magnetic anisotropy in f-electron systems: a crystala...
TRANSCRIPT
Magnetic anisotropy in f-electron systems:A crystal-field perspectiveA crystal-field perspectiveNicola Magnani
(nicola magnani@kit edu)
INSTITUTE OF NANOTECHNOLOGY, KARLSRUHE INSTITUTE OF TECHNOLOGY
KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association www.kit.edu
Outline
Part 1: An introduction to crystal field in f-electron systemsPart 1: An introduction to crystal field in f-electron systems
Part 2: Crystal-field anisotropy in lanthanide-based SMMs y py(selected examples)
Part 3: Crystal-field anisotropy in actinide-based SMMs (selected examples)
Institute of Nanotechnology (INT-KIT)2 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Part 1
An introduction toAn introduction tocrystal field iny
f-electron systems
Institute of Nanotechnology (INT-KIT)3 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
f-electron orbitals: the central field approximation2 2 2
1 1
ˆ2
N Ni
i i ji ij
p Ze eHm r r
ˆ22
N
i rUZepH
ji ij
ˆ2 N e
body)-(one21
0
i
ii
i rUrm
H
b d )(
ˆ1
1
jij
ij
rUreH
body)(o e
1N
r R r Y
body)-(two
1ˆand both commute with i i H L l S s
01
,i i i ii n l i l m i i
i
r R r Y
Institute of Nanotechnology (INT-KIT)4 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
1i ii i
Spherical harmonics Ylm
1l = 0
l = 1
Y
Y
cos312
1
01
00
2
220
235
41
rrzY
l = 1
l = 2
iY expsin23
21
2
11
1
zr
l = 3
Y
151
1cos3541
1
202
l = 4
iY
iY
2expsin215
41
expcossin215
21
222
12
l = 5
l = 6
242
Institute of Nanotechnology (INT-KIT)5 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
l 6
m = 0 m = ±1 m = ±2 m = ±3 m = ±4 m = ±5 m = ±6
The spectra of rare-earth ionsl = 3 2 1 0 -1 -2 -3
4f 3 L 6 S 3/210
ˆˆ HH
2S”+1L”4f 3: L = 6, S = 3/2
0H N
H l
4fn2S’+1L’
HHH ˆˆˆ
i
iiiSO rH1
sl
2S 1
Jmax=L+S
SOHHH 10 Jmin=|L-S|
2S+1L
J i =|L-S|
. . . N 7<
1 JJg J =L+S
. . . N 7>or
Institute of Nanotechnology (INT-KIT)6 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Jmin |L S| 1 JJgJeff Jmax L+S
The crystal field: a simple point-charge example
A magnetic ion placed at the origin of the Cartesian coordinate system, with two equal li d ( f ff ti h Z ) l d t
(0,0,R)
r = (x,y,z)
ligands (of effective charge Ze) placed at the same distance along the z axis.
(0,0,-R)
222
2
222
22
Rzyx
Ze
Rzyx
ZeeZVk k
k
rR
2222
2
//211
//211
RRRRRZe
yy
2222 //21//21 RrRzRrRzR
)(23163355311)()!2(1 765432 xOaaaaaaan n
Institute of Nanotechnology (INT-KIT)7 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
)(10242561281682
1)()!2(1 0
2 xOaaaaaaana n
n
The crystal field: a simple point-charge example
5
4224
4
23
3
22
222 835303
235
231
//211
Rzzrr
Rzrz
Rrz
Rz
RRrRz
7
624426
6
4325
165105315231
8157063
822//21
Rrzrzrz
Rzrzrz
RRRRRRrRz
2r 4260
2352 Y
Rr
045
4
32 Y
Rr0
67
6
132 Y
Rr
...
31
131
514 0
67
60
45
40
23
22
axial YRrY
RrY
RrZeV
Institute of Nanotechnology (INT-KIT)8 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
3135 RRR
The crystal field: a simple point-charge example(0,0,R)
(-R 0 0)
(0 R 0)(0 R 0)
(-R,0,0)
(x,y,z)(0,R,0)(0,-R,0)
(R,0,0)
145
37 4
44
40
45
42
cubic YYYRrZeV
(0,0,-R)
...27
1323
143
46
46
067
6
5cub c
YYYRr
R
Institute of Nanotechnology (INT-KIT)9 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
2132 6667R
Constraints on non-zero matrix elements
formthehaveelementsmatrixitson tocontributieachharmonics, spherical offunction a asrewritten been has ˆ Once CFH
ddsind2"""
*''' rrY mln
mlmln
formthehave elementsmatrix itson tocontributieach
ddid "*'* mmm YYYRR ddsind "*
'""*
''m
lm
lm
llnln YYYrRR therequireson gintegratin,2d Since "'
2"'
e mmmmmmi
nonzero. be result to for the fulfilled be to"'condition
qgg0
mmm
mmm
."'"' andeven bemust "':sconstraintadditional in tworesultson gintegratin Similarly,
llllllll
Institute of Nanotechnology (INT-KIT)10 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
.6 andeven bemust that means this)3"'( electrons For lllf
Constraints on non-zero matrix elements
...06
606
04
404
02
202axial
ii
ii
ii iYrAiYrAiYrAV
iii
145 4
44
40
44
4cubic
i iYiYiYrAV
7
14
4406
4444cubic
i
i
...27 4
64
60
66
6
ii iYiYiYrA
( l ib i f d l l 4)(even less contributions for d electrons: l ≤ 4)
Now all that’s left to do is to calculate these matrix elements!
Institute of Nanotechnology (INT-KIT)11 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Now all that s left to do is to calculate these matrix elements!
Angular momentum space: operator equivalents m
lll
i
ml
mml
li OrYYr ˆ1
ml
mll
lmlCF OrAH ˆ~ˆ
ml ,
Institute of Nanotechnology (INT-KIT)12 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Angular momentum space: operator equivalents
Institute of Nanotechnology (INT-KIT)13 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Angular momentum space: operator equivalents
SLMLSM :statesset Basis
zyx LLLzyx ˆ,ˆ,ˆ,,
mll
l
i
ml
mml
li OrYYr ˆ1
2S”+1L”
y
1ˆ33 e.g. 22202
2 LLLrzYr z
222222
22
2 ˆˆ LLyxYYr
2S’+1L’
22 yx LLyxYYr
JLSJM :statesset Basis4fn
Jmax=L+S
2S+1L
. . . kk JL ˆˆ
Institute of Nanotechnology (INT-KIT)14 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Jmin=|L-S|
Angular momentum space: operator equivalents
ˆ~ˆ~ˆ;,..., with states 12
ml
ml
mll
lmlCF
J
OBOrAH
JJMJ mll
l
i
ml
mml
li OrYYr ˆ1
2S”+1L” cases some(in degeneracy their lifts,,
mlll
mllllCF
2S’+1L’d lilhb dd
nHamiltonia above theof validity The.partially)only
4fn model.ticelectrostathebeyondextends
2S+1L
Institute of Nanotechnology (INT-KIT)15 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
2S+1LJMJ’s
Angular momentum space: operator equivalents
Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF
4040 ˆ21ˆ~ˆ5ˆ~ˆ OOBOOBH 46
066
44
044 215 OOBOOBHCF
111ˆ
2/ˆˆˆ 4444
JJJ MMMJJMJ
JJO
2/389852/1ˆ9852/1ˆ85
2/3ˆ52/3ˆ)2/3)(2/5()2/7)(2/5(2/5ˆ111
2
334
JJJJ
JJ
JJJ
MMMJJMJ4
~B
5124ˆ2/389852/19852/185
44
2
JJ MOM
JJ
Institute of Nanotechnology (INT-KIT)16 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Angular momentum space: operator equivalents
Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF
MJ 5/2 3/2 1/2 -1/2 -3/2 -5/2
5/2 60 0 0 0 60√5 0
MJ 5/2 -3/2 -5/2 3/2 1/2 -1/2
5/2 60 60√5 0 0 0 0120
-240
3/2 0 -180 0 0 0 60√5-3/2 60√5 -180 0 0 0 0
240
1/2 0 0 120 0 0 0
-1/2 0 0 0 120 0 0
-5/2 0 0 60 60√5 0 0
3/2 0 0 60√5 -180 0 04
~B
-3/2 60√5 0 0 0 -180 0
5/2 0 60√5 0 0 0 60
1/2 0 0 0 0 120 0
1/2 0 0 0 0 0 120
Institute of Nanotechnology (INT-KIT)17 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
-5/2 0 60√5 0 0 0 60-1/2 0 0 0 0 0 120
Angular momentum space: operator equivalents
Simple example: 4f 1 (Ce3+), 2F5/2, in a cubic CF
2/1
2F5/2
4~360B
6/2/32/55
4360B
6/2/52/35
Institute of Nanotechnology (INT-KIT)18 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Crystal-field splitting: cerium(III)
Institute of Nanotechnology (INT-KIT)19 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Crystal-field splitting: praseodymium(III)
Kramers’ theorem: all the crystal-field states of an ion with dd b f l t h d
Institute of Nanotechnology (INT-KIT)20 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
an odd number of electrons have even degeneracy.
Zero-field splitting in S-state ions
Ions with a L = 0 ground term should in principle have no spatial anisotropy. However, small mixing with higher-lying p py g g y gmultiplets can give rise to zero-field splitting.
ml
mlZFS OBH ˆ~ˆ S
ml ,Assuming only rank-2 contributions:
SS 2200 ˆ~ˆ~ˆ OBOBH SSS
SS1
21
212
12
22
22
2222
ˆ~ˆ~ˆ~
OBOBOB
OBOBHZFS
cost
~13~
222
2222
202
yxz
SSEDS
SSBSSSB
Institute of Nanotechnology (INT-KIT)21 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
cost. yxz SSEDS
Full configuration diagonalization
2S”+1L”
2S’+1L’4fn
2S+1L
Institute of Nanotechnology (INT-KIT)22 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
2S+1LJ
LS versus jj coupling
J mixing can be taken into account perturbatively:
Institute of Nanotechnology (INT-KIT)23 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
J mixing can be taken into account perturbatively:N. M. et al., PRB 71, 054405 (2005)
End of part 1 - Bibliography
D.J. Newman and B. Ng, Crystal Field Handbook (Cambridge University Press, 2000)
C. Goerller-Walrand and K. Binnemans, in: Handbook on the Physics and Chemistry of Rare Earths Vol. 23 (Elsevier, 1996)
SPECTRA h l l /d l d / t /i d ht l SPECTRA: chmwls.anl.gov/downloads/spectra/index.html
SMMS: cluster.univpm.it/_build/html/productsdir/smms/magnetism.html
Institute of Nanotechnology (INT-KIT)24 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Part 2
Crystal-field anisotropyCrystal field anisotropyin rare-earth based SMMs:
some examples
Institute of Nanotechnology (INT-KIT)25 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Origin of magnetic relaxation in rare-earth-based single-ion complexes
Direct process Orbach processAllowed if Jz is not a good quantum
Ene
rgy
Ene
rgy
g qnumber (like QTM)
Thermally activated
etic
fiel
d
etic
fiel
d
(slow at low T)
Mag
ne
Mag
ne
Institute of Nanotechnology (INT-KIT)26 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Origin of uniaxial anisotropy in SMMs
‘sandwich’ structures naturally accommodatenaturally accommodate oblate 4f orbitals
Institute of Nanotechnology (INT-KIT)27 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
J.D. Rinehart and J.R. Long, Chem. Sci. 2 (2011) 2078
Crystal-field splitting in [Pc2R] complexes
1Ueff~260 cm-1
∆~440 cm-1
Tb might be the best SMM, but Tm is much
Institute of Nanotechnology (INT-KIT)28
Tb might be the best SMM, but Tm is much better for studying the crystal field of the series!
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N. Ishikawa et al., JACS 125 (2003) 8694
Pc2Tm: optical spectroscopy
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N.M. et al, PRB 79 (2009) 104407
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Pc2Tm: neutron spectroscopy
Institute of Nanotechnology (INT-KIT)30 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
Specific heat can help fixing the degeneracy
Doublet: GS 1ES 2ES 3ES
2
2
2
12expln
n
nn
Bm
EgTkC
Institute of Nanotechnology (INT-KIT)31 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Pc2Tm: specific heat
Institute of Nanotechnology (INT-KIT)32 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
Crystal-field states assignment
- States nearest to the GS t h ll J tmust have small Jz to
reproduce low-temperature susceptibility
- States with Jz = ±6 are much lower in energy than expected
- All excited levels detected below 2 meV must correspond to singly degenerate states;to singly-degenerate states; presence of a significant C4contribution is evident
Institute of Nanotechnology (INT-KIT)33 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
Crystal-field states assignment
Institute of Nanotechnology (INT-KIT)34 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Pc2Tm: field-dependent specific heat
Model AModel C
Institute of Nanotechnology (INT-KIT)35 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
Pc2Tm: magnetic susceptibility
Model B
Model A
Institute of Nanotechnology (INT-KIT)36 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
Pc2R: Revised crystal-field parameters
2,0
Ueff
6,0
4,4
4,0
The revised crystal-field model is perfectlycompatible with the relaxation properties.
Institute of Nanotechnology (INT-KIT)37 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
N.M. et al, PRB 79 (2009) 104407
End of part 2 (almost) – Conclusions
The CF parameters for Tm-DD complex were unambiguously determined by a variety of experimental techniques.y y p q
D4d terms are qualitatively in line with those previously determined (signs and orders of magnitude)
NOT quantitatively: A20 smaller by a factor 3!
C symmetry terms small (with respect to the axial potential) but not C4-symmetry terms small (with respect to the axial potential) but not negligible (clear effect on the energy spectra)
Lower-symmetry terms have a much smaller influence on the low- Lower-symmetry terms have a much smaller influence on the low-energy wavefunction, even if present (|A2
2r2| < 0.5 cm-1)
Institute of Nanotechnology (INT-KIT)38 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Addendum: Exploiting magnetically active radicals
(Rinehart et al., Nat. Chem. 2011, 3, 538; J. Am. Chem. Soc. 2011, 133, 14236)
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(Lukens, N.M., and Booth, Inorg. Chem. 2012, 51, 10105)
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
How can the coupling be strengthened further?
I t f di t
π*
Increase t: for direct spin exchange, this means increasing the overlapf the overlap.
Actinides’ 5f shell has a larger
t i th
fπ*
extension than rare earths’ 4fDecrease U,matching thematching the reduction potential of the ligand and metal.
A ti id h h
U2tΔE
2
TS
Actinides have much richer chemistry than lanthanides (exp. more oxidation
Institute of Nanotechnology (INT-KIT)40
more oxidation states)
(Lukens, N.M., and Booth, Inorg. Chem. 2012, 51, 10105)
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
How can the coupling be strengthened further?
58 59 60 61 62 63 64 65 66 67 68 69 70
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm YbCe Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb[Xe]4f15d16s2 [Xe]4f25d16s2 [Xe]4f35d16s2 [Xe]4f45d16s2 [Xe]4f55d16s2 [Xe]4f65d16s2 [Xe]4f75d16s2 [Xe]4f85d16s2 [Xe]4f95d16s2 [Xe]4f105d16s2 [Xe]4f115d16s2 [Xe]4f125d16s2 [Xe]4f135d16s2
Lanthanides with accessible tetravalent states should be coupled with oxidizing radicals
Lanthanides with accessible divalent states should be coupled with
reducing radicalsg g
Actinides have a much richer chemistry!
Institute of Nanotechnology (INT-KIT)41 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Part 3
Anisotropy in actinide-basedAnisotropy in actinide based SMMs: some examplesp
Institute of Nanotechnology (INT-KIT)42 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Why actinides?
Institute of Nanotechnology (INT-KIT)43 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Why actinides?
Switching to actinide-based SMMs could combine the best of both worlds (3d and 4f) since 5f electrons can lead to theof both worlds (3d and 4f), since 5f electrons can lead to the simultaneous presence of strong ligand-field potential and magnetic superexchange coupling.
5New actinide-containing SMMs reported per year
2
3
4
0
1
2
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2009 2010 2011 2012 2013
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
A paradigmatic case: UTp3
Ueff ∆/100 !!!
∆ = 267 cm-1
More complex relaxation processes than expected from the energy spectrum
Institute of Nanotechnology (INT-KIT)45 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
J.D. Rinehart and J.R. Long, Dalton Trans. 41 (2012) 13572
Direct processes are enhanced by the non-axial part of the p y pligand-field potential.
Possible solution: ACTINOCENES (D8h)( 8h)
2
(Karraker et al., JACS 1970, 92, 4841)
PROBLEM: tetravalent U 5f 2
Institute of Nanotechnology (INT-KIT)46 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
“Butterfly” magnetic hysteresis in neptunocene (5f3)
Jz = ±5/2z
237NpI = 5/2
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(N.M. et al., Angew. Chem. Int. Ed. 2011, 50, 1696)
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Exchange coupling and slow relaxation in the heterovalent Np trimetallic [NpVIO2Cl2][NpVO2Cl(thf)3]2
140 K
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(N.M. et al., Phys. Rev. Lett. 2010, 104, 197202)
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Exchange coupling in U(V) complexes
3.36 Å
(Arnold et al., Nat. Chem. 2012, 4, 221)
Institute of Nanotechnology (INT-KIT)49 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
5f1 – 4fn complexes
Ln = Dy
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(Arnold et al., JACS 2013, 135, 3841)
Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
End of part 3 – Conclusions and perspectives
Both the ligand-field anisotropy and the superexchange interaction in actinide-based magnetic molecules are generally stronger than in their lanthanide counterparts. This could make light actinides a better choice to design SMMs, despite their lower magnetic moments…
b t f t t l f d t l i t ti hi h i i t …but unfortunately, some fundamental interactions which give rise to fast relaxation processes (non-axial ligand field, hyperfine coupling, orbital moment reduction) seem to be enhanced too.
Challenge: exploit the richer actinide chemistry (more oxidation states, larger coordination sphere…) to obtain a favorable balance.
Pu(V): strong exchange and large moment?
Institute of Nanotechnology (INT-KIT)51 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013
Acknowledgments
Annie Powell and her group (KIT)
Christos Apostolidis, Roberto Caciuffo and his group (JRC-ITU)
Wayne Lukens, Norman Edelstein, Corwin Booth, Richard Andersen (LBNL)
Clint Sharrad Richard Winpenny and his group (Uni Manchester)Clint Sharrad, Richard Winpenny and his group (Uni-Manchester)
Polly Arnold and her group (Uni-Edinburgh)
Marinella Mazzanti and her group (CEA-Grenoble)
Institute of Nanotechnology (INT-KIT)52 Nicola Magnani – Magnetic anisotropy in f-electron system: A crystal-field perspective12.10.2013