered by the Spanish in the 1500Spanish in the 1500’’s Pt was considered a worthless s Pt was considered a worthless impurity in Au) . Thus, purity has long played a role.impurity in Au) . Thus, purity has long played a role.

Purities of the Elements (a quick look)Purities of the Elements (a quick look)

NIST gold is 99.9999+%NIST gold is 99.9999+%GeGe after zone refining, 1/10after zone refining, 1/101010 impurityimpurity33He can have better than 1/10He can have better than 1/105050 purity purity ––YoonseokYoonseok LeeLee

First metal alloy:First metal alloy:Bronze (88%Cu,12%Sn) Bronze (88%Cu,12%Sn) –– 4000 BC4000 BC

Bronze was replaced by Iron implements only Bronze was replaced by Iron implements only after Steel (Fe alloyed with 0.2after Steel (Fe alloyed with 0.2--2 wt% or 12 wt% or 1--10 10 at% C) was invented ~ 1400 BCat% C) was invented ~ 1400 BCIron is extracted from FeIron is extracted from Fe22OO33 by removing the oxygen by combining it by removing the oxygen by combining it with a preferred chemical partner such as carbon. This process, with a preferred chemical partner such as carbon. This process, known as known as smeltingsmelting, was first applied to metals with lower , was first applied to metals with lower TTmeltmelt like Cu (~1000like Cu (~1000 °°C), C), SnSn (250(250 °°C). Cast ironC). Cast iron——ironiron alloyed with greater than 1.7% carbonalloyed with greater than 1.7% carbon——melts at around 1370melts at around 1370 °°C. All of these temperatures could be reached with C. All of these temperatures could be reached with ancient methods that have been used for at least 6000 years. Sinancient methods that have been used for at least 6000 years. Since the ce the oxidation rate itself increases rapidly beyond 800oxidation rate itself increases rapidly beyond 800 °°C, it is important that C, it is important that smelting take place in a lowsmelting take place in a low--oxygen environment. Unlike copper and tin, oxygen environment. Unlike copper and tin, liquid iron dissolves carbon quite readily, so that smelting resliquid iron dissolves carbon quite readily, so that smelting results in an ults in an alloy containing too much carbon to be called steel. (Wikipediaalloy containing too much carbon to be called steel. (Wikipedia))

Different FeDifferent Fe--C alloys C alloys –– Phase Diagram I (citation: Phase Diagram I (citation: MassalskiMassalski, ASM), ASM)

The maximum solubility of carbon in iron (in austenite) is 2.14The maximum solubility of carbon in iron (in austenite) is 2.14 wt % at 1149wt % at 1149 ooCC

by atom %15 25

bcc

fcc

Carbon and other elements (Cr, Carbon and other elements (Cr, MnMn, V) act as a , V) act as a hardening agent, preventing dislocations in the iron hardening agent, preventing dislocations in the iron atom crystal lattice from sliding past one another. atom crystal lattice from sliding past one another. The heat treatment process for most steels involves The heat treatment process for most steels involves heating the alloy until austenite forms, then quenching heating the alloy until austenite forms, then quenching the hot metal in water or oil, cooling it so rapidly that the hot metal in water or oil, cooling it so rapidly that the transformation to the transformation to ferriteferrite ((αα--Fe) Fe) or or pearlitepearlite does not does not have time to take place. have time to take place. CementiteCementite forms in regions of higher carbon content forms in regions of higher carbon content while other areas revert to ferrite around it. Selfwhile other areas revert to ferrite around it. Self--reinforcing patterns often emerge during this process, reinforcing patterns often emerge during this process, leading to a patterned layering known as leading to a patterned layering known as pearlitepearlite(Fe3C:6.33Fe) due to its pearl(Fe3C:6.33Fe) due to its pearl--like appearance. like appearance. (http://(http://en.wikipedia.orgen.wikipedia.org/wiki/Steel) /wiki/Steel)

What happens to an XWhat happens to an X--ray pattern when a cubic ray pattern when a cubic system (e.g. system (e.g. fccfcc Austenite FeAustenite Fe--C steel) undergoes a C steel) undergoes a

tetragonal distortion (to tetragonal distortion (to bctbct MartensiteMartensite)?)?

202

What is What is MartensiteMartensite? (preview of ? (preview of Lecture II)Lecture II)

MartensiteMartensite, named after the German metallurgist Adolf , named after the German metallurgist Adolf Martens (1850Martens (1850--1914), is any crystal structure that is formed 1914), is any crystal structure that is formed by by displacivedisplacive transformation, as opposed to much slower transformation, as opposed to much slower diffusive transformations. "diffusive transformations. "MartensiteMartensite" most commonly " most commonly refers to a very hard form of steel crystalline structure (the refers to a very hard form of steel crystalline structure (the alloy of iron and carbon) important in some tool steels. The alloy of iron and carbon) important in some tool steels. The bctbct martensitemartensite is formed by rapid cooling (quenching) of is formed by rapid cooling (quenching) of fccfccaustenite which traps carbon atoms that do not have time to austenite which traps carbon atoms that do not have time to diffuse out of the crystal structure. Thus, the phase that diffuse out of the crystal structure. Thus, the phase that should form (Feshould form (Fe33C) is suppressed and a structure C) is suppressed and a structure not shownnot shownin the binary phase diagram (which depicts thermodynamic in the binary phase diagram (which depicts thermodynamic equilibrium) is produced.equilibrium) is produced.Wikipedia (?)Wikipedia (?)

Example of Example of MartensiticMartensitic Transformation:Transformation:AA--15 (cubic) V15 (cubic) V33Si Si →→ tetragonal Vtetragonal V33SiSi

Tm=21.2 K

G. R. Stewart, et al.

The effect of this The effect of this ‘‘martensiticmartensitic’’transformation on Vtransformation on V33Si, its Si, its

superconductivity, the lattice superconductivity, the lattice softening in the elastic constants softening in the elastic constants in the cin the c--axis direction (caxis direction (c4444 and cand c6666

have to have to →→ 0 at the 0 at the transformation) has been transformation) has been

extensivelyextensively studied.studied.

Remember, this kind of Remember, this kind of transformation can be transformation can be

metastablemetastable

G. R. Stewart, et al.

Phase DiagramsPhase Diagrams––WeWe’’ve seen the one below for Feve seen the one below for Fe--C, what about another one to analyze? C, what about another one to analyze?

The maximum solubility of carbon in iron (in austenite) is 2.14The maximum solubility of carbon in iron (in austenite) is 2.14 wt % at 1149wt % at 1149 ooCC

by atom %15 25

bcc

fcc

Phase Diagram #2 for Tin/LeadPhase Diagram #2 for Tin/Lead

wt. %

This 63-37 composition is also known as the eutectic point of the alloy, where the alloy behaves like a pure metal having a single melting (solidification) temperature (176ºC / 349ºF). This is a good operational feature. Once the solder melts on application of heat, it solidifies immediately on removal of heat, without going through a pasty stage like other alloys. This allows for predictable soldering and fast cycle times.

Phase Diagram #3 CopperPhase Diagram #3 Copper--Zinc (Brass)Zinc (Brass)

‘cartridge’ brass ‘Muntz’ or ‘α-β’ brass

‘brazing’ solder

Properties at room T depend on cooling rate!

Phase Diagram #3 CopperPhase Diagram #3 Copper--Zinc (Brass)Zinc (Brass)

‘cartridge’ brass ‘Muntz’ or ‘α-β’ brass

Properties at room T depend on cooling rate!

‘brazing’ solder‘brazing’ solder

(fcc)

(bcc)

(rhombohedralhexagonal)

Phase Diagram #4: Phase Diagram #4: CopperCopper--GoldGold

Massalski, ASM binary phase diagrams

Phase Diagram #4: CopperPhase Diagram #4: Copper--GoldGold

The occurrence of compound-like phases at the AuCu and the AuCu3 stoichiometriccompositions was first observed by [15Kur], who employed thermal analysis, hardness, and X-ray measurements. [23Bail], [23Bai2], and [23Bai3] associated the occurrence of atomic ordering with these compounds on the basis of observed X-ray diffraction lines. [36Joh] discovered an additional order-order transformation in AuCu at higher temperatures, in which an orthorhombicAuCu(II) phase forms from the tetragonal AuCu(I) phase. Prior to this, AuCu(I) was thought to transform directly to the disordered fcc phase (AuCu(D)) at higher temperatures.

Phase Diagram #5: The Phase Diagram #5: The ActinidesActinides

J. L. Smith

Sample PrepSample Prep--an Examplean ExampleA practical example of sample prep from new work A practical example of sample prep from new work on making the iron arsenide superconductor on making the iron arsenide superconductor BaBa0.60.6KK0.40.4FeFe22AsAs22 from from SnSn flux growth. Advantages: flux growth. Advantages: get single crystals for the first time of these new get single crystals for the first time of these new FeAsFeAs‘‘high high TTcc’’ superconductorssuperconductors; avoid ; avoid somewhatsomewhat the the vapor vapor pressure problems of K and As (pressure problems of K and As (PPvaporvapor=10 =10 mm at 443 and 437 mm at 443 and 437 ooCC respectively)respectively)Reference: Reference: ““Anisotropic thermodynamic and transport properties of singleAnisotropic thermodynamic and transport properties of singlecrystalline (Ba1crystalline (Ba1−−xKx)Fe2As2 (x = 0 and 0.45)xKx)Fe2As2 (x = 0 and 0.45)””..by N. Ni, S. L. by N. Ni, S. L. BudBud’’koko, A. , A. KreyssigKreyssig, S. Nandi, G. E. , S. Nandi, G. E. RustanRustan, A. I. Goldman, S. , A. I. Goldman, S. Gupta, J. D. Corbett, A. Gupta, J. D. Corbett, A. KracherKracher, and P. C. Canfield, and P. C. CanfieldarXiv:0806.1874v1arXiv:0806.1874v1

First thing to do: look at binary phase First thing to do: look at binary phase diagrams between diagrams between BaBa, K, Fe, As and , K, Fe, As and SnSn

Note the good solubility of Ba in Sn at relatively low (~ 400 oC) temp.

Massalski, binary phase diagrams, ASM

First thing to do: look at binary phase First thing to do: look at binary phase diagrams between diagrams between BaBa, , KK, Fe, As and , Fe, As and SnSn

Note the ok solubility of K in Sn at relatively low temp.

Massalski, binary phasdiagrams, ASM

First thing to do: look at binary phase First thing to do: look at binary phase diagrams between diagrams between BaBa, , KK, , FeFe, As and , As and SnSn

Note the poor solubility of Fe in Sn at low temp., need 800 oC

Massalski, binary phase diagrams, ASM

First thing to do: look at binary phase First thing to do: look at binary phase diagrams between diagrams between BaBa, , KK, , FeFe, , AsAs and and SnSn

Note the good solubility of As in Sn at low temp.

Massalski, binary phase diagrams, ASM

Why was Why was SnSn chosen instead of Cu, chosen instead of Cu, CdCd, Al, , Al, GaGa, , In, In, PbPb, , SbSb, Bi, . . .? , Bi, . . .?

In order to answer this, would need to go In order to answer this, would need to go look at the binary phase diagrams between look at the binary phase diagrams between BaBa, K, Fe, As and each of these other possible , K, Fe, As and each of these other possible fluxes (all of which are in common use for fluxes (all of which are in common use for preparation of various single crystals.) preparation of various single crystals.) Another consideration is the ease of Another consideration is the ease of removing the metal flux (removing the metal flux (TTmeltmelt).).

LetLet’’s go look !s go look !

CharacterizationCharacterization

See, e. g., Prof. See, e. g., Prof. BiswasBiswas’’ss sitesitehttp://www.phys.ufl.edu/nanofab/equipmenhttp://www.phys.ufl.edu/nanofab/equipment.htmlt.htmlhttp://maic.mse.ufl.edu/http://maic.mse.ufl.edu/

Outline:Outline:I. via XI. via X--raysraysII. via electron microprobeII. via electron microprobeIII. via SEMIII. via SEM

Short Overview of Sample Short Overview of Sample Characterization byCharacterization by

I. I. XX--ray ray DiffractometryDiffractometry

Various resources for this:Various resources for this:http://www.phys.ufl.edu/courses/phy4803L/grhttp://www.phys.ufl.edu/courses/phy4803L/group_I/x_ray/xray.pdfoup_I/x_ray/xray.pdfC. C. KittelKittel, , Introduction to Solid State PhysicsIntroduction to Solid State PhysicsB. D. B. D. CullityCullity, , Elements of Elements of XrayXray DiffractionDiffraction

Sketch of Bragg Reflection of Incident Sketch of Bragg Reflection of Incident

Radiation of WavelengthRadiation of Wavelength λλnnλλ = 2= 2d d sinsinΘΘ (eq. 1)(eq. 1)

λ

Diffraction from various crystal planesDiffraction from various crystal planes

The The spacingsspacings for all possible lattice planes in for all possible lattice planes in the sc lattice can be represented bythe sc lattice can be represented bydd((hklhkl) = ) = n an a00/(/(hh22 + + kk22 + + ll22))1/21/2 where where hklhkl are are integers {called the integers {called the Miller indicesMiller indices} and } and n n is an is an integer. Together with eq. 1integer. Together with eq. 1this leads to this leads to λλ = 2= 2aa00 sinsinΘΘ//(h(h22 + + kk22 + + ll22))1/2 1/2

where awhere a00 is the length of the cube edgeis the length of the cube edge

Diffraction from various crystal planes;Diffraction from various crystal planes;Miller indicesMiller indices

x

y

Specify the orientation of a plane by the indices determined via:

1. Find the intercepts on the axes in terms of the lattice constants a1, a2, a3

2. Take the reciprocals of these numbers and then reduce to three integers having the same ratio, usually the smallest 3 integers. The resulting Miller indices (hkl) are called the index of the plane.

∞,1, ∞ → (0,1,0) 1,1, ∞ → (1,1,0) 2,1, ∞ → 1/2,1,0 → (1,2,0)

In cubic crystals, the direction [hkl] is ⊥ to a plane with Miller indices (hkl)

}a2

{a1

www.stevens.edu

x

y

z

A crystal structure is described by a lattice, e. A crystal structure is described by a lattice, e. g. a simple cubic lattice:g. a simple cubic lattice:

plus a set of atoms (called the ‘basis’) attached to each lattice point. Thus, for a bcc lattice, the basis is two atoms, one at 0,0,0 and one at ½, ½, ½ (uj , vj , wj).A crystal thus consists of a basis of atoms at each lattice point. This can be expressed:lattice + basis = crystal

(Bravais lattice) AND INTERAXIAL ANGLESCUBIC three equal axes, Au, Cu,NaCl,Si,GaAs(Simple cubic, three right anglesBody-centered cubic, a = b = c, α = β = γ = 90o

Face-centered cubic)TETRAGONAL Two of the three axes equal, In, TiO2(Simple tetragonal, three right anglesBody centered tetragonal) a = b ≠ c , α = β = γ = 90o

ORTHORHOMBIC Three unequal axes, Ga, Fe3C(Simple orthorhombic, three right anglesBody centered orthorhombic, a ≠ b ≠ c , α = β = γ = 90o

Base-centered orthorhombic,Face-centered orthorhombic)RHOMBOHEDRAL Three equal axes equally inclined , Hg, Bi(Simple rhombohedral) three equal angles ≠ 90a = b = c, α = β = γ HEXAGONAL Two equal axes at 120o, Zn. Mg(Simple hexagonal) third axis at right anglesa = b ≠ c, α = β =900, γ =1200

MONOCLINIC Three unequal axes, KClO3(Simple monoclinic, one pair of axes not at 90o

Base- centered monoclinic) a ≠ b ≠ c α = γ =90o , β ≠ 90o

TRICLINIC Three unequal axes, Al2SiO5three unequal anglesa ≠ b ≠ c, α ≠ β ≠ γ

Not all reflections are of equal intensity. AsNot all reflections are of equal intensity. Ashklhkl get large, the density of atoms in eachget large, the density of atoms in eachplane decreases and the corresponding peak gets plane decreases and the corresponding peak gets weaker.weaker.

(0,1,0) (1,1,0) (1,2,0)

Because of the j scattering sites within each Because of the j scattering sites within each cell, the intensity of the peak will be cell, the intensity of the peak will be proportional to the square of the magnitude proportional to the square of the magnitude of the of the crystal structure factorcrystal structure factorFF((hklhkl) = ) = ΣΣjj ffjj eexpxp{{22ππii((huhujj++kvkvjj++lwlwjj ))}}The The atomic form factor fatomic form factor fjj above depends onabove depends on the type of atom the type of atom at the site at the site uujj , , vvj j , , wwjj . It also. It also varies with varies with ΘΘ. For. For ΘΘ = 0, it is = 0, it is approximatelyapproximately proportional to the number of electrons in proportional to the number of electrons in thethe atom. For larger reatom. For larger reflflection angles, it decreasesection angles, it decreases due todue tointerinter-- ference eference effffects of scattering fromects of scattering from didifffferent parts of the erent parts of the atom.atom.

Consider a bodyConsider a body--centered cubic crystalcentered cubic crystalwith a two atom basis. Using the sc unit cellwith a two atom basis. Using the sc unit cellwe would have to include one atom at the scwe would have to include one atom at the sclattice site (0lattice site (0, , 00, , 0) and an 0) and an identicalidentical atom atatom atthe bodythe body--centered site (centered site (½½, , ½½, , ½½). The crystal). The crystalstructure factor structure factor ΣΣjj ffjj eexpxp{{22ππii((huhujj++kvkvjj++lwlwjj ))} }

becomes:becomes:FFbccbcc((hklhkl) = ) = ff(1 + (1 + exp{iexp{iππ((hh++kk++ll)}))})

This shows that the crystal structure factorThis shows that the crystal structure factorfor the bcc lattice is nonfor the bcc lattice is non--zero (and equals 2zero (and equals 2ff))only if only if h h + + k k + + l l is even. Thus peaks withis even. Thus peaks withh h + + k k + + l l odd would be entirely missing.odd would be entirely missing.

If have a bcc (only If have a bcc (only h+k+lh+k+l=even) lattice, then =even) lattice, then first lines are (110), (200), (211), (220), (310), first lines are (110), (200), (211), (220), (310), (222), (321), (400), (222), (321), (400),

So the ratio of So the ratio of sinsin22ΘΘnn/sin/sin22ΘΘ11 values would be values would be the ratio of the successive hthe ratio of the successive h22+k+k22+l+l22 values to values to

the lowest angle hthe lowest angle h22+k+k22+l+l22 value (e. g. value (e. g. 2222/(1/(122+1+122)=2, (2)=2, (222+1+122+1+122)/(1)/(122+1+122)= 3, etc. )= 3, etc. to give to give 2,3,4,5,62,3,4,5,6, , ……

Thus, if you are Thus, if you are xrayingxraying an unknown compound, and you detect an unknown compound, and you detect this relationship between the this relationship between the sinsin22ΘΘ values for the diffraction values for the diffraction pattern, you can conclude that you have a bcc lattice (see pattern, you can conclude that you have a bcc lattice (see discussion of discussion of fccfcc following).following).

Consider next a Consider next a fccfcc cubic crystal with a four atom cubic crystal with a four atom basis. Using the sc unit cell we would have to include basis. Using the sc unit cell we would have to include one atom at the sc lattice site (0one atom at the sc lattice site (0, , 00, , 0) and identical 0) and identical atoms at the faceatoms at the face--centered sites (0,centered sites (0,½½,,½½), (), (½½,0,,0,½½), and ), and ((½½,,½½,0). The crystal structure factor becomes: ,0). The crystal structure factor becomes: FFfccfcc((hklhkl)=)=

ff((1+1+exp{iexp{iππ((kk++ll)}+)}+exp{iexp{iππ((hh++ll)}+)}+exp{iexp{iππ((h+kh+k)})}))This shows that the crystal structure factorThis shows that the crystal structure factorfor the for the fccfcc lattice is nonlattice is non--zero (and equals 4zero (and equals 4ff))only if the only if the h,k,lh,k,l are all even or all oddare all even or all odd (i. e., unlike the bcc (i. e., unlike the bcc

case this is not a restriction on the case this is not a restriction on the sumsum of of h+k+lh+k+l, but , but on the values themselves.) on the values themselves.)

Try it and see!Try it and see!

If have If have fccfcc ⇒⇒ hklhkl are either all even or all odd, so can are either all even or all odd, so can have (111), (200) {all the permutations like (020) have have (111), (200) {all the permutations like (020) have the same diffraction angle, i. e. they all lie atop one the same diffraction angle, i. e. they all lie atop one another}, (220), (311), (222), (400), (331), (420), . . . . another}, (220), (311), (222), (400), (331), (420), . . . . where these where these hklhkl are ordered by the lowest size of are ordered by the lowest size of hh22

+ + kk22 + + ll22 first, therefore (111) is a lower angle first, therefore (111) is a lower angle xrayxrayline than (200), similarly (311) is lower than (222).line than (200), similarly (311) is lower than (222).Now take sinNow take sin22ΘΘ =(=(λλ/2a/2a00))22**(h(h22 + + kk22 + + ll22) for the ) for the measured lines, So the ratio of successive measured lines, So the ratio of successive sinsin22ΘΘnn/sin/sin2 2

ΘΘ11 values would bevalues would be 4/34/3 (=2(=222/(1/(122+1+122+1+122), ), 8/38/3(=(2(=(222+2+222)/(1)/(122+1+122+1+122)), )), 11/311/3, , 12/3 = 412/3 = 4, etc. , etc.

Selection rules for the Miller indices

hh + 2+ 2kk = 3= 3nn for odd for odd llll even, even, hh + 2+ 2kk 33nnHexagonal close Hexagonal close packedpackedTriangular latticeTriangular lattice

above, or even & above, or even & hh++kk++ll 4n4n

all: odd, or even & all: odd, or even & hh++kk++ll = 4n= 4nDiamondDiamond, , SiSi, , GeGeDiamond F.C.C.Diamond F.C.C.

hh, , kk, , ll mixed odd and mixed odd and evenevenhh, , kk, , ll all odd or all evenall odd or all evenGoldGold, , NaClNaCl, , Zinc Zinc

blendeblendeFaceFace--centered centered cubiccubic

hh + + kk + + ll oddoddhh + + kk + + ll evenevenBodyBody--centered cubic centered cubic (e. g. Mo)(e. g. Mo)

BodyBody--centered centered cubiccubic

NoneNoneAny Any hh, , kk, , llSimple cubicSimple cubicSimple cubicSimple cubic

Forbidden reflectionsForbidden reflectionsAllowed reflectionsAllowed reflectionsExample Example compoundscompoundsBravaisBravais latticelattice

http://en.wikipedia.org/wiki/Bragg_diffraction

Flux Growth Single CrystalsFlux Growth Single Crystals

Flux Growth Single Crystals Flux Growth Single Crystals –– in the Ovenin the Oven

Work at Seoul National University by YoonseokOh, et al.

Note the interesting arithmetic fact that you cannot get the sum of three integers h2+k2+l2 to equal 7, 15, 23 … - see the simple cubic (sc) case.

http://home.hiram.edu/www/physics/PHYS380/Debye_Scherrer.pdf

h2+k2+l2=

Example of xExample of x--ray (Cu)ray (Cu)

Example of xExample of x--ray (Cu)ray (Cu)Powder diffraction pattern for copper metal shows reflections atPowder diffraction pattern for copper metal shows reflections at 22ΘΘ = = 43.3543.35°°,, 50.5050.50°°, 74.20, 74.20°°, 90.00, 90.00°°, 95.25, 95.25°°, 117.05, 117.05°°, 136.50, 136.50°°, 137.20, 137.20°°, 144.70, 144.70°°and 145.70and 145.70°°. For this pattern, . For this pattern, CuKCuKαα--radiation was used (radiation was used (λλ=1.54178 =1.54178 ÅÅ). ). (When we get to the answer, the lattice constant of copper is kn(When we get to the answer, the lattice constant of copper is known to lie own to lie around 3.6 around 3.6 ÅÅ).).Calculate the corresponding values of sinCalculate the corresponding values of sin22ΘΘ : : 0.1364130.136413, 0.181961, , 0.181961, 0.3638599, 0.5, 0.545751, 0.727384, 0.862687, 0.866865, 0.9080690.3638599, 0.5, 0.545751, 0.727384, 0.862687, 0.866865, 0.908069, 0.913049, 0.913049Now, calculate the ratios of these values to the Now, calculate the ratios of these values to the first value, first value, sinsin22ΘΘ11 : : 0.181961/ 0.181961/ 0.136413 0.136413 ≈≈ 4/3 (1.3330), 8/3 (2.6673), 11/3 (3.6653), 4 (4.0007), 4/3 (1.3330), 8/3 (2.6673), 11/3 (3.6653), 4 (4.0007), 16/3 (5.3322), . . . 16/3 (5.3322), . . .

so this ratio of successive so this ratio of successive sinsin22ΘΘ (proportional to (proportional to (h(h22 + + kk22 + + ll22)) )) tells us tells us (without benefit of knowing a(without benefit of knowing a00) that Cu has the ) that Cu has the fccfcc structure with structure with (calculated from each line) a(calculated from each line) a00= = λλ(h(h22 + + kk22 + + ll22))1/21/2/(2sin/(2sinΘΘ)=3.6151 )=3.6151 ÅÅ, 3.6144 , 3.6144 ÅÅ, 3.6147 , 3.6147 ÅÅ, 3.6158 , 3.6158 ÅÅ, , ……. Systematic errors are smaller as . Systematic errors are smaller as Θ→Θ→9090oo Nelson Nelson ReilleyReilley extrapolation function, see:extrapolation function, see:http://www.iucr.org/iucrhttp://www.iucr.org/iucr--top/comm/cteach/pamphlets/16/node8.htmltop/comm/cteach/pamphlets/16/node8.html

‘‘MysteryMystery’’ xrayxray22ΘΘ=38.3=38.3oo 22ΘΘ=44.4=44.4oo 22ΘΘ=64.9=64.9oo 22ΘΘ=77.6=77.6oo 22ΘΘ=81.7=81.7oo 22ΘΘ=98.1=98.1oo 22ΘΘ=111.1=111.1oo

22ΘΘ=115.4=115.4oo 22ΘΘ=136=136oo

Calculate as an exercise whether this is bcc or fcc and what the lattice parameter is. Assume λ=1.5418 Å

Calculate sinCalculate sin22ΘΘ values:values:0.1076118, 0.1427637, 0.287900290.1076118, 0.1427637, 0.287900290.39263234, 0.4278219, 0.57045062,0.39263234, 0.4278219, 0.57045062,0.6799984, 0.71446757, 0.8596700.6799984, 0.71446757, 0.859670

Calculate ratio of these values to the first one: Calculate ratio of these values to the first one: 1.327 (~4/3), 2.675 (~8/3), 3.649 (~11/3), 1.327 (~4/3), 2.675 (~8/3), 3.649 (~11/3), ……looks looks fcfc

(errors here are due to my inaccurate reading of a (errors here are due to my inaccurate reading of a published published xrayxray pattern.)pattern.)

aa00=4.070 =4.070 ÅÅ, 4.081 , 4.081 ÅÅ, 4.064 , 4.064 ÅÅ, 4.081 , 4.081 ÅÅ, , ……, , 4.075 4.075 ÅÅ, 4.079 , 4.079 ÅÅ, 4.073 , 4.073 ÅÅ (note how the error in reading the(note how the error in reading theangle angle ΘΘ at higher angles has a decreasing effect on the scatteat higher angles has a decreasing effect on the scattein ain a00 -- sin sin ΘΘ becomes flat becomes flat vsvs ΘΘ) (Au is ) (Au is fccfcc, a, a00=4.0782 =4.0782 ÅÅ))

Comparison of Au and Cu xray (Cu xraycompressed to overlay the Au at low angles)

How does a bcc xray look?Moly More Umph (BCC)

-50

0

50

100

150

200

250

300

350

400

0 20 40 60 80 100 120 140

2 Theta

Inte

nsity

Series1

http://www.eng.uc.edu

Peak 2Θn Intensity sin2 Θn hkl ΔΘ(Θn-Θn-1)sin2 Θ1

40.44 347 - 101 -58.32 312 1.99 200 8.9 73.4 366 2.99 211 7.587.96 158 4.04 202 7.3101.88 245 5.05 310 7.0116.32 218 6.04 222 7.2

Note that, as discussed, h+k+l is always even and that h2+k2+l2 (2,4,6,8,10,12,…for the lines shown) is both proportional to sin2Θ and evenly spaced. So the appearance of the diffraction pattern is like a ‘fingerprint’ for the structure. For bcc, the evenly spaced sin2Θcorresponds to a gradual progression in Θ

Diamond (two inter-penetrating fcc lattices) h2+k2+l2=3,8,11,16

http://home.hiram.edu/www/physics/PHYS380/Debye_Scherrer.pdf

Other lattice symmetries and Other lattice symmetries and hklhkl

Tetragonal: 1/dTetragonal: 1/d22 = (h= (h22+k+k22)/a)/a22 + l+ l22/c/c22

Hexagonal: 1/dHexagonal: 1/d22 = = (4/3)(h(4/3)(h22+hk+k+hk+k22)/a)/a22

+ l+ l22/c/c22

Cullity

Hexagonal ZnHexagonal Zn

SEM exampleSEM example--small particles trapped on small particles trapped on a filtera filter

G. R. Stewart

Electron microprobeElectron microprobe

An An electron microprobeelectron microprobe (EMP), also known as an (EMP), also known as an electron electron probe probe microanalysermicroanalyser (EPMA) is an analytical tool used to (EPMA) is an analytical tool used to nonnon--destructively determine the chemical composition of destructively determine the chemical composition of small volumes of solid materials. It works similarly to a small volumes of solid materials. It works similarly to a scanning electron microscope, in which the sample is scanning electron microscope, in which the sample is bombarded with an electron beam and the resulting excited bombarded with an electron beam and the resulting excited core electron core electron xraysxrays that come from the sample are collected. that come from the sample are collected. This enables the elements over the beam diameter (~ 1 This enables the elements over the beam diameter (~ 1 μμ) to ) to be quantitatively be quantitatively analysedanalysed at levels as low as 100 at levels as low as 100 ppmppm..This is done in MAIC with a This is done in MAIC with a EPMA JEOL SUPERPROBE EPMA JEOL SUPERPROBE 733733

wikipediawikipedia

Example of Electron Microprobe Analysis of Example of Electron Microprobe Analysis of a Samplea Sample

In CePtIn CePt33Si, which is superconducting at Si, which is superconducting at TTcc=0.7 K, sometime =0.7 K, sometime a second transition was seen in the specific heat. Annealing a second transition was seen in the specific heat. Annealing increased the size of this transition.increased the size of this transition.

J. S. Kim, et al., Univ. Florida

Was this second transition Was this second transition ‘‘intrinsicintrinsic’’ to to CePtCePt33Si or is it a second phase?Si or is it a second phase?

Do Do metallographymetallography first. What is first. What is metallographymetallography??

MetallographersMetallographers examine polished cross sections of specimens examine polished cross sections of specimens from appropriate locations to determine the grain size, from appropriate locations to determine the grain size, presence and distribution of presence and distribution of second phasesecond phase. Specimens are . Specimens are generally viewed in the asgenerally viewed in the as--polished condition first using polished condition first using brightfieldbrightfield illumination to observe those constituents that illumination to observe those constituents that have a natural color reflectivity difference from the bulk of have a natural color reflectivity difference from the bulk of the metal. This procedure is commonly used to examine the metal. This procedure is commonly used to examine intermetallicintermetallic compounds, . . . Can also use polarized light.compounds, . . . Can also use polarized light.

MetallographyMetallography Picture of Grain Picture of Grain Structure of Rhenium MetalStructure of Rhenium Metal

Result of Result of MetallographyMetallography of of CeCe--poor CePt3Si:poor CePt3Si:

Lighter regions are Lighter regions are second phasesecond phase, with, withcomposition (determined composition (determined by by electron microprobeelectron microprobe) )

~ ~

CeCe : Pt : Si: Pt : Si1 : 7 : 3.4 1 : 7 : 3.4 oror2 :14: ~72 :14: ~7 !!!

J. S. Kim, et al.

And then the And then the ““a ha!a ha!”” moment: moment: TheThe Second Phase Second Phase is is notnot CeCe22PtPt1414SiSi77, , itit’’s the well known, why didns the well known, why didn’’t I t I

think of that, Cethink of that, Ce22PtPt1515SiSi77****

!!!!!!

**Actually, the composition is Ce3Pt23Si11

Conclusion to the microprobe Conclusion to the microprobe detective workdetective work

CeCe22PtPt1515SiSi77, according to further investigation by us, , according to further investigation by us, is an is an antiferromagnetantiferromagnet at 0.4 K with a HUGE at 0.4 K with a HUGE ∆∆C/T C/T

of 22000 of 22000 mJmJ/ Ce/ Ce--moleKmoleK22 ––

Certainly large enough to explain the second Certainly large enough to explain the second ΔΔCC22anomaly at 0.4 K in CePtanomaly at 0.4 K in CePt33Si that occurs with Si that occurs with

annealing (< 1% annealing (< 1% CeCe loss)loss)

XrayXray diffraction to probe structure diffraction to probe structure and impurity phases: detective work and impurity phases: detective work

IIII

The cubic AThe cubic A--15 structure materials (the high 15 structure materials (the high TTcccompounds compounds beforebefore Bednorz and Muller, 1986) like Bednorz and Muller, 1986) like

NbNb33SnSn (used in magnets that have H > 10 T today) and (used in magnets that have H > 10 T today) and NbNb33GeGe (previous record superconducting transition (previous record superconducting transition

temperature temperature TTcc= 22.8 K) have a characteristic = 22.8 K) have a characteristic diffraction pattern of 3 lines, followed by 4 lines, diffraction pattern of 3 lines, followed by 4 lines,

followed by 3 linesfollowed by 3 lines..

XX--ray Trace of Aray Trace of A--15 Nb3Nb15 Nb3Nb

G. R. Stewart, et al.

Thermal Expansion of Thermal Expansion of PuPu Through Its Through Its Six Phase ChangesSix Phase Changes

What happens to an XWhat happens to an X--ray pattern when a cubic ray pattern when a cubic system (e.g. system (e.g. fccfcc Austenite FeAustenite Fe--C steel) undergoes a C steel) undergoes a

tetragonal distortion (to tetragonal distortion (to bctbct MartensiteMartensite)?)?

202

Lecture(sLecture(s) on: The Unusual non) on: The Unusual non--Fermi Liquid Fermi Liquid Behavior in Behavior in AntiferromagneticAntiferromagnetic U(PtU(Pt0.940.94PdPd0.060.06))3 3 –– and all the sample prep. and characterization and all the sample prep. and characterization

problems involved.problems involved.

First, need to learn how to characterize First, need to learn how to characterize materials via resistivity, magnetic materials via resistivity, magnetic

susceptibility, and specific heatsusceptibility, and specific heat

Resistivity, for example, is often used to detect Resistivity, for example, is often used to detect superconductivitysuperconductivity

D. Tanner

Resistivity measurements are conceptually quite easy

i i

Volt meter

But Beware!But Beware!

Contact resistance can play a large role Contact resistance can play a large role (heating)(heating)When a sample is not homogeneous (< 3 d)When a sample is not homogeneous (< 3 d)e.g. TTFe.g. TTF--TCNQ transitionTCNQ transition

. . . . . . . .

And indeed, sometimes the geometry of the sample makes putting 4 spaced contacts easy

G. R. Stewart, et al.

Some Some ‘‘tricks of the tradetricks of the trade’’ for contacting for contacting samplessamples

Silver epoxySilver epoxySilver Silver ‘‘paintpaint’’Gold Gold ‘‘paintpaint’’Spot welding (choice of wire, size)Spot welding (choice of wire, size)Forming a eutectic (e. g. AuForming a eutectic (e. g. Au--GeGe))

Au ball bonding on doped Au ball bonding on doped GeGe thermometersthermometers

Massalski

Noise problems in resistivity measurements Noise problems in resistivity measurements and their treatmentand their treatment

J. S. Kim, et al.

Measurement of Magnetic SusceptibilityMeasurement of Magnetic Susceptibility

χχ = M/H= M/HIn common use today, SQUID In common use today, SQUID magnetometers from Quantum Design, Inc. magnetometers from Quantum Design, Inc. can measure 10can measure 10--44 emu easily (company quotes emu easily (company quotes 1 x 101 x 10--88 emu at 0.25 T) , fields go typically to emu at 0.25 T) , fields go typically to 7 T.7 T.

Measurement of Measurement of Low TemperatureLow Temperature Specific Specific Heat IHeat I

Measurement of Measurement of Low TemperatureLow Temperature Specific Specific Heat IIHeat II

G. R. Stewart

Why do we need low temperatures?Answer: To determine the physics.

For example, when measuring the specific heat, C:

C/T = γ + βT2 / γ ~ N(EF), the density of electron states at

the Fermi energy.

β ~ 1/ΘD3 , where ΘD ~ lattice stiffness

If cannot cool the sample down to low enough temperature

(typically < ΘD/20) so that a.) there is no δT4 additional lattice term

b.) the electronic term γ is not invisibly small vs the lattice βT2 term.

then have trouble determining γ.

Determining Physical Properties from Low T Specific Heat: the Debye temperature, ΘD, and the electronic

density of states at the Fermi energy, N(EF)

C/T

T2

Intercept of C/T data with y-axis at T=0 gives γ ~ N(EF)

ΔC/T

ΔT2

Slope = ΔC/T÷ΔT2 = β

β ~ 1/ΘD3

In the BCS theory of superconductivity, Tc ~ exp(-1/ N(EF)*V)

x

xx

xx

x

xx

x

xx

Noise Problems in the Measurement of CNoise Problems in the Measurement of C

Where does the noise in the lower tau Where does the noise in the lower tau curve come from?curve come from?

Next, discuss: Preparation of UPtNext, discuss: Preparation of UPt33 alloyed alloyed with Pd via arcwith Pd via arc--meltingmelting

Preparation of UPtPreparation of UPt33 alloyed with Pd alloyed with Pd via arcvia arc--meltingmelting

Melting Points, TMelting Points, Tmm, and Vapor Pressures at T, and Vapor Pressures at Tmm of of U, Pt, PdU, Pt, Pd

TTmeltmelt = 1132, 1774, 1555 = 1132, 1774, 1555 ooCC (U, Pt, Pd)(U, Pt, Pd)

TTvaporvapor PP= <<0.001 mm, <0.001 mm, 0.01 mm= <<0.001 mm, <0.001 mm, 0.01 mm

Phase DiagramsPhase Diagrams

Massalski, ASM

Phase DiagramsPhase Diagrams

Massalski, ASM

Note that UPtNote that UPt33 has a different structure has a different structure (DO19 hexagonal) than that of UPd(DO19 hexagonal) than that of UPd33 (DO24 (DO24 double hexagonal). The solubility of Pd in double hexagonal). The solubility of Pd in the DO19 structure is aboutthe DO19 structure is about11 20%.20%.

11. G.R. Stewart and A.L. . G.R. Stewart and A.L. GiorgiGiorgi, , ““Suppression of Spin Fluctuations by Suppression of Spin Fluctuations by

Alloying in UPt3,Alloying in UPt3,”” J. Low Temp. Phys. J. Low Temp. Phys. 5959, 185 (1985)., 185 (1985).

At the composition U(PtAt the composition U(Pt0.80.8PdPd0.20.2))33, the spin fluctuation , the spin fluctuation term, ~ Tterm, ~ T33logT, is suppressed in the specific heatlogT, is suppressed in the specific heat

G. R. Stewart, et al.

What happens at U(PtWhat happens at U(Pt0.940.94PdPd0.060.06))33 ??

It has been known since 1991 and Seaman, et It has been known since 1991 and Seaman, et al.al.’’s discovery of the unusual properties of s discovery of the unusual properties of UU0.20.2YY0.80.8PdPd33 that that nonnon--Fermi liquid behaviorFermi liquid behaviorexists experimentally.exists experimentally.

What is Fermi Liquid Behavior?What is Fermi Liquid Behavior?

From a more theoretical point of view, a metal From a more theoretical point of view, a metal that is a that is a ‘‘Fermi liquidFermi liquid’’ at at low temperatureslow temperatures is the is the same as the same as the Fermi gasFermi gas of weakly correlated of weakly correlated electrons in the metal at electrons in the metal at room temperatureroom temperature, , only only –– due to the strong correlations between due to the strong correlations between the electrons at low temperatures the electrons at low temperatures –– the low the low temperature temperature ‘‘statestate’’ of the electrons can be of the electrons can be viewed as a collection of nonviewed as a collection of non--interacting interacting ‘‘quasiquasi--particlesparticles’’ that have an increased effective mass, that have an increased effective mass, mm**..

What is Fermi Liquid Behavior What is Fermi Liquid Behavior Experimentally ?Experimentally ?

A metal that is a A metal that is a ‘‘Fermi liquidFermi liquid’’ at at low low temperaturestemperatures has the same physical properties as has the same physical properties as a a Fermi gas:Fermi gas:

both the specific heat/temperature C/Tboth the specific heat/temperature C/Tand and χχ the magnetic susceptibility the magnetic susceptibility →→ constant at constant at low T while low T while ρρ the electrical resistivity ~ aTthe electrical resistivity ~ aT22

butbut with the constants increased with the constants increased ∝∝ mm**

What is nonWhat is non--Fermi Liquid Behavior ?Fermi Liquid Behavior ?

Trivially, a metal that is a Trivially, a metal that is a ‘‘nonnon--Fermi liquidFermi liquid’’ at at low temperatures does low temperatures does notnot have C/T have C/T and and χχ →→constant at low T.constant at low T.Further, Further, ρρ ~ ~ aTaTαα, but , but αα ≠≠ 22

Okay, now we know what properties Okay, now we know what properties nonnon--Fermi Liquids donFermi Liquids don’’t have. What t have. What

properties properties dodo they have?they have?Often a Often a ‘‘nonnon--Fermi liquidFermi liquid’’ at low temperatures at low temperatures has C/T and has C/T and χχ ~ ~ logTlogT or Tor T--αα (divergent). The (divergent). The resistivity is a little more varied, with resistivity is a little more varied, with ρρ ~ ~ aTaTαα, , αα sometimes 1.0 or 1.5. However, almost sometimes 1.0 or 1.5. However, almost always (stronger than always (stronger than ‘‘usuallyusually’’), a ), a nonnon--Fermi Fermi liquid liquid does not occur does not occur belowbelow a magnetic a magnetic transition!transition!

Where does Where does nFlnFl behavior occur?behavior occur?At a Quantum Critical PointAt a Quantum Critical Point

Standard Picture of a QCPStandard Picture of a QCP

QCP Behavior where TQCP Behavior where TNN (2(2ndnd order) order) --> 0> 0

Quantum Critical Point (quantum fluctuations dominate a large region (both in δ and to higher temperature) in the phase diagram.)

Non-Fermi Liquid Behavior

U(PtU(Pt0.940.94PdPd0.060.06))33, however, has , however, has nonnon--Fermi liquidFermi liquidbehavior below behavior below TTNeelNeel=6 K=6 K

This fact has been known since 1992 (J. S. This fact has been known since 1992 (J. S. Kim, B. Kim, B. AndrakaAndraka, and G. R. Stewart, , and G. R. Stewart, ““Possible Marginal Fermi Liquid Behavior in Possible Marginal Fermi Liquid Behavior in Doped UPt3,Doped UPt3,”” Phys. Rev. Phys. Rev. B45B45 12081 (1992)) 12081 (1992)) from measurements of the low temperature from measurements of the low temperature specific heat.specific heat.

MagneticMagnetic Phase Diagram of Phase Diagram of U(PtU(Pt11--xxPdPdxx))33

de Visser, et al.

Low temperature C/T of Low temperature C/T of U(PtU(Pt0.940.94PdPd0.060.06))33

J. S. Kim, et al.

Low temperature (T < Low temperature (T < TTNeelNeel) C/T of ) C/T of U(PtU(Pt0.940.94PdPd0.060.06))33 shows C/T ~ shows C/T ~ logTlogT !!

J. S. Kim, et al.

Another look at the low temperature C/T of Another look at the low temperature C/T of U(PtU(Pt0.940.94PdPd0.060.06))33

J. S. Kim, PRB 78, 035130 (2008).

A last look at the low temperature C/T of U(PtA last look at the low temperature C/T of U(Pt0.940.94PdPd0.060.06))33

Distinguishing Distinguishing logTlogT from Tfrom T--1+1+λλ(a digression for experts)(a digression for experts)

Low temperature data are important.

graph from de Andrade, et al. PRL 81, 5620 (1998)

C/T ~ C/T ~ logTlogT ⇒⇒ m* diverges as Tm* diverges as T→→00

What about the other measureable quantities What about the other measureable quantities like magnetic susceptibility, like magnetic susceptibility, χχ, and the , and the electrical resistivity, electrical resistivity, ρρ, ? Do they show , ? Do they show nonnon--Fermi liquid Fermi liquid behavior?behavior?

Magnetic Susceptibility, Magnetic Susceptibility, χχ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33

Magnetic Susceptibility, χ, of U(Pt0.94Pd0.06)3 –initial results were not quite so problem free

Magnetic Susceptibility, Magnetic Susceptibility, χχ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33Implications/MeaningImplications/Meaning

Note that Note that χχ ~ T~ T22 as Tas T→→0, i. e. 0, i. e. Fermi liquid Fermi liquid behavior! What does this mean about the qbehavior! What does this mean about the q--vector dependence of the magnetic vector dependence of the magnetic fluctuations that cause the fluctuations that cause the nonnon--Fermi liquid Fermi liquid behavior in the specific heat?behavior in the specific heat?

Electrical ResistivityElectrical Resistivity, , ρρ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33

Electrical ResistivityElectrical Resistivity, , ρρ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33--initial sampleinitial sample’’s results werens results weren’’t so goodt so good

Electrical ResistivityElectrical Resistivity, , ρρ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33--Initially, low T Initially, low T rho(Hrho(H) data were noisy (H=0)) data were noisy (H=0)

Electrical ResistivityElectrical Resistivity, , ρρ, of , of U(PtU(Pt0.940.94PdPd0.060.06))33--Additionally, low T Additionally, low T rho(Hrho(H) data were ) data were very very noisy (H=18 noisy (H=18

T)T)

Electrical ResistivityElectrical Resistivity, , ρρ, of , of U(PtU(Pt0.940.94PdPd0.060.06))3 3 Implications/MeaningImplications/Meaning

ρρ ~ T~ T22 , i. e. , i. e. Fermi liquid Fermi liquid behavior again, just like in behavior again, just like in χχTherefore, in summary, even though the specific Therefore, in summary, even though the specific heat shows nonheat shows non--Fermi liquid behavior, both the Fermi liquid behavior, both the magnetic susceptibility and the electrical resistivity magnetic susceptibility and the electrical resistivity show show Fermi liquidFermi liquid behavior. This contradiction behavior. This contradiction must mean (but this should be checked) that both must mean (but this should be checked) that both q=0 (determines q=0 (determines χχdcdc) and large q (determines the ) and large q (determines the dominant scattering processes in dominant scattering processes in ρρ) magnetic ) magnetic fluctuations in fluctuations in U(PtU(Pt0 940 94PdPd0 060 06))33 are Fermi liquid in are Fermi liquid in