Are Ge-based skutterudites promising thermoelectrica? Ground state - - PowerPoint PPT Presentation

ARW Workshop Properties and Applications of Thermoelectric Materials September 20 - 26, 2008 Hvar, Croatia

Are Ge-based skutterudites promising thermoelectrica? Ground state properties, electronic and thermal transport

• E. Bauer

Institute of Solid State Physics A - 1040 Wien, Austria

• 24. September 2008

ARW Workshop Properties and Applications of Thermoelectric Materials September 20 - 26, 2008 Hvar, Croatia

Are Ge-based skutterudites promising thermoelectrica? Ground state properties, electronic and thermal transport

• E. Bauer

Institute of Solid State Physics A - 1040 Wien, Austria

• 24. September 2008

in co-operation with:

in co-operation with:

• G. Hilscher, H. Kaldarar, H. Michor, E. Royanian; T.U. Vienna,

Austria

• A. Grytsiv, Xing-Qiu Chen, N. Melnychenko-Koblyuk, M. Rotter, R.

Podloucky & P. Rogl Uni. Vienna, Austria Work supported by the Austrian FWF, P19165 and by the EU Complex metallic alloys, CMA

Contents

• Formation and general properties of skutterudites
• Novel skutterudites {Ba, Sr, Eu, Th, U}Pt4Ge12
• Normal state and superconducting properties
• Thermoelectric properties of Ge-based skutterudites
• Summary and Outlook

More details:

• Phys. Rev. Lett. 99, 217001, (2007);
• Adv. Mat. 20, 1325 (2008)
• J. Phys. Soc. Jpn., 77 121, (2008);
• Phys. Rev. B (2008), in press.

Filled skutterudites

Filled skutterudites

a b c x y z

el.positive element (e.g. Pr, Nd ) Sb, (P,As) d-element (Fe, Co, Rh ...)

2a-site: Ep 8c-site: transition metal 24g-site: P, As, Sb, Ge

• structure type: LaFe4P12

(CoAs3-structure).

• lattice parameter:

a = 9.127 ˚ A (PrFe4Sb12)

• a strongly dependent on

pnictogen atom (change as large as 15 % )

• RE ion sixfold co-ordinated

by X.

• extremely large atomic dis-

placement parameter of fil- ler elements; increases with increasing cage volume; in- creases with decreasing io- nic size.

Formation of skutterudites EpT4X12

4 12

transitional element in the 8c site p-element in the 24g site electopositive element in the 2a site

until recently: No skutterudites entirely formed by Pt (Pd) and Ge

Ba-Pt-Ge phase diagram

BaPt BaPt4

4Ge

Ge12

12

~Ba ~Ba8

8Pt

Pt4

4Ge

Ge42

42

• a = 0.87 nm; lat-

tice parameter much smaller than for X = P, As, Sb;

• BaPt4Ge12

exhibit the strongest de- viation from Zintl count (6 uncompen- sated electrons per formula) among all known skutterudites!

• only example having

simultaneously cla- thrate and skutteru- dite members.

Structural and electronic properties of EpPt4Ge12

X-ray diffraction in BaPt4Ge12

8 18 28 38 48 58 68 78 88 98

• 30000
• 10000

10000 30000 50000 70000 90000 110000

Intensity [counts]

Iobs Icalc

• I

I

• bs calc

Bragg_position

2 [ ]

BaPt Ge

4 12

R F

a = 0.86889(1) nm

= 0.017;

Ba Ge Pt

Im-3;

• known members: Ep . . . Sr, Ba; La, Ce, Pr, Nd, Eu; Th, U;
• a = 0.87 nm; much smaller than for X = P, As, Sb;

Is there a rattling mode in EpPt4Ge12?

• reduced lattice parameter prohibits “rattling” (at least) in

BaPt4Sb12.

• however: atomic displacement parameter of Eu in EuPt4Ge12

twice as large as of Ba in BaPt4Ge12

Oftedal relation in EpPt4Ge12

• in general for pnictogen: position parameter y + z ≈ 0.5
• largest deviation from Oftedal relation in EpPt4Ge12

Stability and Bonding in BaPt4Ge12

Thermodynamical stability of XPt4Ge12 (X=Ba,Sr) by density functional theory (DFT) using the Vienna ab initio simulation package (VASP)

• total energy for hypothetical Pt4Ge12.
• bonding energy EX for guest atom X from relation

EX = U DF T

XPt4Ge12 − U DF T Pt4Ge12 − U DF T X

U DF T . . . total energy of compound or elemental solid.

• EBa = -3.24 eV and ESr = -3.38 eV

⇒ stabilizing effect of the Ba and Sr guest atoms.

• binary Pt4Ge12 not stable; different to “ordinary” skutterudites

Electronic density of states of {Sr, Ba}Pt4Ge12

• DOS at EF

do- minated by Ge-p states; small Pt- d contributions; vanishing Ba,Sr states

• small

differences between relativi- stic (s-o coup- ling) and non- relativistic calcu- lations.;

• significant

el-ph enhancement ⇒ SC!?

Electron density in BaPt4Ge12

Isosurfaces of the electron density (0.14 electrons per ˚ A3) for the states of the DOS peak at Fermi energy; view along [1,1,1]

• tubes connecting Ge atoms

visualizing strongly directed bonds;

• spherical shapes around Pt

atoms representing metal-like states formed by the Pt 5d-like states.

Superconductivity in EpPt4Ge12

Electrical resistivity of {Sr, Ba}Pt4Ge12

T [K]

50 100 150 200 250 300

ρ [μΩcm]

20 40 60 80 100

T [K]

1 2 3 4 5 6

ρ [μΩcm]

10 20 30

0 T 0.1 T 0.5 T 1 T 1.25 T 1.5 T 2 T

SrPt4Ge12

(b)

BaPt4Ge12

T [K]

50 100 150 200 250 300

ρ [μΩcm]

20 40 60 80 100

0 T 0.5 T 1 T 1.5 T 2 T 2.25 T 2.5 T 2.75 T 3 T 3.25 T

ρ [μΩcm]

(a)

1 2 3 4 5 6 10 20 30 40 50

T [K]

• Woodward & Cody: ρ = ρ0+ρ1T +ρ2 exp(−T0/T); T0 ≈ 130 K
• Tc = 5.1 and 5.35 K for Sr and Ba-based compounds

Specific heat of BaPt4Ge12

• γ ≈ 42 mJ/molK2

θD ≈ 245 K

• ∆Cp/T(T = Tc) ≈ 58mJ/molK2 ⇒ ∆Cp/(γnTc) ≈ 1.35,

BCS theory: ∆Cp/(γTc) ≈ 1.43.

Specific heat of {Sr, Ba, Ca}Pt4Ge12

Tc/T 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Ces/γTc

0.1 1 BaPt4Ge12 Δ(0) = 9.7(1) K

(b) T [K]

1 2 3 4 5 6 7

Cp/T [J/(mol K2)]

0.00 0.05 0.10 0.15 0.20

BaPt4Ge12

0 T 3 T 0 T 3 T

SrPt4Ge12

(a)

SrPt4Ge12 Δ(0) = 9.4(1) K

0 T 3 T

Ba0.8Ca0.2Pt4Ge12

Ba0.8Ca0.2Pt4Ge12 Δ(0) = 8.8(1) K

• Tc = 5.1 and 5.35 K for Sr and Ba-based compounds
• γ(Ba) ≈ 42 mJ/molK2

γ(Sr) ≈ 41 mJ/molK2

• θD(BaPt4Ge12) ≈ 245 K;

θD(SrPt4Ge12) ≈ 220 K!!!

SC features of {Ba, Sr}Pt4Ge12

• Electron - phonon enhancement factor λ ≈ 0.7

⇒ SC beyond weak coupling limit

• smaller mass of Sr-compound does not produce larger Debye

temperature; smaller atomic volume of Sr responsible for weaker bonding in cage; less oscillator strength; larger thermal displacement, smaller Tc

• mostly Ge-p states at the Fermi energy! SC due to Ge-electrons
• comparable cage-forming compounds like {Ba, Sr}8Si46: states

at EF predominantly formed from Ba-5d and Sr-4d states Tc(Ba8Si46) ≈ 8 K.

Lattice contribution to {Ba, Sr}Pt4Ge12

Cph(T) = R ∞ F(ω) ( ω

2T )2

sinh2( ω

2T ) T [K]

2 5 20 50 200 10 100

(Cp-γT)/T3 [mJ/g-atomK4]

0.00 0.05 0.10 0.15 0.20 0.25

(4/5)Rπ4ω−2F(ω) [mJ/(g-atomK4)] 2 4 6 (ω/4.93) [K] SrPt4Ge12 BaPt4Ge12 BaCa0.2Pt4Ge12

Sr Ba Ca

least squares fit

• low lying pho-

non mode responsible for superconducti- vity? (Junod et al., 83)

Upper critical field of {Sr, Ba}Pt4Ge12

Tc [K]

1 2 3 4 5 6

μ0Hc2 [T]

0.0 0.5 1.0 1.5 2.0 specific heat magnetisation resistivity

• 0.46 T/K

SrPt4Ge12 BaPt4Ge12

specific heat magnetisation resistivity

• 0.53 T/K
• 0.275 T/K
• 0.31 T/K
• BaPt4Ge12:

µ0Hc2 ≈ 1.8 T; µ0H′

c2 = −0.46 T/K

• SrPt4Ge12:

µ0Hc2 ≈ 1.0 T; µ0H′

c2 = −0.27 T/K

• Hake et al:

H′

c2 ∝ 1/vF

• free electrons

vF = (N/V)(1/3)

• volume

increases from Sr to Ba ⇒ vF decreases!

Upper critical field of {Sr, Ba}Pt4Ge12

Tc [K]

1 2 3 4 5 6

μ0Hc2 [T]

0.0 0.5 1.0 1.5 2.0 specific heat magnetisation resistivity

• 0.46 T/K

SrPt4Ge12 BaPt4Ge12

specific heat magnetisation resistivity

• 0.53 T/K
• 0.275 T/K
• 0.31 T/K

Maki parame- ter:

• BaPt4Ge12:

α ≈ 0.18

• SrPt4Ge12:

α ≈ 0.14

• WHH model applicable;
• Contribution of α compensated by λso!

Pressure response of Tc of SrPt4Ge12

SrPt4Ge12

T [K]

50 100 150 200 250 300

ρ [μΩcm]

20 40 60 80 100 SrPt4Ge12 Pressure [kbar]

5 10 15 20

Tc [K]

5.0 5.2 5.4 1.0 bar 3.5 kbar 8.6 kbar 13.5 kbar 17.5 kbar 19.2 kbar 20.5 kbar

• θD = 220 K; pressure inde-

pendent; normal state region: model of Woodward & Cody T0 = 123 K

• non-monotonous variation of

Tc(p)

p [kbar]

5 10 15 20

DOS(E=EF) [states/(ev.f.u.)]

2 4 6 8 10 12 14

SrPt4Ge12

• Bogolyobov et al.:

Tc = 1.14¯ hωD exp

• −

1 λ − µ∗

• ,

λ = N(EF)V

Ground state of {Th, U}Pt4Ge12

T [K]

50 100 150 200 250

ρ [μΩcm]

50 100 150 200 2 4 6 8 10 12 14 1 2 3 4 5

ρ [μΩcm] T [K]

ThPt4Ge12 UPt4Ge12 ThPt4Ge12

(a) (b)

• ThPt4Ge12; T > Tc: model of

Woodard & Cody T0 = 121 K; Tc = 4.8 K

• UPt4Ge12; spin fluctuations;

T ≪: ρ(T) = ρ0 + AT n with ρ0 = 14.5 µΩcm, A = 0.42 µΩcm/K1.5; n = 1.5.

T [K]

1 2 3 4 5 6 7

Cp/T [J/(mol K2)]

0.00 0.05 0.10 0.15 0.20 0.25 0.30

T [K]

1 2 3 4 5

μ0H [T]

0.00 0.05 0.10 0.15 0.20 0.25

ThPt4Ge12

γ = 35 mJ/mol K2

Hc2' = -0.064 T/K Hc2(0, theor.) = 0.21 T α = 0.017 λso = 15 (a) (b)

μ0Hc2 WHH model μ0Hc

0 T 0.05 T 0.1 T 0.2 T

• γ = 35 mJ/molK2; θD =

260 K

• λe−ph = 0.66
• µ0H′

c2 = −0.064 T/K

SC in {Sr, Ba, Th}Pt4Ge12: facts and figures

property BaPt4Ge12 SrPt4Ge12 ThPt4Ge12 lattice parameter a @300 K [nm] 0.86928(3) 0.86601(3) 0.85931(3) Ge 24g site: y 0.15302 0.15197 0.1515(3) Ge 24g site: z 0.35683 0.35536 0.3556(3) RF 2 0.019 0.018 0.057 transition temperature Tc [K] 5.35 5.10 4.75 upper critical field µ0Hc2 [T] 1.8 1 0.22 slope of µ0Hc2, µ0H′

c2(Cp) [T/K]

• 0.46
• 0.275
• 0.067
• thermod. critical field µ0Hc [mT]

53 52 50 Fermi velocity vF [m/s] 52500 67000 136000 coherence length ξ [nm] 14(1) 18(1) 40(2) penetration depth λ [nm] 320(10) 250(10) 120(1) G.L. parameter κ 24(1) 14(1) 3(1) ltr/ξ 0.5(2) 0.7(2) 3.2

• ltr/ξ ≈ 1: {Ba, Sr}Pt4Ge12 dirty limit superconductors;
• κ ≈ 10 − 20 pronounced type II superconducting behaviour.

Similarities of SC in {Sr, Ba, Th}Pt4Ge12?

κ0=1.9

κ κ0 (γ ρ0) √γ [J/m3 K2].ρ0[Ωcm]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

κ

5 10 15 20 25 30

Abrikosov's: κGL=Hc2/√2Hc(0) Gor'kov-Goodman

κ κ0 (γ ρ0)

µ H [T]

μ0 μ0 −μ0

α λ

κ=κ0+0.0237(γ1/2ρ0) Sr

Th Ba

Gor’kov - Goodman model

• κ = κ0 + 0.0237ρ0√γ
• κ0 = 1.9;

{Sr, Ba, Th}Pt4Ge12

• n universal line;

Low temperature behaviour of EuPt4Ge12

EuPt4Ge12

B [T]

2 4 6 8 10 12

ρ(B)/ρ(0)

0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02

EuPt4Ge12

T [K]

50 100 150 200 250

ρ [μΩcm]

20 40 60 80 100

T [K] 2 4 6 8 10

ρ [μΩcm]

25.0 25.5 26.0 26.5

Bloch-Grüneisen law + Mott-Jones term (T3)

0 T 4 T 8 T 12 T

T=2 K T=4 K T=6 K T=8 K T=10 K T=12 K T=15 K T=30 K

(a) (b) TN

T=0.5 K

• magnetic ordering at TN =

1.7 K

• small magnetoresistance
• γ = 220 mJ/molK2; θD =

212 K

• µeff = 7.6 µB

[ ] [

[

• γband = 21 mJ/molK2; → si-

gnificant fluctuations!

• AFM

and FM state more stable (0.32 eV) than non- magnetic ground state!

• PBE+U calculation yields Eu-

moment: 7 µB

Magnetism of EuPt4Ge12

T [K]

50 100 150 200 250 300

1/χ [mol/emu]

10 20 30 40

• mod. Curie-Weiss law

EuPt4Ge12

μeff = 7.5 μB θp = -11.5 K 1 Tesla

• Curie Weiss behaviour;

µeff = 7.5 µB, θp = −11.5 K

• no CEF effect!
• low

AFM transi- tion temperature (TN ≈ 1.7 K); note: Tmag(EuFe4Sb12) = 84 K [Fe4Sb12] sublattice magnetic; [Pt4Ge12] nonmagnetic!

Transport and thermoelectric features of EpPt4Ge12

Transport in EpPt4Ge12

T [K]

50 100 150 200 250 300

ρ[μΩcm]

20 40 60 80 100 120 140

EuPt4Ge12 SrPt4Ge12 Ba0.8Ca0.2Pt4Ge12 BaPt4Ge12

least squares fits parallel resistance model

• differences in residual resisti-

vities (slight changes in filling grade)

• S-shape indicates deviations

from simple metals Parallel resistance model 1 ρ(T) = 1 ρsat + 1 ρideal ρideal = ρ0 + ρBG ρBG = ce−phθD T θD 5 θD/T z5dz (exp(z) − 1)(1 − exp(−z))

Transport in EpPt4Ge12

T [K]

50 100 150 200 250 300

ρ[μΩcm]

20 40 60 80 100 120 140

EuPt4Ge12 SrPt4Ge12 Ba0.8Ca0.2Pt4Ge12 BaPt4Ge12

least squares fits parallel resistance model

T [K]

5 10 15 20 25 30

ρ [μΩcm]

60 61 62 63 64 65 66

BaPt4Ge12

least squares fit, parallel resistance model Woodard & Cody

Woodard & Cody: ρ = ρ0 + ρ1T + ρ2 exp(−T0/T); T0 ≈ 130 K

• resistivity at low temperatures significantly deviates from simple

metal ⇒ superconductivity!?

Thermopower of EpPt4Ge12

T [K]

50 100 150 200 250 300

S [μV/K]

2 4 6 8 10 12

{Ba,Sr,Ca,Eu}Pt4Ge12

SrPt4Ge12 EuPt4Ge12 Ba0.8Ca0.2Pt4Ge12 BaPt4Ge12

• predominatly ho-

les als charge carriers!

• carrier

concen- tration n ≈ 2.5× 1022 1/cm3 (S ∝ 1/n)

• deviation from li-

nearity due to electronic corre- lations.

Thermal conductivity of EpPt4Ge12

λ = λe + λph λe = LeT ρ ≈ L0T ρ L0 = 2.45 × 10−8 WΩ/K2; L0 . . . Lorenz number

T [K]

50 100 150 200 250 300 350

λ[mW/cmK]

20 40 60 80 100 120 140 160

EuPt4Ge12 SrPt4Ge12 Ba0.8Ca0.2Pt4Ge12 BaPt4Ge12

• overall

relatively large conductivity

• radiation losses at

high temperatures in steady state techniques

Thermal Conductivity: Analysis

Debye approximation for lattice thermal conductivity: λph = CT 3 θD/T τcx4 exp(x) [exp(x) − 1]2dx x = ¯ hω/kBT, ω (phonon frequency), θD (Debye temperature); τc . . . overall relaxation time: τ −1

c

= τ −1

B + τ −1 D + τ −1 U

+ τ −1

e

τb, τD, τU, τe relaxation times for boundary scattering, defect scattering, Umklapp processes and electron scattering, respectively. τ −1

B

= B τ −1

D = Dω4

τ −1

U

= Uω2T exp(−θD/3T) τ −1

e

= Eω

Is the rattling mode missing?

ARTICLES

Breakdown of phonon glass paradigm in La- and Ce-filled Fe4Sb12 skutterudites

MICHAEL MAREK KOZA1*, MARK ROBERT JOHNSON1, ROMAIN VIENNOIS2, HANNU MUTKA1, LUC GIRARD3 AND DIDIER RAVOT3

1Institut Laue Langevin, 6 Rue Jules Horowitz, B.P

. 156, 38042 Grenoble, Cedex 9, France

2DPMC, Universite de Geneve, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland 3LPMC, Universite de Montpellier II, pl. Eugene Bataillon, 34095 Montpellier, France

*e-mail: koza@ill.fr

Published online: 31 August 2008; doi:10.1038/nmat2260

NO!!!???

Is the rattling mode missing?

ARTICLES

Avoided crossing of rattler modes in thermoelectric materials

MOGENS CHRISTENSEN1*

†, ASGER B. ABRAHAMSEN2, NIELS B. CHRISTENSEN2,3,4, FANNI JURANYI3,

NIELS H. ANDERSEN2, KIM LEFMANN2*, JAKOB ANDREASSON5, CHRISTIAN R. H. BAHL2* AND BO B. IVERSEN1†

1Center for Energy Materials, Department of Chemistry and Interdisciplinary Nanoscience Center iNANO, University of Aarhus, DK-8000 Aarhus C, Denmark 2Materials Research Department, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark 3Laboratory for Neutron Scattering, ETHZ and PSI, CH-5232, Villigen PSI, Switzerland 4Nano-Science Center, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark 5Department of Applied Physics, Chalmers University of Technology, 412 96 G¨

• teborg, Sweden

*Current address: The Bragg Institute, ANSTO, NSW 2234 Menai, Australia (M.C.); The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark (K.L.); Fuel Cells and Solid State Chemistry Department, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark (C.R.H.B.)

†e-mail: moc@ansto.gov.au; bo@chem.au.dk

Published online: 31 August 2008; doi:10.1038/nmat2273

YES???!!!?

Lattice thermal conductivity: scattering processes

τ −1

c

= τ −1

U

+ τ −1

D + τ −1 B + τ −1 ph−el

T [K] 100 200 300

λ [mW/cmK]

200 400 600 800 1000 T [K] 100 200 300

Wph [cmk/mW]

0.00 0.01 0.02 0.03 0.04 0.05

umklapp scattering boundary scattering el-ph scattering λph / Wph point defect scattering

Minimal thermal conductivity

λmin = 3n 4π 1

3 k2

BT 2

¯ hθD θD/T x3ex (ex − 1)2dx

T [K]

50 100 150 200 250 300

λ [mW/cmK]

2 4 6 8 10

minimum thermal conductivity dependence on the Debye temperature θD θD = 100 K θD = 400 K n = 2n0 n = n0

• D. G. Cahill and R.
• O. Pohl (1989)

n = N/V . . . number of atoms per unit volume. Lower limit of λph in glass-like

• r

amorphous systems

Thermal conductivity of EpPt4Ge12

λ = λe + λph λe = LeT ρ ≈ L0T ρ L0 = 2.45 × 10−8 WΩ/K2; L0 . . . Lorenz number

SrPt4Ge12

T [K]

50 100 150 200 250 300

λ [mW/cmK]

20 40 60 80 100 120 140

λmeas λe λl Callaway Fit λl T3 corr.

λmin

• electron

and phonon contri- bution of about same size

• no

significant rattling mode; λ(T) away from thermal conductivity limit

Thermal conductivity of EpPt4Ge12

BaPt4Ge12

T [K]

50 100 150 200 250 300 350

λ[mW/cmK]

20 40 60 80 100 120 140

λmeas λe λl Callaway Fit λl T3 corr.

Radiation losses

EuPt4Ge12

T [K]

50 100 150 200 250 300 350

λ [mW/cmK]

20 40 60 80 100 120 140 160

λmeas λe λl Callaway Fit λl T3 corr.

Radiation losses

Summary

• novel family of skutterudites; Ge instead of volatile and/or toxic

As, P, Sb;

• BCS type superconductivity with medium-strong electron -

phonon coupling (λe−ph ≈ 0.7);

• SC dominated by Ge-p states!
• metallic behaviour; poor thermoelectric performance.

What to do?

• inserting 3+ rare earth elements (correlations and magnetism)
• reduce charge carrier density

• exchange Pt

Model of Werthammer et al. WHH model

Two mechanisms limiting µ0Hc2: orbital pair breaking and Pauli

• limiting. Additionally non-magnetic spin-orbit scattering

WHH model ln t = Ψ 1 2

• − 1

2

• 1 + λso/4

X

• Ψ

1 2 + Y + λso/4 − X t

• (1)

− 1 2

• 1 − λso/4

X

• Ψ

1 2 + Y + λso/4 + X t

• Ψ is digamma function. Y = 2h/π2, t = T/Tc,

h = Bc2(T)/ ∂Bc2 ∂T |T =TcTc

• and X =
• (λso/4)2 − 4h2α2/π41/2

.

Model of Werthammer et al. WHH model

T/Tc

0.0 0.2 0.4 0.6 0.8 1.0

h*

0.0 0.2 0.4 0.6 0.8

α = 0, λso = 0 α = 1, λso = 0 α = 3, λso = 0 α = 3, λso = 5

model of Werthamer et al.

α = 3, λso = infinite

• two parameters,

– α, . . . Pauli parama- gnetic limitation (Maki parameter) – λso . . . spin-orbit scat- tering. α from γ and ρ0: α = (3e2¯ hγρ0)/(2mπ2k2

B)

Increasing value of α reduces Bc2 from upper limit h = 0.693. BaPt4Ge12: α ≈ 0.18 SrPt4Ge12: α ≈ 0.14 Spin-orbital scattering compensates for decrease due to paramagnetic li- mitation and restores h∗ ≈ 0.693 for λso → ∞. increasing atomic number: λso incre-

• ases. 5d systems: λso ≈ 10 − 20