UNIVERSIT`
A DEGLI STUDI DI
PARMA
DOTTORATO DI RICERCA IN FISICA
CICLO XXXV
Cohe en and incohe en dynamics o
molecula nanomagne s in es iga ed by
magne ic esonances and inelas ic
neu on and x- ay sca e ing
Coo dina o e:
Chia .mo P o . S e ano Ca e a
Tu o e:
Chia .mo P o . S e ano Ca e a
Co- u o e:
D .ssa Elena Ga la i
Chia .mo P o . Giuseppe Allodi
Do o ando:
Simone Chicco
Anni Accademici 2019/20 - 2022/23
Abs ac
The ocus o his hesis is he cha ac e iza ion o he cohe en and incohe en dynam-
ics o molecula nanomagne s. Fo his pu pose, we will exploi se e al s a e-o - he-
a expe imen al echniques, such as magne ic esonances and inelas ic sca e ing o
neu ons and X- ays. By means o nuclea magne ic esonances we will cha ac e -
ize he nuclea elaxa ion imes and he pa ame e s o he spin Hamil onian o wo
V-based molecula qudi s, demons a ing, in addi ion, he capabili y o cohe en ly
manipula e hei nuclea s a es. This p oo -o -concep s expe imen s ep esen an
impo an i s s ep owa ds he implemen a ion o molecula qudi s in quan um
in o ma ion p ocessing. Mo eo e , by X- ay inelas ic sca e ing, we will in es iga e
he phonon dispe sions o one benchma k molecula qudi and hei ole in i s elax-
a ion dynamics. We will also ocus on he key ac o s go e ning he phonon-induced
elaxa ion in Dy-based single molecule magne s by s udying, h ough inelas ic neu-
on sca e ing, he changes induced in hei phonon densi y o s a es by chemical
subs i u ions o s uc u al de o ma ions. Finally, a syne gis ic app oach combining
elec on and nuclea magne ic esonance will gi e us insigh s on he elec onic e-
laxa ion dynamics o a sup amolecula assembly linking an isola ed nuclea qudi
and an elec onic spin qubi .
1
Lis o Collabo a o s
Uni e si y o Pa ma (IT)
–Supe iso and Co-Supe iso s: E. Ga la i, G. Allodi, S. Ca e a.
–Collabo a o s: A. Chiesa, P. Bon `a, I.J. Onuo ah, R. De Renzi, P. San ini.
Uni e si y o Flo ence (IT)
–F. San anni, A. Albino, F. To i, L. So ace, R. Sessoli.
Uni e si y o Tu in (IT)
–E. Sal ado i, M. Chiesa.
Lab.Na . Champs Magn´e iques In enses, G enoble (FR)
–M. A zo i.
Uni e si y o Manches e (UK)
–F. S. J. Lockye , A. B ook ield, J. Skel on, F. Tuna, D. Mills, N. Chil on,
E. McInnes, R. Winpenny.
T ini y College Dublin (IE)
–A. Nguyen, A. Lunghi.
Uni e si y o Glasgow (UK)
–A. Ma omagoulos, A. B. Canaj, M. Mu ie.
Uni e si y o Came ino, (IT) and ISIS Neu on and Muon
sou ce, Didco , OX (UK)
–T. Guidi.
Eu opean Synch o on Radia ion Facili y, G enoble (FR)
–L. Paolasini.
Ins i ue Laue Lange in, G enoble (FR)
–M. Jimenez Ruiz, A. Pio ano, A. I ano .
B ookha en Na ional Labo a o y, Up on, NY (USA)
–C. Mazzoli, C. Yong.
2
Con en s
1 In oduc ion 15
1.1 Molecula nanomagne s . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Di Vincenzo c i e ia o Molecula Magne s . . . . . . . . . . . . . . . 18
1.3 Spin Hamil onian o malism . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 Elec onic and Nuclea Zeeman in e ac ions . . . . . . . . . . 24
1.3.2 The Hype ine in e ac ion . . . . . . . . . . . . . . . . . . . . 25
1.3.3 The Nuclea quad upola coupling . . . . . . . . . . . . . . . 28
1.3.4 Second o de e ec s . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Lindblad Mas e equa ion . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Expe imen al echniques 33
2.1 Resonance echniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Nuclea Magne ic Resonance . . . . . . . . . . . . . . . . . . . 33
2.1.2 Elec on Pa amagne ic esonance . . . . . . . . . . . . . . . . 37
2.2 Sca e ing echniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 Inelas ic sca e ing . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.2 Inelas ic Neu on Sca e ing . . . . . . . . . . . . . . . . . . . 41
2.2.3 Inelas ic X ays Sca e ing . . . . . . . . . . . . . . . . . . . . 46
3 Ex ensi e b oadband NMR s udy o Vanadium-based molecula
qudi s 50
3.1 Vanadium-based qudi s . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 NMR in es iga ion o he [VO(TPP)] molecula qudi . . . . . . . . . 51
3.3 Comp ehensi e s udy o an O ganome allic spin Qudi wi h Radio e-
quency and Mic owa e echniques . . . . . . . . . . . . . . . . . . . . 61
3.4 Encoding a QEC algo i hm on he [VO(TPP)] molecula qudi . . . . 75
3.5 Conclusions ................................ 77
4 Un eiling phonons in a p o o ypical molecula qubi by IXS 79
4.1 S a eo a ................................ 79
4.2 Inelas ic X- ay sca e ing expe imen . . . . . . . . . . . . . . . . . . 80
4.2.1 Resul s............................... 83
4.3 The c i ical ole o ul a-low ene gy ib a ions . . . . . . . . . . . . . 86
4.4 Conclusions ................................ 89
5 E ec o chemical and s uc u al modi ica ions on he phonon in-
duced magne iza ion dynamics o Dysp osium-based Single-Molecule
3
Chap e 0
Magne s 90
5.1 The e ec o chemical composi ion on he pDOS o new Dy-based
SMMS................................... 91
5.1.1 Inelas ic neu on sca e ing expe imen . . . . . . . . . . . . . 96
5.1.2 Resul s............................... 98
5.2 Measu ing phonon modes o a Dy-based SMM in applied p essu e . . 99
5.2.1 Inelas ic neu on sca e ing expe imen . . . . . . . . . . . . . 101
5.2.2 Resul s...............................103
5.3 Conclusions ................................105
6 Combined EPR-NMR s udy o he Sup amolecula {C 7Ni}-Cu dy-
namics 106
6.1 Elec onic spin-la ice and spin-cohe ence imes . . . . . . . . . . . . 107
6.2 Spin dynamics p obed by 1H-NMR ...................110
6.3 Elec onic spin Manipula ions . . . . . . . . . . . . . . . . . . . . . . 113
6.4 Conclusions ................................114
Gene al Conclusion 115
4 Simone Chicco

Lis o Figu es
1.1 On he le and on he igh side o he igu e, he chemical s uc u e
o wo iconic single molecule magne s. On he le he i s SMMs o
exhibi magne ic hys e esis a he molecula le el and slow elaxa ion
o magne iza ion, Mn12 [1] ( o mally {Mn12O12[O2CCD2C(CD3)3]16
(CD3OD)4}). On he igh , he i s a e-ea h based SMMs o display
magne ic hys e esis up o liquid ni ogen empe a u e, Dysp osoce-
nium [2] ([(Cp )2Dy][B(C6F5)4]. In he middle, a ske ch o a double
well po en ial o he e e sal o he magne iza ion o a high-spin S
complex, wi h a high-spin g ound s a e mS=±S............ 16
1.2 S uc u e o some o he mos ep esen a i e molecula qubi eme ged
o each class desc ibed in he main ex . (a) {C 7Ni} o d-block
clus e molecules. [3] (b) Vanadium and Vanadyl-based molecula
qubi [4,5,6] (c) Te bium and Y e bium based single ion qubi o
a e-ea h based class. [7,8] (d) Two example o s uc u e linking
oge he mo e han one molecula qubi : a a e-ea h based dime
[CeE ] [9] and a sup amolecula chain deco a ed wi h wo {C 7Ni}
qubi linked wi h h ee coppe based qudi [10] ............ 18
1.3 Bloch sphe e showing he |0⟩and |1⟩basis o a qubi and all he
possible linea combina ion a|0⟩+b|1⟩ illing he sphe e su ace. . . . 20
1.4 Scheme o wo easible cooling me hods: he mal ini ializa ion (le ),
based on he slow cooling o he sys em o a empe a u e low enough
o isola e he g ound s a e om he exci ed ones; p ojec ion measu e-
men s ( igh ), based on esonan e.m. pulses o apidly p ojec he
sys em o a desi ed s a e (when he he mal popula ion is negligible). 21
1.5 In a o a ing ame sys em, p ecessing spins a e ep esen ed by s a ic
Bloch sphe e s a es. On he le , he sys em e olu ion om a pe -
u bed in-plane s a e owa d he he modynamical equilib ium s a e
|0⟩, imed by T1 elaxa ion ime. On he igh , he spin dephasing
p ocess o e ime (blue and g een a ow ep esen s spins wi h in-
c eased o dec eased nu a ing equency espec i ely), imed by T2
dephasing ime............................... 22
1.6 On he le , he S=1/2 elec onic g ound s a e degene acy spli ing
induced by he elec onic Zeeman in e ac ion HZ, as a unc ion o he
ield. On he igh , he |−1/2⟩→|1/2⟩elec onic ansi ion enabled
by his Hamil onian con ibu ion obse ed in a canonical spec al
analysis .................................. 25
5
Chap e 0
1.7 On he le , he nuclea degene acy spli ing induced by nuclea Zee-
man in e ac ion HZn, as a unc ion o he ield in a S= 1/2, I= 3/2
sys em. On he igh , he mIdegene a e |3/2⟩ → |1/2⟩,|1/2⟩ →
| − 1/2⟩,| − 1/2⟩ → | − 3/2⟩nuclea ansi ion enabled wi hin he
same elec onic mul iple ∆mS= 0. The spec a is ex ended o dis-
play bo h elec onic and nuclea ansi ions. . . . . . . . . . . . . . . 26
1.8 On he le he spli ing e ec induced by he pa allel Hype ine com-
ponen A∥when he ex e nal ield is applied pa allel o i ; On he igh
he expec ed spec a showing he degene a e nuclea and he esol ed
elec onic ansi ions, a ising om he Hype ine enso exp essed in
inse ..................................... 27
1.9 On he le he spli ing e ec induced by an axial Hype ine en-
so (inse ) when he ex e nal ield is applied pe pendicula o he
s onges componen ; On he igh he expec ed spec a whe e all
he nuclea and elec onic ansi ions a e esol ed. . . . . . . . . . . . 27
1.10 On he le he spli ing o he s a e wi h di e en |mI|induced by
a Quad upola coupling Pwi h componen s only along ˆzwhen he
ex e nal ield is applied pa allel o i ; on he igh he esul ing spec-
al componen s o he spin Hamil onian wi h only Quad upola ,
elec onic and nuclea Zeeman couplings. The nuclea ansi ions a e
pa ially esol ed, while he in e ac ion lea es he elec onic gaps un-
pe u bed.................................. 28
2.1 Ske ch o he ”HyReSpec ” spec ome e ha dwa e and o he LC-
Resonan p obe. T ansmi e and Recei e s ages ha dwa e a e be e
de ailed in [11]. .............................. 37
2.2 Sca e ing p ocess wi h incoming and sca e ed beam ene gies Ei, E
and wa e ec o s 
ki,
k , and geome ical ep esen a ion o he ans e
momen um 
Qand he sca e ing angle θ................. 40
2.3 The layou o he IN8 spec ome e a ILL. The pic u e is ep oduced
om he acili y webpage. ........................ 44
2.4 Top (le ) and side ( igh ) iews o he IN1 spec ome e layou . The
pic u e is ep oduced om he ILL webpage............... 45
2.5 Schema ic ep esen a ion o he ID28 spec ome e layou . The pic-
u e is ep oduced om he ESRF websi e ............... 48
3.1 [VO(TPP)] molecula s uc u e. The p incipal o hogonal symme y
di ec ions o he molecula c ys al a e shown independen ly: he po -
phy in lying in he ab plane and he oxido anadium bond di ec ion
along he caxis.............................. 51
3.2 Spec a collec ed a di e en applied ield B0along he molecule sym-
me y di ec ions: he ab plane (a), and he caxis (b). In inse , a spu-
ious peak is iden i ied om he absence o any ime-domain spin-echo
signals. Indeed black line shows he echo signal in quad a u e o a
eal nuclea exci a ion in co espondence o τ(dashed line), whe eas
a spu ious appea s as he pe sis en g ay oscilla ions. Rep oduced
om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 52
6 Simone Chicco
Chap e 0
3.3 Red do s co espond o he measu ed ansi ion equencies in he
b oadband NMR spec a o 51V. Black lines ep esen he calcula ed
e olu ion o he ansi ion equencies as a unc ion o he ield B0
applied (a) in he ab-plane, and (b) along c-axis. Wi h capi al le e s
AB# and C# we label he ∆mI=±1 ansi ions o each di ec ion.
Shaded a eas we e no expe imen ally explo ed. Rep oduced om
Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . . . . 53
3.4 Ene gy le els as a unc ion o he applied ield om he diagonal-
iza ion o he spin Hamil onian wi h s a ic ield B0applied (a) in
plane ab and (b) along c-axis. The ed and blue shades highligh
he mS=±1/2 mul iple s, espec i ely. On he le o each plo , wo
inse s show a zoom in o he nuclea le el spli ing o he lowe and up-
pe mul iple s, wi h he le els ma ked by he nuclea spin componen s
mIalong he ield. Ve ical ma ks highligh he nuclea ansi ions
iden i ied in he spec a o igu e 3.2. Rep oduced om Re . [4] wi h
pe mission om he Royal Socie y o Chemis y. . . . . . . . . . . . . 54
3.5 Di e ence in subsequen nuclea ansi ions ene gies δ(mI)=(EmI+1−
EmI)−(EmI−EmI−1) in he mS= 1/2 mul iple (c osses), as a unc-
ion o he applied ield Bˆ
x. These alues a e in sound ag eemen wi h
wha expec ed om he diagonaliza ion o he pseudo-quad upola
Hamil onian (black line). Rep oduced om Re . [4] wi h pe mission
om he Royal Socie y o Chemis y. . . . . . . . . . . . . . . . . . . 55
3.6 Echo in ensi y decay as a unc ion o he delay be ween he Hahn echo
exci ing and e ocusing pulses. The expe imen al da a o a selec ed
nuclea ansi ion ((a) AB2, (b) C1) a e plo ed oge he wi h he
elaxa ion a e T2single exponen ial i o di e en applied ields.
Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o
Chemis y.................................. 56
3.7 Nuclea phase memo y ime o di e en nuclea ansi ions, mea-
su ed wi h a e ocusing Hahn-echo sequence, as a unc ion o he
applied ield B0(a) in he ab-plane and (b) along c-axis. Rep oduced
om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 56
3.8 Nuclea Rabi oscilla ions o e a wide ime in e al, o ansi ions
AB1 and AB7, co esponding o he ansi ion mI=−3/2→ −1/2
o bo h he lowe and highe -ene gy elec onic mul iple s mS=
±1/2, espec i ely. Rep oduced om Re . [4] wi h pe mission om
he Royal Socie y o Chemis y. . . . . . . . . . . . . . . . . . . . . . 57
3.9 Rabi oscilla ions as a unc ion o pulse B1a enua ion, on he an-
si ion labeled AB1 be ween he s a es mI=−3/2→ −1/2 o he
lowes elec onic mul iple mS= 1/2. The powe is de ined wi h
an inc easing a enua ion om a e e ence alue (0 dB). Rep oduced
om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 58
7 Simone Chicco
Chap e 0
3.10 Simula ed e ec o a pulse esonan wi h a nuclea ansi ion ha
inc ease he mI alue by ∆mI= +1, when he sys em is ini ialized in
mI= 1/2,−1/2,−3/2 (a,d; b,e; c, espec i ely). On he le (a-c) he
di e ence be ween a ge le el popula ion as a unc ion o ime. On
he igh (d- ) he colo map ep esen s he popula ion e olu ion wi h
ime o all he nuclea s a es. The pulse is applied along he molecule
ˆxaxis wi h in ensi y B1= 5 G. The a ge ed ansi ions a e: (a,d)
mI= 1/2→3/2, (b,e) mI=−1/2→1/2, (c, ) mI=−3/2→ −1/2,
wi h ∆mS= 0. The ocus he e is on he mS= 1/2 spin mul iple ,
which is he only one popula ed because o he s a es ini ializa ion.
Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o
Chemis y.................................. 60
3.11 Simula ed e ec o a pulse esonan wi h a nuclea ansi ion ∆mI=
+1, on he he mally popula ed sys em. On he le (a-c) he di e -
ence be ween he a ge le els popula ions as a unc ion o ime. On
he igh (d-i) he colo map ep esen s he popula ion e olu ion o e
ime o all he nuclea s a es o bo h he elec onic spin mul iple s
mS=±1/2. The pulse is applied along he molecule ˆxaxis wi h in en-
si y B1= 5 G. The a ge ed ansi ions a e: (a-d) mI= 1/2→3/2,
(b-e) mI=−1/2→1/2, (c- ) mI=−3/2→ −1/2, wi h ∆mS= 0.
Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o
Chemis y.................................. 61
3.12 On he le , he [V(Cp)2Cl2] ben -me allocene s uc u e. The molec-
ula e e ence ame is shown, wi h he y-axis bisec ing he Cl-V-Cl
angle, he z-axis pe pendicula o he Cl-V-Cl plane (axial di ec ion)
and he x-axis unequi ocally de ined. On he igh he c ys allo-
g aphic uni cell wi h he wo inequi alen molecules labeled as V1
and V2. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021
Ame ican Chemical Socie y. . . . . . . . . . . . . . . . . . . . . . . . 62
3.13 Pic u e o he c ys al collec ed om mic oscopy (le ) and a ske ch
o he c ys al shape ( igh ). The wo a e compa ed o show he edge
leng h hie a chy and he ela i e indexing o c ys al aces wi h he co -
esponding c ys allog aphic planes. Rep in ed wi h pe mission om
Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. . . . . . . . . . 63
3.14 The c ys allog aphic planes (111), (010) and (10-1), ha co espond
o he c ys al aces, a e highligh ed in he uni cell. This ske ch assis
in he isualiza ion o he applied ield di ec ions wi h espec o he
molecula e e ence ame. Rep in ed wi h pe mission om Re . [12].
Copy igh 2021 Ame ican Chemical Socie y. . . . . . . . . . . . . . . 63
8 Simone Chicco
Chap e 1
In oduc ion
Molecula magne ism has ep esen ed a e y ac i e esea ch opic o e he las ew
decades. The ui ul collabo a ion o scien is s belonging o di e en backg ounds,
such as Physics and Chemis y, has con ibu ed in pa ing he way o he imple-
men a ion o hese sys ems in quan um in o ma ion p ocessing. This hesis ocuses
on se e al aspec s o he expe imen al cha ac e iza ion o molecula magne s spin
dynamics. By exploi ing di e en echniques we deepen he unde s anding o e-
laxa ion dynamics mechanisms in a ious molecula magne s, we ully cha ac e ize
he model spin Hamil onian o some p o o ypical sys ems, we de ec he e ec o
physical and chemical s imuli on hei phonon-induced elaxa ion and inally we
implemen a ge ed manipula ion o he sys em spin s a es by adio equency and
mic owa e pulses, demons a ing he possibili y o exploi ing hese complexes o
quan um in o ma ion p ocesses.
1.1 Molecula nanomagne s
The disco e y, a he beginning o he 1990s, o magne ic hys e esis o igina ing a
he single molecule le el by R. Sessoli and cowo ke s [1] igge ed he de elopmen
o a b and new esea ch ield [19] which nowadays pe mea es in o se e al b anches
o science, om quan um in o ma ion heo y o supe -dense in o ma ion s o age,
om nanoscopic sensing and nanoscopic p obing o elec ic anspo e c: Molecu-
la magne ism.
The p ogeni o o his compounds class, displaying magne ic hys e esis and slow
elaxa ion o he magne iza ion, was he so-called Mn12 (aka [Mn12O12(CH3COO)16
(H2O)4]). [1,20] This sys em, and he class o compounds ha eme ged in he ol-
lowing yea s (as Fe8o Ni10), a e clus e s o exchange-coupled d-block me als, whose
magne ic beha io can be app oxima ed o a single gigan ic o al spin Sslowly e-
laxing because o an e ec i e double well po en ial a ising om he la ge g ound
s a e spin and a s ong magne ic aniso opy. [21,22] The p esence o an ene gy
ba ie o he eo ien a ion o he magne ic momen leads o he bis abili y o he
g ound s a e o al spin S( ig. 1.1). This e ec made hese molecules in e es ing
candida es o he ul ima e minia u iza ion o he memo ies, whe e each molecule
can encode a single ”bi ” o memo y ( om he e he name Single-Molecule-Magne s).
Howe e , he magne ic bis abili y o he gian spin g ound s a es wi h mS=±Sin
unde mined by di e en elaxa ion p ocesses: some a e empe a u e independen ,
15

Chap e 1
such as he quan um unneling o he magne iza ion (QTM) below he ene gy ba -
ie , o he s a e ins ead he mally ac i a ed, due o he coupling o he spin wi h
molecula ib a ions, gene a ing o he elaxa ion pa hs such as he O bach mul i-
s ep mechanisms o e he ene gy ba ie and non- esonan Raman. Consequen ly,
he magne ic hys e esis in hese i s SMMs can be obse ed only when QTM is no
ac i a ed/ine ec i e and a ex emely low empe a u es, whe e he mal luc ua ions
a e supp essed. [19,23,24,25]
Figu e 1.1: On he le and on he igh side o he igu e, he chemical s uc u e o
wo iconic single molecule magne s. On he le he i s SMMs o exhibi magne ic
hys e esis a he molecula le el and slow elaxa ion o magne iza ion, Mn12 [1]
( o mally {Mn12O12[O2CCD2C(CD3)3]16 (CD3OD)4}). On he igh , he i s a e-
ea h based SMMs o display magne ic hys e esis up o liquid ni ogen empe a u e,
Dysp osocenium [2] ([(Cp )2Dy][B(C6F5)4]. In he middle, a ske ch o a double
well po en ial o he e e sal o he magne iza ion o a high-spin Scomplex, wi h
a high-spin g ound s a e mS=±S.
A ema kable s ep in he magne iza ion blocking empe a u e o SMMs ha e been
achie ed in he ollowing yea s wi h a second gene a ion o molecula nanomagne s,
based on a single -block magne ic ion. [26] He e indeed, he s ong spin-o bi cou-
pling o he a e-ea h elemen s (e.g. Lan hanides) combined wi h a p ominen axial
ligands c ys al ield, p oduces a la ge aniso opy ba ie , ha s abilize he g ound
s a e spin e en a high empe a u e ( ig. 1.1). The ene gy ba ie s o se e al hun-
d eds o Kel in deg ees high o mos o hese single molecule magne s is howe e no
he only ac o ha go e ns he pe sis ence o he hys e esis a high empe a u es.
Indeed, he complex in e play o se e al elaxa ion p ocesses in e e es wi h he
magne ic s abili y and an ex ensi e knowledge on hese mechanisms is undamen al
o he imp o emen o he pe o mances. A e y low empe a u e, he mos e ec-
i e elaxa ion mechanism is he quan um unneling o he magne iza ion (QTM),
[19] which is empe a u e independen bu can be made ine icien in p onounced
axial ligands ield en i onmen . [27] Mo eo e , a highe empe a u e, whe e la ice
and molecula ib a ions a e ac i a ed, he coupling o phonons wi h he magne ic
momen s induce an incohe en elaxa ion dynamics in hese sys ems. As a conse-
quence, he enginee ing o hese ib a ions and he s udy o hei coupling wi h he
spin is a key s ep o he imp o emen o he pe o mances o molecula sys ems.
[28,29,30] Taking in o conside a ion all hese ac o s, a eco d b eaking magne ic
blocking empe a u es o 80 K ha e been ecen ly epo ed in Dysp osium-based
molecules ([(Cp )2Dy][B(C6F5)4], wi h Cp = 1,2,4- i- e -bu ylcyclopen adienyl)
wi h double-decke axial ligands en i onmen (see ig. 1.1). [2,15] He e, he p omi-
16 Simone Chicco
Chap e 1
nen axiali y o he ligands ield, besides inducing a g ea isola ion o he g ound
s a e K ame double s om he exci ed c ys al ield s a es, con ibu e signi ican ly
o he supp ession o he quan um unneling o he magne iza ion unde he ba ie .
Mo eo e , he s ong bonds in he ligands en i onmen o he a e-ea h ion p e en s
e icien spin coupling wi h molecula ib a ions, hus enhancing he pe sis ence o
he magne ic bis abili y a highe empe a u e.[15]
Ve y ecen ly a cos -e ec i e semi-ab ini io app oach ha e been de eloped o un-
de s and he elaxa ion dynamics o hese sys ems. Physical insigh s on he o igin
o hese elaxa ion mechanism can indeed gi e undamen al in o ma ion in o de
o de elop he nex class o eco d b eaking molecula complexes, wi h blocking
empe a u es abo e he liquid ni ogen limi . [17,31] In pa icula non- esonan
wo phonon Raman p ocesses we e disco e ed o play he key ole in igge ing he
magne iza ion elaxa ion o hese SMM class in he ange o empe a u es c ucial
o he de e mina ion o TB(see also sec . 5).
Because o he long spin li e ime and he ema kable cohe ence in insic o se e al
molecula magne s, hese sys ems o e s also an a ac i e op ion o he ealiza ion
o quan um bi s. Indeed, he wo quan um basis s a e |0⟩and |1⟩can be easily
encoded in o he wo s a es o a single spin S= 1/2 molecule. The majo obs a-
cle o he exploi a ion o hese sys ems in quan um in o ma ion p ocessing is he
pe sis ence o he cohe ence o a gi en supe posi ion o s a es in each qubi uni ,
de ined by he decohe ence ime T2(see sec . 1.2). The e o e, since la ice ib a-
ions, oge he wi h elec ic and magne ic in e ac ions wi h he en i onmen , make
quan um supe posi ions o s a es e y agile, lo o e o ha e been o ien ed in he
las decades o he syn hesis o molecula complexes wi h long cohe ence imes and
o he in es iga ion o he decohe ence mechanisms.[5,28,30,32,33,34,35,36,37]
The i s p oposed molecula qubi in he 2000s wi h a cohe ence ime T2>1µs was
again a clus e wi h an e en numbe o d-block a oms an i e omagne ically (AF)
coupled: a [C 7NiF6Pi 16] ing. Indeed, he subs i u ion o a C magne ic cen e
in an Homome allic S= 0 ing [38] wi h a di alen Ni ca ion causes he unbalance
o he AF coupling and he eme gence o a S= 1/2 double g ound s a e. [3,39]
As well as long cohe ence ime (T2), hese clus e ed sys ems we e also capable o
implemen he i s one-qubi ope a ions, wi h clea Rabi oscilla ions o spin popula-
ions demons a ed in he benchma k C 7Ni ing and in Vanadium-based molecula
clus e s.[40,41]
Recen ly, complexes wi h a single ansi ion-me al ion a e in ac eme ging as highly
cohe en spin qubi s (see ig. 1.2). In pa icula , se e al Vanadium-based sys ems
display cohe en oscilla ions o spin s a e popula ions also app oaching oom em-
pe a u e, [35] as well as empe a u e- esilien cohe ence imes, o he o de o he
milliseconds in he op imal condi ions.[5] Vanadium complexes ha e also been e-
cen ly s udied as p omising coupled qubi -qudi sys ems. Indeed, he coupling o
he elec onic Sand nuclea Ispin deg ees o eedom p o ides a 2I+ 1 mul ile el
s uc u e called qudi (i.e., quan um digi s ea u ing mo e han wo le els d > 2)
ha can be exploi ed o expand he compu a ional space o o implemen quan um
e o co ec ion algo i hms on single molecula objec s. [4,12,42]
Ra e-ea h ions play a ole also in he de elopmen o molecula qubi s. Indeed, he
ones ea u ing a g ound s a e K ame double , well sepa a ed om he exci ed s a es
by a signi ican c ys al ield spli ing, can be desc ibed as sys ems wi h an e ec i e
spin Se = 1/2 [8,33,43] o qubi encoding (see ig. 1.2). In o he cases, owing
17 Simone Chicco
Chap e 1
Figu e 1.2: S uc u e o some o he mos ep esen a i e molecula qubi eme ged
o each class desc ibed in he main ex . (a) {C 7Ni} o d-block clus e molecules.
[3] (b) Vanadium and Vanadyl-based molecula qubi [4,5,6] (c) Te bium and
Y e bium based single ion qubi o a e-ea h based class. [7,8] (d) Two example
o s uc u e linking oge he mo e han one molecula qubi : a a e-ea h based
dime [CeE ] [9] and a sup amolecula chain deco a ed wi h wo {C 7Ni}qubi
linked wi h h ee coppe based qudi [10]
an elec onic spin S > 1/2, a e-ea h based molecula qubi s can open new possi-
bili ies o he expansion o he a ailable compu a ional space. This can be done by
exploi ing all he elec onic spin s a es o encode mo e han a single qubi in each
molecule [44] o a single qudi wi h se e al ds a es. [7,45] In addi ion, a e-ea h
based sys ems embedding mo e han one asymme ic magne ic uni , as dime s [46] o
ime s [47] ha e been p oposed as single-molecule quan um ha dwa es wi h chem-
ically coupled qubi s. A simila app oach has been pu sued o chemically-linking
oge he se e al ansi ion me al clus e s (i.e. ings) in sup amolecula assembly
(see examples in ig. 1.2). [10,48,49]
The inal s ep o he ac ual implemen a ion o molecula sys ems o quan um
compu a ion is he deposi ion o single objec s and hei wi e-up in ”p ocesso -
like” s uc u es. [42,50,51] Consequen ly, se e al s udies ha e ecen ly ocused on
he deposi ion o magne ic molecules on su aces,[52,53,54] and on he coupling
wi h sys ems ha enable he manipula ion and ead-ou o hese single quan um
objec s,[55] e.g. wi h supe conduc ing esona o s o nanos uc u ed junc ions.[7]
1.2 Di Vincenzo c i e ia o Molecula Magne s
Molecula spin sys ems ha e been iden i ied as a p omising e olu ion o Noisy
In e media e-Size Quan um de ices (NISQs) in o de o push o wa d he capa-
bili ies o mode n quan um echnologies. [56,57,58]
Indeed, among a a ie y o possible sys ems s udied in he pas decades, as ul a-
cold a oms,[59] pho ons,[60] supe conduc ing ci cui s, [61,62] Ni ogen and o he
18 Simone Chicco
Chap e 1
acancies in solid c ys als [63,64,65,66] e c., molecula sys ems appea o be pa ic-
ula ly sui able o he ealiza ion o quan um compu a ion, [50,51,67,68,69]. This
because hey po en ially ul ill he equi emen s se by Di Vincenzo and cowo ke s
a he beginning o he millenium o he ealiza ion o a quan um compu ing pla -
o m. [70] Indeed, despi e sho e wi h espec o o he pla o ms (e.g. see [71]),
he cohe ence imes o many molecula qubi s ha e been demons a ed o be long
enough o he spin s a e manipula ion wi hou signi ican losses in cohe ence. [39]
In he ollowing, we will deepen each o hese basic c i e ia o he ac ual ealiza-
ion o quan um in o ma ion p ocessing wi h physical pla o ms, highligh ing how
molecula spin sys ems sa is y hese equi emen s and which aspec s a e cu en ly
he mos c i ical. [42]
Scalable de ined Qubi
Any sys em, o being exploi ed as a quan um bi , mus be well cha ac e ized. This
means ha he pa ame e o i s Hamil onian, oge he wi h he coupling wi h o he
nea by qubi s and wi h physical (pho ons, phonons, ex e nal ields, e c.) s imuli
exploi ed o s a e manipula ion, mus be known wi h accu acy. He e, magne ic
molecules ha e he ad an ages o be pe ec ly ep oducible quan um objec s, chem-
ically syn he ized and ”enginee able”, and easily cha ac e ized by bulk spec oscopic
esonan echniques as Nuclea Magne ic Resonance and Elec on Spin Resonance
o single molecula -c ys als (as de ailed in sec . 2.1.1 and 2.1.2). [45,72] In-
deed, e en o a la ge ensamble o indi idual molecules, such as a single c ys al
a angemen , he s ain o he single molecule pa ame e s (g-s ain and D-s ain as
de ined in [73]) is gene ally limi ed and esul s in a mode a e b oadening o spec al
linewid h (cen e ed a ound he esonan equency o he mean pa ame e alues).
Thus, despi e s ain e ec s, o he pu pose o elec omagne ic ield manipula ions,
molecules in a c ys al can be conside ed p ac ically iden ical. In addi ion, some
o hese molecules na u ally posses a eal (o e ec i e) S= 1/2 g ound s a e ha
ou line a e y simple ealiza ion o a qubi , since he wo mS=±1/2 spin s a es
na u ally ep esen he |0⟩and |1⟩s a e ha de ine he qubi s a e. [3,5,8]
Mo eo e , besides being well de ined, a physical sys em o being applied as a
qubi in a quan um ha dwa e mus be scalable. The scaling up, o expand he com-
pu a ional space, can be done in se e al ways. [50] By chemical design, o example,
i is possible o c ea e molecula s uc u e ha hos s mo e ha one magne ic cen e ,
as dime s o ime s o qubi s. [9,47] Simila ly, AF ings, such as [C 7Ni], a e well
known and ho oughly s udied sys ems ha ea u e he possibili y o be bonded
oge he in sup amolecula s uc u es, combined wi h o he spin 1/2 complexes,
[10,48,49,74,75,76] and o be deposi ed on su aces, e aining hei magne ic
p ope ies. [53,77,78]
Ano he op ion, mo e ecen ly de eloped, is o exploi he in e nal deg ees o ee-
dom o he sys em, o expand he compu a ional space. The esul ing quan um
sys em can be seen as a qubi ha ea u e mo e han wo disc e e s a es (qudi ) o
as a collec ion o e ec i e wo-le el qubi embedded wi hin he same molecula uni .
This happen o example in S > 1/2 complexes, as he S= 7/2 o Gd3+ ions whe e
d= 8 elec onic spin le els a e a ailable. [43,44] Ra he han he elec onic spins,
he nuclea spin deg ees o eedom o he molecule me al ion can be exploi ed as
19 Simone Chicco
Chap e 1
Figu e 1.3: Bloch sphe e showing he |0⟩and |1⟩basis o a qubi and all he possible
linea combina ion a|0⟩+b|1⟩ illing he sphe e su ace.
ano he powe ul esou ce o quan um compu a ion, because o hei isola ion om
he en i onmen . The nuclea spin is indeed e icien ly coupled o he elec ons by
Hype ine in e ac ion, esul ing in a spli o he le el degene acy and in a conside -
able speed up o he equi ed manipula ion ime, ega dless o i s isola ion om he
en i onmen . [4,12,45]
Howe e , he ad an ages gained by inc easing he numbe o elec onic and nuclea
s a es o he qudi is no su icien o co e complex algo i hm a single molecule
le el. The e o e, i is necessa y o design mul i-qubi pla o ms ha enables he
ealiza ion o wo- o many-qubi s ga es. As p oposed in his e iew [50] by S. Ca -
e a e al., a possibili y p o ided by molecula qubi s, hanks o hei s abili y (e en
isola ed) and chemical unabili y, is o link he molecules chemically, by in oducing
swi chable molecula linke s [76,79] wi h ul a- as swi ching- ime (well below he
hund ed o ns, depending on he sys em) [76,80], o physically by pe iodic deposi-
ion on ac i e subs a es, as supe conduc ing on-chip esona o . [51,81] Bo h hese
app oaches allows o con ol cohe en ly and indi idually he spin s a es. [50]
Qubi Ini ializa ion
F om a compu a ional poin o iew i is undamen al o ha e a clea pic u e o
he qubi s s a e a he beginning o he quan um in o ma ion p ocessing. This e-
qui emen is ul illed i he physical sys em chosen o he compu a ion can be easily
ini ialized o a well known s a e. [70] Fo he case s udy o molecula magne s, ini-
ializa ion o he s a e can be easily achie ed by cooling s a egies. Gi en a molecula
sys em wi h an e ec i e spin S= 1/2 g ound s a e, by na u ally cooling a he mK
empe a u e ( u inely achie ed in NISQ pla o ms) we ini ialize he sys em by pop-
ula ing only he lowes ene gy spin s a e ( ig. 1.4). [82] Fas ini ializa ion du ing
compu a ion p ocesses can also be implemen ed by a i icial cooling wi h p ojec ing
measu emen s in o he desi ed s a e (e.g. wi h elec omagne ic pulses). Ano he
s a egy sui able o molecula sys ems is he exploi a ion o cooling algo i hms,
[83] ha allows o each ini ialized s a es wi h pulse sequences, e en a empe a u e
in which all he s a es a e he mally popula ed.
20 Simone Chicco

Chap e 1
Figu e 1.4: Scheme o wo easible cooling me hods: he mal ini ializa ion (le ),
based on he slow cooling o he sys em o a empe a u e low enough o isola e
he g ound s a e om he exci ed ones; p ojec ion measu emen s ( igh ), based on
esonan e.m. pulses o apidly p ojec he sys em o a desi ed s a e (when he
he mal popula ion is negligible).
Qubi Cohe ence imes
When he physical quan um sys em is placed in con ac wi h he en i onmen ,
he gene ic qubi s a e |ψ⟩=a|0⟩+b|1⟩degene a es, in a ime we de ine as de-
cohe ence ime, in o he classical mix u e desc ibed by he sys em densi y ma ix
ρ=|a|2|0⟩⟨0|+|b|2|1⟩⟨1|. The e o e, his decohe ence ime ep esen s he li e ime
o he quan um supe posi ion s a e in which he qubi is placed o compu a ional
pu poses.
Fo a p o icien implemen a ion o a quan um algo i hm, he decohe ence ime o
he physical pla o m on which i is implemen ed is no supposed o las longe han
he en i e compu a ion i sel . Indeed, he e o induced on quan um s a es by de-
cohe ence can be co ec ed by e o co ec ion p o ocols, as p oposed by Sho and
S eane in 1995 and 1996 espec i ely. [84,85] As a consequence, a good comp omise
o he decohe ence o a sys em is se by he ga e ime (”Clock ime” in [70]) i
Quan um E o Co ec ion (QEC) is applied. In o de o implemen p o icien ly
any quan um ga e, wi h negligible cohe ence losses due o elaxa ion, he sys em
decohe ence ime mus be longe han he ime needed o implemen a single ga e
(a leas 104 imes he ”clock” o comple e aul - olle an compu a ion [70]).
In his ega ds, magne ic molecules display unique ad an ages in uning he elax-
a ion mechanisms. The p incipal sou ces o decohe ence come om he in e ac ion
o molecula spins wi h molecula ib a ions (phonons), ligands nuclea spins and
nea by elec onic spins. [51] In he las decades, a e he iden i ica ion o hese elax-
a ion mechanisms, se e al s a egies ha e been p oposed o ema kably imp o e he
decohe ence imes o molecula spins. The iden i ica ion o he speci ic ib a ional
modes mos s ongly coupled wi h he spin (de imen al o quan um cohe ence)
enables he supp ession o hose elaxa ion pa h by chemical op imiza ion o he
s uc u e, e.g. by s i ening o ligands o g oups emo al (see sec . 4). [29,30,86]
Mo eo e , decohe ence e ec s induced by magne ic dipola in e ac ion ha e been
ema kably educed by exploi ing s ong magne ic dilu ion o molecula magne s in
diamagne ic hos s uc u es. Finally, he in e ac ion o elec onic spins wi h he
nuclei hos ed in o he molecule ligands s uc u e canno be easily supp essed, since
21 Simone Chicco
Chap e 1
Hyd ogen is he one ha in e e e he mos and also he mos abundan elemen in
his class o molecules. Howe e , he molecula s uc u e can be op imized in o de
o educe his con ibu ion o elaxa ion by sepa a e as much as possible he sou ces
o decohe ence and educing hei abundance (e.g. by deu e a ion o luo ina ion).
[87]
Two di e en cha ac e is ic imes a e used o quan i y he decohe ence o a molec-
ula qubi : T1and T2. The i s desc ibe he in e ac ion o he spin wi h he c ys al
s uc u e (phonons) and i is hen called spin-la ice elaxa ion ime, while he sec-
ond is called spin-spin elaxa ion ime (o phase memo y ime) and accoun s o
he in e ac ion wi h nea by spins, deno ing he cohe ence ime o he qubi . I we
ep esen he spin mo ion o he sys em in he o a ing ame ( ha allows us o ge
id o nu a ion mo ion) as a Bloch sphe e, he wo elaxa ion mechanisms can be
dis inguished in wo di e en mo ions as shown in igu e 1.5.
Figu e 1.5: In a o a ing ame sys em, p ecessing spins a e ep esen ed by s a ic
Bloch sphe e s a es. On he le , he sys em e olu ion om a pe u bed in-plane
s a e owa d he he modynamical equilib ium s a e |0⟩, imed by T1 elaxa ion
ime. On he igh , he spin dephasing p ocess o e ime (blue and g een a ow
ep esen s spins wi h inc eased o dec eased nu a ing equency espec i ely), imed
by T2dephasing ime.
T1indica e he ime needed o he sys em o e-es ablish he he modynamical equi-
lib ium a e a pe u bing s imulus, hus p obing i s coupling wi h he en i onmen
wi h which i exchanges ene gy. T2ins ead, indica es he decay a e o spin phase
cohe ence wi hin he sys em.
The possibili y o implemen ing QEC, o p o ec he in insic agili y o quan um
in o ma ion, on a single mul ile el quan um objec ep esen s ano he undamen al
ad an age o exploi ing magne ic molecules as building blocks o quan um com-
pu a ion pla o ms. Indeed he qudi a chi ec u e, ea u ing mul iple elec onic o
nuclea le els in a single molecule, enables he encoding o e o p o ec ed ”logical”
qubi s in o a single quan um objec . [88] The ap i ude o he implemen a ion o
QEC pe mi s o elease he s ic equi emen s o ex emely long elaxa ion ime
o molecula qubi s, since he e o s a ising om dephasing can be mi iga ed by
a i icial co ec ions.
22 Simone Chicco
Chap e 1
Uni e sali y
The ou h equi emen se by Di Vincenzo o a p o o ypical quan um pla o m is
he capabili y o implemen ing a Uni e sal se o single and mul i-qubi quan um
ga es. Since any quan um algo i hm can be decomposed as a sequence o uni a y
ans o ma ions Ui,[89] he equi emen is ansla ed in iden i ying sys em in which
is possible o implemen any uni a y ans o ma ion. When e e ing o a physical
sys em, his condi ion is ul illed i all he manipula ion o he sys em can be iden i-
ied by a se o Hamil onians, ha gene a e all he equi ed uni a y ans o ma ions
Ui=eiHi /¯h. Implemen ing he uni e sal se o uni a y ans o ma ion enable he
simula ion o any quan um ime e olu ion o a sys em.
In his ega d, molecula magne s a e easily manipulable sys ems, whe e he molecule
spin is pushed in in e ac ion wi h an ex e nal elec omagne ic (e.m.) s imulus o
manipula e he elec onic and nuclea s a es, wi hou any signi ican cons ain .
The e o e, we can easily assume ha gi en a magne ic molecule, i s spin s a e can
be manipula ed in o de o co e uni o mly he whole Bloch sphe e ( ig. 1.3). The
expansion o he compu a ional space o mul i-qubi ga es, mus go h ough he
coupling o se e al molecula uni s. The e o e, as discussed o he scalabili y, o e-
alize unc ional wo and mul i-qubi en angling ga es, a ious molecula uni s mus
be wi ed up by coupling wi h ex e nal supe conduc ing on-chip esona o s [81] o
by in oducing swi chable molecula couple s o linke s, sensible o ex e nal s imuli.
[76,79]
Qubi measu emen s
The inal s ep o a quan um compu a ion p ocess is he eadou o he esul . The
ideal physical pla o m mus allow he measu emen o he s a e o each speci ic
qubi , wi hou al e ing he s a e o he o he s.
This aspec is cu en ly he mos c i ical, since i po en ially equi e he ealiza ion
o a single molecule ead ou mechanism. Rega ding he wo possible a chi ec u e
p oposed abo e, ha ing single-molecule qubi s spa ially sepa a ed and selec i ely
coupled by plana esona o s can allow he indi idual eadou o a desi ed p o-
cesso egion wi h e.m. s imuli, wi hou causing he collapse o he ull sys em
wa e unc ion. [50,55] Howe e , up o da e his app oach poses s ic expe imen al
limi a ion, due o he e y weak spin-pho on coupling (see [90]). Mo eo e , o he
pla o ms pe mi s single qubi eadou by elec ic anspo measu emen s [42] o
ligh i adia ion. [91] Elec ic and magne ic ield a e also exploi ed o he eadou
o single molecule qudi , when he popula ion o a single couple o le els can be
ex ac ed by esonan exci a ions, wi hou a ec ing nea by s a es.
1.3 Spin Hamil onian o malism
The Spin Hamil onian app oach enables he desc ip ion o he s a ic p ope ies and
cohe en dynamics o magne ic molecules and i ep esen s he mos e ec i e o -
malism o he in e p e a ion o expe imen al da a om spec oscopic echniques.
Gi en a molecula sys em i pe mi s o desc ibe accu a ely he ene gy le els o he
sys em and he hie a chy o magne ic in e ac ions. [19,92,93]
This app oach pe mi s o desc ibe all he con ibu ion o he sys em Hamil onian
23 Simone Chicco
Chap e 1
in e ms o spin-only ope a o s, s a ing om he assump ion ha each ion in he
molecule can be desc ibed as an e ec i e spin ˆsi. The la es assump ion is accu a e
o sys ems wi h quenched o bi al angula momen um (as in almos all 3d sys ems),
o when he o bi al angula momen um is no quenched (as in 4 sys ems), bu he
c ys al ield spli ing o he Jquan um numbe yields a well isola ed g ound s a e
mul iple .
The mos gene al spin Hamil onian desc ibing molecula nanomagne s can be w i -
en adding se e al con ibu ions: [19,92]
H=Hex +HCF +Hdip +HZ+Hn.(1.1)
He e, Hex ep esen s he exchange in e ac ion be ween mangne ic ions in he molecule,
HCF he e ec o he local c ys al ield, and Hdip he h ough-space and h ough-
bond dipola coupling. Finally, HZmodels he Zeeman in e ac ion wi h an ex e nal
magne ic ield and Hnde ined as ” ine s uc u e” Hamil onian, accoun s o he
in e ac ion wi h he nuclea deg ees o eedom. In his hesis we will mainly ocus
on single ion molecules wi h e ec i e o in insic spin S= 1/2, in which he ex-
change in e ac ion Hex be ween ions in he molecule is absen and he c ys al ield
exci ed s a es a e igno ed because o he conside able isola ion om he g ound s a e
K ame double . Mo eo e , by d as ic dilu ion ( up o 2%) o he MNM in i s c ys al
s uc u e wi h i s diamagne ic analogue, he in e -molecula dipola con ibu ion o
he spin Hamil onian in 1.1 is also s ongly supp essed. The e o e, o hese sys ems,
he spin Hamil onian in 1.1 can be uni e sally exp essed in i s educed o m:
H=HZ+Hn=HZ+HHyp +Hq+HZn,(1.2)
whe e in he ine s uc u e Hamil onian we made explici all he con ibu ion due o
he in e ac ion wi h he nuclea momen um: he Hype ine coupling HHyp, he nu-
clea quad upola in e ac ion Hqand he nuclea Zeeman e ms HZn. These e ms
a e exp essed in e ms o p oduc o spin ope a o s, as de ailed in he ollowing
sec ions.
1.3.1 Elec onic and Nuclea Zeeman in e ac ions
The g ound s a e K ame degene acy ( o bo h sys ems cha ac e ized by eal o
e ec i e spin 1/2 g ound s a es) is emo ed by applying an ex e nal magne ic ield
B, as a esul s o i s Zeeman coupling wi h he elec onic spin S. The Hamil onian
o he elec onic Zeeman e ec is hus gi en by he scala p oduc :
HZ=µBˆ
S·gS·B,(1.3)
whe e µB= 1.3996 ×104MHz/T is he Boh magne on and gS he spec oscopic
spli ing enso , ha a ies o di e en ions and s uc u es. I we ocus on he
coo dina e e e ence ame o he molecule, he enso ial p oduc is educed as:
HZ=µB(gxSxBx+gySyBy+gzSzBz),(1.4)
wi h he ield componen s Bx,y,z de ined in pola coo dina es wi h espec o he
molecule ame (Bx=Bsin θcos ϕ,By=Bsin θsin ϕ,Bz=Bcos θ). When he
24 Simone Chicco
Chap e 1
con enien o desc ibe he sys em by ocusing on he educed Hilbe space o he
spin Ssys em, which con ains all he spin dynamics in o ma ion. The ope a o s
ˆ
Xac ing on his educed Hilbe space a e de ined by means o he pa ial ace
o malism:
⟨ˆ
X⟩= Snˆ
XˆρSo,(1.20)
whe e he educed densi y ma ix ˆρS= B(ˆρ) is ob ained om he pa ial ace o
he o al densi y ma ix, o e he en i onmen deg ees o eedom. The equa ion o
mo ion o he open sys em becomes hen:
d
d ˆρS( ) = −i
¯h Bhˆ
H o ,ˆρ( )i.(1.21)
I we assume ha he co ela ion ime o he en i onmen dynamics a e negligibly
sho e hen he ime scale o he sys em co ela ion (Ma ko app oxima ion), and
ha he ime dependen coupling ˆ
Vis a small pe u ba ion o he sys em (weak-
coupling app oxima ion), we can hen o mula e he quan um mas e equa ion in
in e ac ion pic u e as he Lindblad equa ion:
d
d ˆρS( ) = −i
¯hhˆ
H o ,ˆρS( )i+D(ˆρS( )),(1.22)
whe e he i s e m o he equa ion ep esen s he uni a y cohe en dynamics gen-
e a ed by he sys em Hamil onian ˆ
Hunpe u bed by B, while he second e m
D(ˆρS( )) (Dissipa o ) accoun s o he non-uni a y e ec o he in e ac ion wi h he
en i onmen and is w i en as: [94]
D(ˆρS( )) = X
m,n Lm,n ˆρS( )L†
m,n −1
2(L†
m,nLm,n ˆρS( ) + ˆρS( )L†
m,nLm,n),(1.23)
wi h Lm, n de ined as Lindblad ope a o s.
Spin sys ems a e subjec o he pu e dephasing mechanism, due o he coupling wi h
he en i onmen . Fo aking in o accoun he non-uni a y e ec o dephasing, he
Lindblad ope a o s Lm, n mus be w i en as: [94,95]
Lm,n =δm,n γm
2|m⟩⟨m|,(1.24)
and he Lindblad equa ion becomes: [94]
d
d ˆρS( ) = −ihˆ
H o ,ˆρS( )i+
X
m,n
γmδm,n |m⟩⟨m|ˆρS( )|m⟩⟨m|− 1
2(|m⟩⟨m|ˆρS( ) + ˆρS( )|m⟩⟨m|),
(1.25)
whe e |m⟩a e he eigens a e o he sys em. This dissipa o causes he decohe ence
o he o -diagonal elemen s o he densi y ma ix ρm,n( ) wi h a a e τm,n = (γm+
γn)/2), whils he diagonal na u e o he Lindblad ope a o keeps unchanged he
diagonal popula ion ρm,m( ) (no ene gy exhange be ween sys em and en i onmen ).
31 Simone Chicco

Chap e 1
Fo he nuclea spin sys em ha will be discussed in sec ion 3, he eigens a es a e
labelled by he co esponding componen o he nuclea spin |mI⟩and he dephasing
a e co esponds o he nuclea phase memo y ime τm,n = 1/TmI,m′
I
2. [4] Mo eo e ,
o he case-s udy o molecula qubi s i could be use ul o simula e he e ec o an
ex e nal s imulus on a spin sys em al eady coupled wi h he en i onmen . This can
be simply achie ed by adding a ime dependen e m ˆ
H1( ) o he Hamil onian o
he mas e equa ion ha ules he uni a y e olu ion o he sys em densi y ma ix.
The Lindblad mas e equa ion becomes:
d
d ˆρS( ) = −ihˆ
H0+ˆ
H1( ),ˆρS( )i+
X
m,n
γmδm,n |m⟩⟨m|ˆρS( )|m⟩⟨m|− 1
2(|m⟩⟨m|ˆρS( ) + ˆρS( )|m⟩⟨m|).
(1.26)
By nume ically sol ing his mas e equa ion by using ealis ic pa ame e s o he
spin Hamil onian, he ime dependen pe u ba ion and he dephasing a es gi es
undamen al insigh in o he dynamics o p o o ypical molecula qubi s.
32 Simone Chicco
Chap e 2
Expe imen al echniques
The expe imen al s udy o molecula nanomagne s in his hesis has been done
by exploi ing se e al echniques: esonan echniques, such as b oadband Nuclea
Magne ic Resonance, Elec on Pa amagne ic Resonance, and sca e ing echniques,
such as neu on and X- ays inelas ic sca e ing. In he ollowing sec ions we will
gi e a b ie in oduc ion o all o hem.
2.1 Resonance echniques
In his sec ion we will ocus on wo esonance echniques which ha e se e al simila -
i ies, om he poin o iew o he heo e ical backg ound, and one impo an di e -
ence, he physical a ge . Bo h Nuclea Magne ic Resonance (NMR) and Elec on
Pa amagne ic Resonance (EPR) a e in ac esonan expe imen al echniques ex-
ploi ing he in e ac ion o magne ic momen s wi h an ex e nal elec omagne ic i a-
dia ion, pe u bing he sys em. In he i s case, he magne ic momen s µI=µNgNI
a ise om he nuclea spin Io he sys em, while, o he second echnique, he o-
cus is on he magne ic momen s µe=µsgs·Sa ising om he unpai ed elec ons
spin S. Since we a e dealing wi h esonan echniques, his implies ha he elec o-
magne ic i adia ion mus be uned o a na u al equency o he sys em in o de
o be e ec i e, which is, depending on he di e en physical a ge , he equency
o he magne ic momen s gy oscopic p ecession unde an ex e nal s a ic magne ic
ield. These magne ic esonance equencies alls ypically in he adio- equency
( ) egion o he nuclei, while o he elec ons in he mic owa e (mw) one, hus,
we will e e o and mw i adia ion depending i we a e desc ibing an NMR o an
EPR expe imen . In he ollowing chap e we will discuss in de ails he B oadband
NMR (sec . 2.1.1) and EPR (sec . 2.1.2) echniques.
2.1.1 Nuclea Magne ic Resonance
In he nuclea magne ic esonance echnique we a e dealing wi h nuclea mag-
ne ic momen s µI, a ising om he a omic g ound s a es o se e al nuclei ea u ing
I= 0. Because o he small magne ic momen µN(compa ed o he elec onic
one µe/µI≈103÷4), and he absence o a s ong exchange mechanism, nuclea
e omagne ism is limi ed o he mK empe a u es. Thus, in a NMR expe imen
unde applied s a ic ield, he a ge sys em will be a spin pa amagne s ea u ing
se e al s a es Em(depending on he spin I) co esponding o he di e en alues
33
Chap e 2
o he quan um numbe mI=Izo he spin quan ized along he applied ield.
The popula ion o hese nuclea s a es will be go e ned by he Bol zmann s a is ic
Pm∝e−Em/kT . Nuclea magne ic esonance is he e o e he spec oscopic echnique
ha enables o s udy hese ene gy le els and he ansi ions induced among hem
by a esonan adia ion. The e exis wo possible esonan exci a ion me hods:
ield-sweep and equency-sweep. He e we ocus on he equency-sweep me hod,
used o he NMR in es iga ions o his hesis. I consis s in a ge ing he sys em
wi h a - ield o a iable equency. When his equency ν ma ches one o he
cha ac e is ic gaps o he nuclea s a es s uc u e ∆ ≈hν , he ansi ion p ob-
abili y is enhanced and he p ocess de ec ed. The de ec ion o hese ene gy gaps
and he measu emen o he spin elaxa ion p ocesses owa ds equilib ium a e he
i adia ion a e he in o ma ion commonly accessible om an NMR expe imen
on a magne ic nucleus. The NMR expe imen can be in e p e ed in wo di e en
ways: as a o ced p ecession o he nuclea magne ic momen in a ield ( om F.
Bloch [96,97,98]) o as he esul o he compe i ion be ween he elaxa ion o
he he mal equilib ium (Bol zmann popula ions) and he exci a ion induced by he
elec omagne ic i adia ion. [92,99]
In a quan um mechanical app oach, he eigen alues o he angula momen um
ˆ
I= ¯hIcomponen Izalong he ex e nal ield di ec ion ˆ
z, coincide wi h all he
2I+ 1 possible con igu a ions m=−I, −I+ 1, ...I −1, I. Thus, gi en he ela-
ionship be ween he magne ic momen s and he angula momen um componen s
⟨Im|µz|Im′⟩=µNgN⟨Im|Iz|Im′⟩, he e ec o an applied ex e nal s a ic ield H0ˆ
z
is he Zeeman in e ac ion ene gy:
H=µ·B=γ¯hBIz=µNgNBIz,(2.1)
ha spli s he nuclea spin s a es, cha ac e ized by he eigen alues Em o each m
s a e:
E=−µNgNBm. (2.2)
To induce exci a ions be ween hese Zeeman le els and hus pe o ming he NMR
expe imen , he sys em mus be a ge ed wi h a pe u ba ion ¯hω esonan o one o
he gaps wi hin he Ems a es.
¯hω =Em′−Em.(2.3)
In his case, he pe u ba ion consis s in a - ield B1⊥B0 ha induces an addi ional
in e ac ion e m B1=−µNgNB0,xIxcos(ω ) ha d i es he ansi ion be ween close
nuclea s a es m→m±1 wi h ∆m=±1. By conside ing he case o a spin I= 1/2
sys em, he ansi ion p obabili y be ween he wo mInuclea s a es is de ined as
|⟨+|Ix|−⟩|2, whe e |+⟩and |−⟩ co espond o he s a es o componen mI=±1/2.
Fo s imula ing he ansi ion i is necessa y o ha e a popula ion di e ence be ween
he wo a ge s a es. Indeed, by de ining he numbe o spin pe le el as N±( ) and
he ansi ion a e as W, he exci ed spin pe uni ime is de ined by he p oduc
N±( )Wand he s a es popula ion a ia ion is desc ibed by he a e equa ion:
dN±
d =WN∓( )−WN±( ),(2.4)
whose solu ion in e ms o popula ion di e ence n( ) = N+( )−N−( ) = n0e−2W
gi es an exponen ial decay owa d a condi ion in which he e is no popula ion di e -
ence. Indeed, he ime dependen ene gy o he sys em E( ) = N−E−+N+(E++¯hω)
34 Simone Chicco
Chap e 2
sa u a es wi h an abso p ion a e:
dE/d = ¯hωWn( ) = n0¯hω(We−2W ).(2.5)
Howe e , a e he -s imulus is emo ed, he nuclea spin sys em in e ac s wi h
he he mal ba h en i onmen and elaxes owa ds he ini ial he mal equilib ium
s a e, ollowing a Bol zmann dis ibu ion:
N−
N+
=e−∆E+,−
kT .(2.6)
Thus, he elaxa ion o he popula ion di e ence induced by he ield, which
is go e ned by he in e ac ion wi h he en i onmen al he mal ba h (la ice), is
modeled by a elaxa ion a e 1/T1called spin-la ice elaxa ion a e:
n( ) = n0(1 −e− /T1).(2.7)
This, combined wi h he a e equa ion 2.4, gi es an ene gy abso p ion a e om he
nuclea spin sys em:
dE
d =n0¯hω W
1+2WT1.(2.8)
The nuclea spin ime e olu ion du ing and a e he -manipula ion can be de-
sc ibed by he phenomenological Bloch equa ions o he nuclea magne iza ion.
[96,97,98] This app oach enables he de ini ion o he elaxa ion cons an s ha a e
usually employed o model spin elaxa ion. We s a om a nuclea spin sys em
in i s he modynamical equilib ium when a s a ic ield B=Bˆ
zis applied. This
condi ion co esponds o a null ans e se magne iza ion Mx,y = 0 and a sa u a ed
longi udinal magne iza ion Mz=M0=χnH0, wi h χnnuclea suscep ibili y. The
classical p ecession o he nuclea spins in his ield, plus a elaxa ion e m, is de-
sc ibed by he Bloch equa ions:
(dMz
d =µNgN(M×B)z+M0−Mz
T1
dMx,y
d =µNgN(M×B)x,y +M0−Mx,y
T2
,(2.9)
whe e T1and T2accoun o he elaxa ions o he longi udinal and ans e se mag-
ne iza ion, espec i ely.
When a exci ing ield B1⊥Bis added o he spin sys em, in a e e ence ame
ˆx ˆy ˆz o a ing a he same equency o B1, he nuclea spins ime e olu ion is s ill
desc ibed by a p ecession, bu in his con igu a ion he p ecession is a ound an e -
ec i e ield Be =B1ˆ
x + (B−ω/γ)ˆ
z, sum o he wo applied ields. The Bloch
p ecession equa ions become:





dMz
d =µNgNB1My−Mz−M0
T1
dMx
d = (µNgNB0−ω)My−Mx
T2
dMy
d =−(µNgNB0−ω)Mx+µNgNB1Mz−Mx
T2
.(2.10)
As al eady ou lined abo e, he elaxa ions imes o he longi udinal ˆzand ans-
e se ˆx, ˆymagne iza ion componen s a e kep dis inc . This because hese cons an s
ep esen wo di e en p ocesses. As seen o he de ini ion o he abso p ion a es
35 Simone Chicco
Chap e 2
o a he mal-like elaxa ion p ocess, T1is ela ed o he eco e y o he he mal
equilib ium o he nuclea s a e popula ion, induced by he coupling wi h a he -
mal ba h cons i u ed by he elec onic and c ys al en i onmen o he nuclea spins
(spin-la ice elaxa ion). Con e sely, he ans e se elaxa ion consis s in a dephas-
ing ha occu s du ing spin p ecession in he xy-plane (spin-spin elaxa ion).
I he ield is a sho pulse o du a ion P(sho enough o a oid simul aneous
decohe ence) esonan wi h he sys em na u al equency ω=µNgNB, he e ec o
he pe u ba ion is a nu a ion o an angle θ=µNgNB1 Pa ound he e ec i e ield
B1ˆ
x. Consequen ly, by uning P, i is possible o ealize π/2 o a ions on o he xy
plane o πin e sion o he nuclea magne iza ion. In pa icula , a e a π/2 pulse
he nuclea magne iza ion e ol es ollowing Bloch equa ions:
Mxy( ) = Mxy(0)e− /T2(2.11)
which is he so-called ee induc ion decay ha ollows he esonan pe u ba ion.
The decohe ence induced by s a ic ields once he nuclea magne iza ion is on he
xy plane can be e ocused by applying a second πpulse o he spins sys em ha
e e se he dephasing, gene a ing an echo-like signal om he e ocusing o he
spin p ecession (dynamical con ibu ions a e i e e sible phenomena ha de ines
he in insic T2). Con e sely, he longi udinal elaxa ion consis in a eco e y o he
ini ial s a e M0by he exchange o ene gy wi h he he mal ba h. F om he Bloch
equa ions he eco e y ollows he end:
Mz( ) = M0+ (Mz(0) −M0)e− /T1(2.12)
T2and T1can be measu ed expe imen ally by p ope pulse sequence on well de ined
spin ansi ions (a ixed ω). By e ocusing he echo wi h a π/2−τ−πpulse
sequence o example, as a unc ion o he delay τwe can measu e he decay o he
ans e se magne iza ion (T2spin elaxa ion). By ”hea ing” he nuclea spin sys em
wi h a pulse ain o he mos diso de ed s a e and hen e ocusing a e a delay
τ, we measu e he eco e y o he longi udinal sa u a ion in a ”cooling-down”-like
p ocess (P ain − −π/2−τ−π). Mo eo e , by collec ing he esonance signals
o e a wide equency ange, om he collec ed NMR spec a we can link he ene gy
gaps o he exci ed ansi ions o he le els s uc u e o he spin Hamil onian, and
i i s pa ame e s, as ou lined in sec ion 1.3.
HyReSpec b oadband spec ome e
The pa amagne ic s a e o an o dina y magne ic ma e ial is cha ac e ized by a spin
dynamics ha ea u es as Hype ine ield luc ua ions. Being hese luc ua ions
as e hen he nuclea La mo ωL equency, he esul ing nuclea elaxa ion, co e-
la ed o he hype ine ield luc ua ions, is ul a- as and p e en he de ec ion o a
esonance signal. Molecula NanoMagne s, howe e , a low empe a u es a e cha -
ac e ized by spin-la ice elaxa ion a es much longe han he a he sho ime
needed o he sys em manipula ion wi h pulses ( as ened by s ong hype ine
couplings). Thus, du ing he sho ime-scales o a spin-echo sequence, he a ge
nucleus expe iences a quasi-s a ic elec onic en i onmen . [100]
In his con ex , he NMR appa a us mus be sui able o he exci a ion and de ec ion
o as elaxing signals. Fo he expe imen s epo ed in his hesis, we exploi ed a
pulsed b oadband NMR spec ome e named ”HyReSpec ” (”Hype ine Resonance
36 Simone Chicco

Chap e 2
Spec ome e ”), en i ely designed a he Uni e si y o Pa ma and op imized o he
in es iga ion o magne ic ma e ials, spanning bo h equency and ield. [11] This
Figu e 2.1: Ske ch o he ”HyReSpec ” spec ome e ha dwa e and o he LC-
Resonan p obe. T ansmi e and Recei e s ages ha dwa e a e be e de ailed in
[11].
expe imen al appa a us (ske ched in ig. 2.1) ea u es a wide equency span wi h a
la esponse in he 8-800 MHz ange, co e ing he na u al equency ange o se e al
nuclei, such as ansi ion me als (V, Cu, Mn, e c.). I is also equipped wi h a as
swi ching, wi h pulses ha a e p og ammable in s eps o 12 ns, b oadband ecei e
±3 MHz and an e ec i e expe imen al i adia ing bandwid h limi ed o 1-2 MHz
due o he LC esona o ini e passband. The ecei e bandwid h, oge he wi h a
sho ins umen al dead ime and a high speed digi al pulse , enables he gene a ion
o pulse sequences cha ac e ized by sho pulses and delays, necessa y o he de ec-
ion o sho li ing signals wi h a ine ime esolu ion. Fas e signal a e aging wi h
espec o comme cial NMR spec ome e s enables also he collec ion o op imal
signal o noise a io on sho e expe imen imescales. The used p obehead is a LC
esonan ci cui in which he sample is placed wi hin he coil and inse ed in o a
Maglab EXA supe conduc ing c yomagne .
2.1.2 Elec on Pa amagne ic esonance
As al eady highligh ed in he in oduc ion, Elec on Pa amagne ic Resonance (EPR)
is complemen a y o NMR, since i gi es undamen al and non edundan in o ma-
ion on he coupled elec o-nuclea spin sys em. Elec onic esonances indeed in es-
iga e he elec onic magne ic momen s a ising in sys ems wi h unpai ed elec ons.
[101,102] The EPR expe imen can be desc ibed exac ly in he same heo e ical
amewo k used o he NMR expe imen , wi h he only di e ence o he magne ic
momen µe=µBgS·S, which o igina es om he unpai ed elec ons o he sys em.
Thus, in he case o an elec onic spin mul iple S= 1/2, o a s a ic ield Bˆ
zaligned
wi h he ˆ
zaxis o he g enso , he Zeeman in e ac ion:
H=µe·B=µBgSZBSz,(2.13)
37 Simone Chicco
Chap e 2
causes he spli ing o he ene gy le els, labeled by he wo elec onic spin componen
along he s a ic ield di ec ion mS=±1/2
EmS=−µBgSZBmS.(2.14)
Consequen ly, o his simple model sys em, he ene gy o he mw-EPR s imulus
inducing he ansi ion o he esonan expe imen be ween he wo mSs a es mus
be:
¯hω =gSZµBB. (2.15)
EPR expe imen s a e usually pe o med a ixed mw- equency and he ansi ion
ene gies a e a ied by sweeping he s a ic ield Bun il he Zeeman gaps ma ch he
i adia ion ene gy. This because he esona o quali y ac o s ongly depends on
he a ge equency and hus limi s he equency span ange. Fo his eason, EPR
can be pe o med wi h di e en cha ac e is ic equencies epo ed in able 2.1, each
o hem pa icula ly sui ed o speci ic expe imen s/samples.
EPR equency bands
L-band 0.8-1.2 GHz
S-band 3.4-3.8 GHz
X-band 9-10 GHz
Q-band 34 GHz
W-band 94 GHz
Table 2.1: EPR equency bands. X and Q bands a e he mos commonly exploi ed
o he s udy o molecula magne s spin Hamil onians and decohe ences.
In EPR spec oscopy he spin sys em can be a ge ed bo h by a con inuous mw- ield
(con inuous wa e EPR) o by a pulsed mw- ield (pulsed EPR). In his sec ion we
will ocus only on he pulsed EPR echnique, which is he one used o he mea-
su emen s o spin elaxa ion epo ed in sec ion 6. When he sys em is a ge ed
by he mw- ield B1o a pulsed EPR expe imen , in analogy wi h wha shown in
sec ion 2.1.1 o NMR, he elec onic spin magne ic momen s p ecess ollowing he
Bloch equa ions 2.1.1. Thus, i is possible o manipula e, h ough mw-pulses he
elec onic spins, inducing uned nu a ions o angles θ=µBgxB1 P. This enables
he cha ac e iza ion o he sys em ene gy spec a, oge he wi h he elec onic spin
la ice eT1and spin-cohe ence eT2 elaxa ion imes. The o me is de ec ed wi h
a sligh ly di e ence pulse sequence wi h espec o NMR. He e he pulse ain is
subs i u ed by a simple in e sion πpulse and he esul ing sequence is known as
in e sion- eco e y sequence π−π/2−π. The de ec ion is done by e ocusing he
elec onic spins p e iously pushed in o he ”in e ed” non equilib ium s a e −MZ,
as a unc ion o he ime equi ed o eco e he equilib ium condi ion M0=MZ.
Pulsed-EPR spec a as a unc ion o he s a ic ield Bin ixed equency egime en-
ables he i ing o he spin Hamil onian model. This analysis is complemen a y o
he i ing o he NMR spec a, since he wo echniques ea u e di e en sensi i i y
o he Hamil onian pa ame e s o elec onic o nuclea o igin espec i ely (e.g., he
bes i o he spec oscopic spli ing enso gis achie ed wi h EPR, while NMR
ea u es be e sensi i i y on he hype ine enso componen s).
38 Simone Chicco
Chap e 2
Elexsys-E580 Pulsed spec ome e
In o de o pe o m he measu emen s epo ed in he p e ious sec ion, he EPR
appa a us mus ea u e high sensi i i y and high speed manipula ions. Fo his
pu pose, we exploi ed a B uke Elexsys-E580 pulsed-EPR spec ome e om he
EPSRC Na ional Se ice o Elec on Pa amagne ic Resonance Spec oscopy Labo-
a o y a he Uni e si y o Manches e . This pulsed-EPR spec ome e is equipped
wi h a Supe Q-FT mic owa e b idge o pulsed Q-band (νEPR = 34 GHz) expe i-
men s combined wi h a dedica ed Q-band esona o . The spec ome e is supplied
wi h a 1.8 T elec omagne and a c yogen- ee c yos a inco po a ing a closed he-
lium ci cui .
The sample is placed in o a modula B uke Flexline esona o , sui able o he mul i-
esonance Elexsys pla o m, ea u ing high ca i y illing ac o and sho dead imes.
The ca i y quali y Q- ac o is a iable and enables expe imen op imiza ion. I
could be c i ically coupled o con inuous wa e (CW) in es iga ions, o kep low o
sho dead ime and la ge i adia ion bandwid h o ma ched o he sample elaxa ion
ime o inc ease he signal o noise a io.
The wide de ec ion bandwid h (800 MHz) and he as a e aging signal p ocessing
a e gua an eed by he combina ion o he Pa e nJe -II pulse p og amme , which
deli e s a high ime esolu ion (up o 1 ns), and he as SpecJe -III digi ize , which
enables high speed acquisi ion wi h 1 ns esolu ion, ze o dead ime be ween sho s and
on boa d signal p ocessing. This wo digi al uni s in pa icula allowed o a e age
FIDs and as -decaying spin echoes wi hin he sample T1limi . The SpinJe -AWG
(a bi a y Wa e o m Gene a o ) addi ional modulus, enables high speed manipula-
ion and de ec ion o pulses wi h non- i ial shaping o e a mul iple channel a chi-
ec u e.
2.2 Sca e ing echniques
S udying he sca e ing o a cohe en adia ion on a sample and he ene gy ex-
changed du ing he p ocess gi es unique insigh s in o he a omic s uc u e and
cohe en dynamics o he sys em. Indeed, sca e ing echniques a e o undamen-
al in e es in molecula magne ism, since he adia ion in e ac ion wi h he nuclei
allows o di ec ly p obe he molecula and la ice ib a ions igge ing hei phonon-
induced elaxa ion dynamics. [30,31] Mo eo e , o p obes ea u ing a non-ze o spin,
i.e. neu ons, he magne ic in e ac ion wi h spin and o bi al momen a enables also
he cha ac e iza ion o he sys em spin Hamil onian and i s cohe en spin dynamics.
[103]
In his hesis, X- ay and neu on sca e ing expe imen s a e used o p obing he
phonon spec a and dispe sions in Molecula nanomagne s (MNMs). Thus, in he
ollowing sec ions, we will gi e a b ie in oduc ion o he Inelas ic Neu on Sca e -
ing (INS) and he Inelas ic X- ay Sca e ing (IXS) echniques, highligh ing he im-
po an in o ma ion ha can be ex ac ed om he expe imen s. [104,105,106,107]
39 Simone Chicco
Chap e 2
2.2.1 Inelas ic sca e ing
An inelas ic sca e ing p ocess, as ske ched in igu e 2.2, is de ined by he wo
conse a ion laws o momen um and ene gy:

Q=
k −
ki,(2.16)
¯hω =E −Ei,(2.17)
wi h he squa ed momen um depending on he sca e ing angle:
Q2=k2
i+k2
−2kik cos(θ).(2.18)
To desc ibe he sca e ing p ocess we s a de ining an inciden lux o he incoming
Figu e 2.2: Sca e ing p ocess wi h incoming and sca e ed beam ene gies Ei, E
and wa e ec o s 
ki,
k , and geome ical ep esen a ion o he ans e momen um 
Q
and he sca e ing angle θ.
beam pe uni o pe pendicula sample su ace and pe second I0. The sca e ing
c oss sec ion σsis hen de ined as he numbe o sca e ed pa icle pe lux uni :
σs=numbe o sca e ed pa icles
I0
.(2.19)
I we now es ic o a gi en di ec ion iden i ied by a small solid angle dΩ and o a
inal ene gy o he sca e ed pa icles in a na ow ange (E , E +dE ), he local
double di e en ial c oss sec ion becomes:
∂2σs
∂Ω∂E =sca e ed pa icles in o solid angle dΩ wi h ene gy (E ,E +dE )
I0dΩdE
.
(2.20)
The double di e en ial c oss sec ion includes all he physical in o ma ion ha can
be ex ac ed om he sca e ing expe imen .
All he sca e ing p ocesses can be desc ibed wi hin he Fe mi Golden Rule o -
malism, de ining a ansi ion a e be ween wo iden i ied s a es, by means o he
in e ac ion o he p obe wi h he sample exp essed by a po en ial V. Le ’s con-
side a sca e ing e en in a dΩ solid angle, de ining a inal wa e ec o 
k , in which
he ene gy o he sca e ing sys em changes om Eλ o Eλ′. The p obe-sample
s a es changes consequen ly om |
ki, λ⟩ o |
k , λ′⟩. The esul ing double-di e en ial
c oss-sec ion can be w i en as:
∂2σs
∂Ω∂E =1
I0dΩdE X
k in dΩ
W|
ki,λ⟩→|
k ,λ′⟩.(2.21)
40 Simone Chicco
Chap e 2
in o he sca e ing unc ion:
S(
Q, ω) =Scoh(
Q, ω) + Sincoh(
Q, ω) =
1
2π¯hX
k,k′
2Z∞
−∞⟨e−i
Q·
Rk(0)ei
Q·
Rk′( )⟩e−iω d +
1
2π¯hX
k
2Z∞
−∞⟨e−i
Q·
Rk(0)ei
Q·
Rk( )⟩e−iω d . (2.42)
In oducing a small displacemen uk( ) o each ka oms, we de ine he same op-
e a o s U=−i
Q·uk(0) and V=i
Q·uk′( ) o equa ion 2.2.2 and we de i e he
one-phonon sca e ing unc ion om he i s o de Taylo expansion o he he -
mal a e aged exponen ial exp(⟨UV ⟩). The cohe en one phonon X- ays sca e ing
unc ion hus becomes:
S(
Q, ω)∝X
s,q
1
ωs(q)|Fs
1−ph(
Q)|2
[(ns(q) + 1)δ(ω−ωs(q))δ(
Q−(τ +q)) + ns(q)δ(ω+ωs(q))δ(
Q−(τ −q))],
(2.43)
wi h he s uc u e ac o Fs
1−ph(Q) de ined as:
Fs
1−ph(
Q) = X
k
k(
Q)
√2Mk
e−2Wk(
Q)(
Q·es(q))ei
Q· k,(2.44)
whe e we no e he explici dependence on he phonon modes ene gies ωs(q) (eigen-
alue) and pola iza ion es(q) (eigen ec o s), which can be di ec ly p obed in he
sca e ing expe imen .
We inally no e ha , di e en ly om a neu on sca e ing expe imen , he e he en-
e gy o he exci ed phonon modes a e much smalle han he inciden pho on ene gy
(¯hω ≪Ei), hus he exchanged momen um Qand inciden pho on momen um ki
a io depends only on he sca e ing angle θ(in ig. 2.2). Thus, he e a e no kine-
ma ic es ic ion on he ene gy ans e depending on he Q alues. Mo eo e , he
IXS c oss sec ion is mainly composed by he cohe en sca e ing unc ion ex ac ed
abo e, wi h he incohe en sca e ing ha con ibu es a highe ene gies and, in
he mode a e low ene gy phonon egime, can be neglec ed. Finally, elemen s con-
ibu e di e enl y o he IXS and INS c oss sec ions. This is due o he in insic
di e ence be ween he X- ays o m ac o and he neu on sca e ing leng h, which
ea u e dis inc dependence on he momen um ans e Qand on he sca e ing sys-
em elec on numbe Z. A mo e de ailed analysis o he di e ences and simila i ies
be ween INS and IXS is epo ed below, as a subsec ion o his chap e .
ID28
The Inelas ic X- ay sca e ing measu emen s shown in sec ion 4 we e collec ed a
he Eu opean Sync o h on Radia ion Facili y (ESRF) on he beamline ID28. ID28
is a iple axis spec ome e dedica ed o he s udy o phonon dispe sions in con-
densed ma e in a wide ange o ans e ed momen um 
Qand exchanged ene gy
47 Simone Chicco

Chap e 2
¯hω, e en o samples o e y small dimensions. Measu emen s can also be pe o med
in a a ie y o sample en i onmen s, such as high- acuum, c yogenic empe a u es,
high-p essu e, e c. This beamline is a one-o -a-kind because, hanks o i s unique ca-
pabili ies in e ms o lineshape and esolu ion, enables he cha ac e iza ion o op ical
and acous ic phonons o molecula c ys als. The beamline layou is shown in igu e
2.5. The spec ome e ea u es an almos exac backsca e ing Si monoch oma o
Figu e 2.5: Schema ic ep esen a ion o he ID28 spec ome e layou . The pic u e
is ep oduced om he ESRF websi e
(89.98◦) ha exploi di e en Si(n, n, n) e lec ion o de s. Each o hese e lec ions
selec s an ini ial ene gy Eiand a lineshape wid h ∆E( esolu ion), o a X- ays lux
ha dec ease by inc easing he e lec ion o de n(see able below).
ID28 monoch oma o s
no de Ei(keV) ange ∆E(meV) lux (pho ons/s)
Si(777) 13.84 5.3 10.5×1010
Si(888) 15.817 4.4 9 ×1010
Si(999) 17.794 2.2 2.7×1010
Si(111111) 21.747 0.83 6.6×109
Si(121212) 23.725 0.73 5.8×109
Si(131313) 25.704 0.5 1.47 ×109
A e he monoch oma ion, he beam is ocused on he sample, posi ioned on a o-
a ing goniome e wi h o a ional, il ing and ansla ional deg ees o eedom, back
in o he expe imen al cabin. He e lays he second ”axis” o he spec ome e , a 7
me e s a m, equipped wi h nine analyze , each o hem cons i u ed by 1200 single
c ys als ben in a sphe ical shape, o ocusing he beam on he de ec o s o e an
angula accep ance o 40 m ad2. The nine analyze s a e placed in he ho izon al
sca e ing plane, wi h an angula displacemen o 0.75◦, and selec nine di e en
momen um ans e 
Qin he ange 1-80 nm−1wi h a momen um esolu ions o 0.03
nm−1. The indi idual Si de ec o s cons i u e he hi d ”axis” o he spec ome e
and selec he inal sca e ing ene gy E . They a e in eg a ed on a monoli hic Si
chip, oge he wi h hei p eampli ie and he Pel ie cooling modulus ( o elec onic
noise educ ion). The leng h o he spec ome e a ms is jus i ied by he backsca -
e ing geome y and he need o o se he incoming whi e beam wi h espec o he
48 Simone Chicco
Chap e 2
monoch oma ed and ocused one Ei.
INS and IXS, di e ences and simila i ies
Since in his hesis bo h INS and IXS echniques a e used, i is wo h men ioning
some o he main di e ences be ween he wo p obes. In pa icula , we will ocus on
he aspec s ha makes X- ays a o able wi h espec o neu ons o he in es iga-
ion o phonon dispe sions in molecula c ys als. The i s aspec ela es o he c oss
sec ion ha o X- ays is highly cohe en , while in neu on sca e ing some elemen s
like 1H, abundan ly p esen in molecula complexes, display an highly incohe en
c oss sec ion ha causes he blu ing o he expe imen al phonon dispe sions due o
incohe en sca e ing. Con e sely in IXS, he incohe en sca e ing in ol es ene gies
way highe han he ypical phonon ene gies o molecula magne s we a e in e es ed
in. Thus, i is no necessa y o, e.g., deu e a e he c ys als employed. Toge he
wi h he absence o incohe en sca e ing, mul iple sca e ing e en s can also be
neglec ed, making IXS a basically backg ound ee echnique. Thus, wi h espec
o INS he da a educ ion p ocedu e esul s smoo he . Con e sely o INS, IXS do
no su e o s ong dynamical limi a ion wi hin he (Q,E) space and he o de o
magni ude di e ence be ween he inciden and he exchanged ene gies leads o he
decoupling o ene gy and momen um ans e (which depends only on he sca e -
ing angle) and causes he esolu ion o he expe imen o be independen om he
ene gy. Mo eo e , he almos absence o X- ay magne ic c oss sec ion makes his
p obe a o able o he s udy o phonon dispe sions and pola iza ion ec o s, since
he magne ic exci a ions wi hin he ene gy ange o in e es a e na u ally emo ed.
Fo INS ins ead, whe e he magne ic c oss sec ion is compa a ble a low Q o he
nuclea one, he sub ac ion mus be done manually by measu ing a diamagne ic
analogue o he s udied c ys al.
Howe e , he wo echniques ea u es also some simila i ies, such as he coupling
s eng h, which is compa able be ween X- ay-elec on and neu on-nucleus. Fi-
nally, he easie ocusing o he X- ay beam and i s in insic highe lux makes IXS
bes sui ed o he s udy o c ys als o small dimensions in easonable imes, wi h
espec o INS ha , in absence o la ge c ys als, equi es long coun ing imes.
49 Simone Chicco
Chap e 3
Ex ensi e b oadband NMR s udy
o Vanadium-based molecula
qudi s
In his chap e we ocus on he cha ac e iza ion o Vanadium-based molecula qubi s.
By exploi ing he b oadband NMR echnique, we access a comp ehensi e desc ip ion
o he coupled elec o-nuclea spin Hamil onian and o he spin cohe ence imes o
hese sys ems. By u he combining nuclea and elec onic esonance echniques,
we u he inc ease he ex en and he accu acy o he desc ip ion o his complexes,
achie ing complemen a y and non edundan esul s. Mo eo e , we demons a e
ha a ge ed adio- equency ( ) and mic owa e (mw) pulses can be exploi ed o im-
plemen monoch oma ic cohe en manipula ions o he elec onic and nuclea s a es
o hese sys ems. Finally, he abili y o manipula e nuclea s a e wi h pulses is
pushed u he by implemen ing he elemen a y encoding ope a ions o a Quan um
E o Co ec ion algo i hm, unde he guide o ealis ic simula ions. This capabili y
demons a es ha hese complexes a e p omising uni s o quan um in o ma ion
p ocesses applica ions.
3.1 Vanadium-based qudi s
Molecula complexes embedding a single ansi ion me al ion ha e been in ensi ely
s udied in he las decade o quan um in o ma ion p ocessing applica ions. In-
deed, he chemical abili y o ailo he ligands en i onmen and hus o enginee
he quan um p ope ies o he sys em makes hese complexes an ideal es bed o
disce n he di e en con ibu ion o elaxa ion and decohe ence. Among plen y o
possibili ies, Vanadium and Oxido anadium(IV) based complexes a e eme ging as
p omising candida es, because o an inna e obus quan um cohe ence. [5,6,35,67]
Mo eo e , hese Vanadium-based sys ems ea u e a c ucial ad an age. The Vana-
dium nuclea spin I=7
2can in ac be exploi o expand he a ailable compu a ional
space wi hin a single quan um objec . Indeed, he hype ine coupling be ween he
nuclea spin deg ees o eedom and he elec onic spin double gene a es a mul-
ile el s uc u e which is called Qudi , i.e. a quan um digi ea u ing mo e han
wo le els. [42,51] Encoding in o ma ion on his mul ile el s uc u e expands he
dimensions o he compu a ional Hilbe space om a wo-dimensional qubi o a
50
Chap e 3
d-dimensional (d > 2) qudi .
In his ame, esonan echniques ha exploi elec omagne ic pulses (as NMR and
EPR, espec i ely in sec . 2.1.1 and 2.1.2) play a c ucial ole, since hey allow bo h
he cha ac e iza ion and he cohe en manipula ion o hese mul ile el s uc u es.
[55,100] In pa icula , in he ollowing, we demons a e why b oadband NMR is
an ideal echnique o cohe en ly manipula ing he nuclea s a es o p o o ypical
molecula qudi s and how his echnique can be used o cha ac e ize hem. [4,12]
3.2 NMR in es iga ion o he [VO(TPP)] molec-
ula qudi
The i s p o o ypical Vanadium-based molecula qudi s udied he e is Vanadyl
e aphenylpo phy ina e [VO(TPP)]. The complex is composed by a single Vana-
dium ion bound o an oxygen o o m a Vanadyl g oup which is inse ed a he
cen e o a po phy ine ing. The molecule c ys allizes as a squa e py amid wi h a
e agonal I4 space g oup. In he c ys al cell, he po phy ine g oup lies in he c ys-
allog aphic ab-plane, wi h he O-V bond de eloping along he c-axis (see ig. 3.1).
[4,14] Mo eo e , he squa e py amid basis ma ches he c ys allog aphic ab-plane,
while he py amid heigh co espond o he c ys allog aphic c-axis.
Figu e 3.1: [VO(TPP)] molecula s uc u e. The p incipal o hogonal symme y
di ec ions o he molecula c ys al a e shown independen ly: he po phy in lying in
he ab plane and he oxido anadium bond di ec ion along he caxis
The NMR spec a o 51Vnuclei a e measu ed by he home-buil b oadband spec-
ome e ”HyReSpec ” desc ibed in sec ion 2.1.1 and in Re .[11]. The da a we e
collec ed a T= 1.4 K, in o de o supp ess as much as possible decohe ence e -
ec s induced by empe a u e, on a single c ys al o [VO(TPP)] dilu ed a 2% in o
i s isos uc u al diamagne ic analogue [TiO(TPP)]. Dilu ion ep esen s he mos
e icien way o supp ess he dipola in e -molecula in e ac ions as a sou ce o deco-
he ence. Howe e , dilu ion mus also be uned in o de o pe mi he de ec ion o he
NMR signal om he a ge ed nuclei. The dilu ion pe cen age used o [VO(TPP)]
measu emen s he e o e ep esen s a good ade-o be ween hese wo ac o s. [100]
The single dilu ed c ys al is placed on a cus om 3D p in ed plas ic sample-holde
and hold in place by acuum g ease. This sample holde pe mi s o easily o ien he
51 Simone Chicco
Chap e 3
c ys al in o de o apply he ex e nal s a ic ield along speci ic symme y di ec ions.
The i s s ep o he analysis was he cha ac e iza ion o he model spin Hamil onian
pa ame e s by an ex ensi e b oadband NMR s udy as a unc ion o he applied ield.
We measu ed 51V spec a by collec ing Hanh-echoes o e a wide equency ange a
a ious applied s a ic ields in he ange B0= 0.05−0.3 T. The pulse sequence used
is he uncon en ional 2π
3−τ−2π
3sequence which p o ides an op imized esonance
signal,[100,110] wi h pulses du a ion o Tpulse = 0.202 µs and delay τ≈6µs. The
disc e e equency s eps can be hickened in da a pos -p ocessing by me ging he
o e lapping equency domains o spin-echoes as Fou ie ans o m o subsequen
equency shi ed s eps. [111]
In Figu e 3.2 we show wo examples o 51Vspec a, measu ed by applying he
s a ic ield along he wo nonequi alen and o hogonal symme y di ec ions o he
[VO(TPP)] molecule, he po phy in plane ab and he oxido anadium bond di ec ion
c.
Figu e 3.2: Spec a collec ed a di e en applied ield B0along he molecule sym-
me y di ec ions: he ab plane (a), and he caxis (b). In inse , a spu ious peak is
iden i ied om he absence o any ime-domain spin-echo signals. Indeed black line
shows he echo signal in quad a u e o a eal nuclea exci a ion in co espondence
o τ(dashed line), whe eas a spu ious appea s as he pe sis en g ay oscilla ions.
Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.
The [VO(TPP)] molecule can be desc ibed by a simple coupled elec o-nuclea spin
Hamil onian composed as ollows:
H0=ˆ
I·A·ˆ
S+pˆ
I2
z+µBˆ
S·gS·B0+µNgNˆ
I·B0.(3.1)
The i s e m in eq. 3.2 accoun s o he Hype ine coupling be ween he elec-
onic ˆ
Sand he nuclea ˆ
IVanadium spins, while he second one exp esses a small
axial nuclea quad upola con ibu ion (see sec . 1.3). Finally, he las wo e ms
ep esen he nuclea and elec onic Zeeman in e ac ions. As seen in sec ion 1.3,
he pa ame e s o he spin Hamil onian de e mine he ield e olu ion o he nuclea
esonances. The e o e, by compa ing he measu ed ield dependence o he spec a
peaks wi h he simula ed spec a, we can i he spin Hamil onian pa ame e . By
conside ing he c ys allog aphic axes as e e ence ame abc =xyz, we assume he
Hamil onian enso s o be diagonal, axial and collinea . Mo eo e , his choice o
axes combined wi h he c ys al shape simpli y he de ini ion o he di ec ion o he
52 Simone Chicco

Chap e 3
applied ields 
B0,
B1, because he xy plane ma ches he squa e py amid basis while
he z axis coincides wi h he e ex.
Figu e 3.3: Red do s co espond o he measu ed ansi ion equencies in he b oad-
band NMR spec a o 51V. Black lines ep esen he calcula ed e olu ion o he
ansi ion equencies as a unc ion o he ield B0applied (a) in he ab-plane, and
(b) along c-axis. Wi h capi al le e s AB# and C# we label he ∆mI=±1
ansi ions o each di ec ion. Shaded a eas we e no expe imen ally explo ed. Re-
p oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.
The bes ag eemen be ween he calcula ions and he expe imen al spec a is ound
wi h an axial Hype ine enso and a small axial quad upola coupling (see able 3.1).
[4] The elec onic spec oscopic spli ing enso gsis kep ixed o alues ob ained
om a p e ious EPR s udy [14], since ha echnique ea u e an highe sensi i i y
on he de e mina ion o he elec onic g- alues. Finally, he nuclea g- ac o is ixed
o he cons an iso opic alue o he 51Vnucleus µNgN=−11.213 MHzT−1. [112]
The bes i pa ame e o [VO(TPP)] spin Hamil onian a e lis ed in able 3.1.
The ag eemen achie ed be ween he expe imen al esonance equencies and he
model Hamil onian cu es is excellen o bo h he o hogonal ield di ec ions (plane
ab and caxis), as shown in igu e 3.3. By diagonalizing he spin Hamil onian 3.2
53 Simone Chicco
Chap e 3
[VO(TPP)] spin Hamil onian
A(MHz) p(MHz) gS(MHz)
x−170 ±1 0 1.9865
y−170 ±1 0 1.9865
z−480 ±1−0.35 ±0.07 1.963
Table 3.1: Bes i pa ame e o he [VO(TPP)] spin Hamil onian 3.2. [4]
wi h he ob ained pa ame e s we can access he eigen alues o he sys em and plo
i s ene gy le els diag am as a unc ion o he applied ield ( ig. 3.4).
The calcula ed ene gy le els a e labeled as |mS, mI⟩, hus wi h he componen o
he elec onic Sand nuclea Ispins pa allel o he applied s a ic ield B0. This
a ibu ion is accu a e abo e B0= 0.25 T, whe e gS,xµBB0>|Az|wi h he ield ap-
plied along he xaxis o he molecule ame. In his egime, he sys em eigens a es
a e ac o ized o mo e han 98%.
Figu e 3.4: Ene gy le els as a unc ion o he applied ield om he diagonaliza ion
o he spin Hamil onian wi h s a ic ield B0applied (a) in plane ab and (b) along
c-axis. The ed and blue shades highligh he mS=±1/2 mul iple s, espec i ely.
On he le o each plo , wo inse s show a zoom in o he nuclea le el spli ing o he
lowe and uppe mul iple s, wi h he le els ma ked by he nuclea spin componen s
mIalong he ield. Ve ical ma ks highligh he nuclea ansi ions iden i ied in he
spec a o igu e 3.2. Rep oduced om Re . [4] wi h pe mission om he Royal
Socie y o Chemis y.
Fo he pu pose o exploi ing he nuclea s a es o his sys em as he mul i-le els o
a qudi , hey mus ul ill wo undamen al equi emen s. Fi s ly, he nuclea spin
dynamics mus be condi ioned by he elec onic spin s a e, which means ha he
ene gies o he nuclea ansi ions belonging o an elec onic spin s a e a e di e -
en om he ene gies o he same nuclea ansi ions o he o he elec onic spin
mul iple . As a consequence o his, any manipula ion o he nuclea s a e is condi-
ioned by he elec onic s a e. The second equi emen conce ns he add essabili y
o indi idual nuclea ansi ions. Indeed, any quan um ope a ion on he nuclea
spin s a es can be exp essed as a combina ion o ansi ions be ween single pai s
o nuclea consecu i e le els mI. To ensu e his, he di e ence be ween subsequen
nuclea gaps δmI= (EmI+1 −EmI)−(EmI−EmI−1) mus be bigge hen he i a-
54 Simone Chicco
Chap e 3
dia ion bandwid h used o he nuclea manipula ion. Indeed, his ea u e enables
one o add ess single nuclea ansi ions, wi hou pe u bing nea by s a es.
The [VO(TPP)] complex ul ills bo h he equi emen s epo ed abo e. In pa ic-
ula , he pseudo quad upola coupling desc ibed in sec ion 1.3.4 plays a key ole,
since he di e ences in he ansi ions ene gies a e mos ly impu able o his e ec .
Indeed, he di e ences 2ζbe ween he ansi ion ene gies ob ained by diagonalizing
he e ec i e Hamil onian o sec ion 1.3.4 (composed by he de ined pseudo-Zeeman
and pseudo-quad upola con ibu ions), ma ch hose calcula ed by diagonalizing he
eal spin Hamil onian o he sys em δ(mI) (see ig. 3.5). I is he e o e demons a ed
ha his pseudo-con ibu ion is pa icula ly e ec i e in sys ems wi h s ong Hype -
ine couplings.
Figu e 3.5: Di e ence in subsequen nuclea ansi ions ene gies δ(mI) = (EmI+1 −
EmI)−(EmI−EmI−1) in he mS= 1/2 mul iple (c osses), as a unc ion o he
applied ield Bˆ
x. These alues a e in sound ag eemen wi h wha expec ed om he
diagonaliza ion o he pseudo-quad upola Hamil onian (black line). Rep oduced
om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.
The dephasing a es o he nuclea spin 1/TmI,mI′
2a e also in es iga ed by means
o he 2π
3−τ−2π
3Hahn pulse sequence wi h inc easing τ, as a unc ion o he ap-
plied s a ic ield B0along molecule c-axis and ab-plane. The echo in ensi y decays
ollowing a single exponen ial law as a unc ion o τ, o all he measu ed ansi-
ions ( ig. 3.6). The ex ac ed ime cons an TmI,mI′
2is he spin phase memo y ime
o he sys em, co esponding o he li e ime o he induced s a es supe posi ion.
The alues ob ained o [VO(TPP)] a e epo ed in igu e 3.7 as a unc ion o B0
and show an inc emen om 10 µs o 60 µs wi h inc easing ield. The esul s a e
ema kable, since [VO(TPP)] T2 alues a e among he highes in his ca ego y o
complexes. Mo eo e , he phase memo y imes esul much longe han he ime
expec ed o implemen ing elemen a y ope a ions on he qudi sys em. Thus, hese
equisi es make his complex e y p omising o Quan um In o ma ion P ocessing
(QIP) applica ions.
55 Simone Chicco
Chap e 3
Figu e 3.6: Echo in ensi y decay as a unc ion o he delay be ween he Hahn echo
exci ing and e ocusing pulses. The expe imen al da a o a selec ed nuclea an-
si ion ((a) AB2, (b) C1) a e plo ed oge he wi h he elaxa ion a e T2single
exponen ial i o di e en applied ields. Rep oduced om Re . [4] wi h pe mis-
sion om he Royal Socie y o Chemis y.
Figu e 3.7: Nuclea phase memo y ime o di e en nuclea ansi ions, measu ed
wi h a e ocusing Hahn-echo sequence, as a unc ion o he applied ield B0(a) in
he ab-plane and (b) along c-axis. Rep oduced om Re . [4] wi h pe mission om
he Royal Socie y o Chemis y.
In a dilu ed sample he dipola coupling be ween elec onic spins and he in e -
ac ion o he nuclei wi h su ounding elec on spins a e bo h s ongly supp essed.
The e o e, he nuclea spin decohe ence is go e ned by he in e ac ion wi h nea by
magne ic nuclei media ed by he i ual exci a ion o he elec onic spins. In ou
[VO(TPP)] molecule, his mechanism domina es o e he di ec nucleus-nucleus
mechanism as epo ed o o he simila compounds. [45] The elec o-nuclea mix-
ing o he sys em wa e unc ions p omo es his elaxa ion mechanism. Thus, he
inc emen o he measu ed phase memo y imes wi h he applied ield, is jus i ied
by an inc ease in he ac o iza ion o he elec onic and nuclea componen s o he
wa e- unc ion. Mo eo e , he inc emen in he elec onic pola iza ion by inc easing
B0supp esses he elec onic spin luc ua ions and u he educe hei in e ac ion
wi h he p obed nuclei. [113,114]
Nuclea spin s a e popula ions can be manipula ed by means o pulses. Indeed,
56 Simone Chicco
Chap e 3
(M) (see ig. 3.13). Howe e , because o he c ys allog aphic inequi alence, each o
he p e iously de ined di ec ions comp ises wo di e en con igu a ions o he ield
wi h espec o he wo molecules ( ig. 3.14).
Figu e 3.13: Pic u e o he c ys al collec ed om mic oscopy (le ) and a ske ch o he
c ys al shape ( igh ). The wo a e compa ed o show he edge leng h hie a chy and
he ela i e indexing o c ys al aces wi h he co esponding c ys allog aphic planes.
Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical
Socie y.
Figu e 3.14: The c ys allog aphic planes (111), (010) and (10-1), ha co espond
o he c ys al aces, a e highligh ed in he uni cell. This ske ch assis in he isual-
iza ion o he applied ield di ec ions wi h espec o he molecula e e ence ame.
Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical So-
cie y.
To de e mine he spin Hamil onian pa ame e o his sys ems a hyb id NMR-EPR
app oach was used o he i s ime. Indeed, he wo echniques (see sec . 2.1.2 and
2.1.1) ea u es di e en esolu ion and sensibili y o each o he spin Hamil onian
e ms. The s a ing poin is he elec o-nuclea spin Hamil onian, w i en as:
H0=µBˆ
S·gS·B0+ˆ
I·A·ˆ
S+µNgNˆ
I·B0+ˆ
I·P·ˆ
I.(3.3)
He e, he i s wo e ms co espond o he elec onic Zeeman e m and o he Hy-
pe ine coupling be ween he elec onic and he nuclea spins. The o he wo e ms
cons i u e he ine-s uc u e Hamil onian o sec ion 1.3 and co espond o he nu-
clea quad upola and he nuclea Zeeman couplings.
63 Simone Chicco

Chap e 3
Con inuous wa e (CW) X-band and Q-band EPR spec a (9.405 GHz and 33.7 GHz
espec i ely) we e measu ed by ou collabo a o s a he Uni e si y o Flo ence and
a he Labo a oi e Na ional des Champs Magn´e iques In enses (LNCMI) in G eno-
ble, (see he Lis o Collabo a o s) on he dilu ed powde s (shown in ig. 3.15-(a,b)
o sample 3c). The spec a clea ly show he eigh spec al lines induced by he
Hype ine coupling wi h he 51V nuclea spin, co esponding o he eigh elec onic
ansi ions mS=±1/2 a ixed nuclea componen s ∆mI= 0.
Figu e 3.15: (a) Con inuous Wa e (ν= 9.405 GHz, black) and Echo De ec ed Field
Sweep (ν= 9.70 GHz, blue) X-band EPR powde spec a o sample 3c measu ed a
T= 20 K. The spec a a e bo h simula ed ( ed) using he Hamil onian pa ame e
o able 3.2. EDFS is ansla ed in B o be e compa ison. (b) CW-EPR Q band
spec um (ν= 33.7 GHz, black). (c,d) 2D colo map (blue nega i e, ed posi i e) o
3c single c ys al X-band EPR spec a as a unc ion o θAand θB o a ions. Black
lines co espond o he simula ed spec a o he wo inequi alen molecules. I is
e iden he o e lap o he wo spec a along he e e ence ame axis (θA,B = 0◦,90◦)
wi hin he EPR esolu ion. Rep in ed wi h pe mission om Re . [12]. Copy igh
2021 Ame ican Chemical Socie y.
The X and Q band EPR spec a a e i ed only wi h he i s wo componen s o
he spin Hamil onian 3.3, i.e. he elec onic Zeeman and he Hype ine couplings,
e ining he spec oscopic gand Hype ine A enso s. This because i was no pos-
sible o deduce he small nuclea in e ac ion e ms om he expe imen al esolu ion
o his echnique. To ep oduce he da a he wo enso s a e conside ed hombic
and collinea . The inal alues a e shown in able 3.2 a e he subsequen NMR e-
64 Simone Chicco
Chap e 3
inemen s. In e es ingly, i was no necessa y o simula ing he expe imen s esul s
o assume di e en pa ame e s o he wo molecules in he uni cell. This p o es
ha he wo molecules di e s only in he o ien a ions wi hin he c ys al, which a e
a e aged ou by measu ing powde samples.
The single c ys al EPR in es iga ion enables he de e mina ion o he ela i e molecules
o ien a ions. S a ing wi h he ex e nal ield applied along he (111) c ys al ace,
he sample has been o a ed s ep by s ep inside he ca i y and he ield dispe sion
o he esonances has been eco ded in he anglua ange θ= 0◦ o θ= 180◦( ig.
3.15). The o a ional axes we e he edge be ween (111) and (010) planes (θA) and
he edge be ween (111) and (¯
101) planes (θB). The o a ional axes de ined abo e
co espond wi h he molecula ame de ined in igu e 3.12, wi h θAco esponding
o he o a ion om x o zand θBco esponding o he o a ion om y o z. Fo
bo h he o a ional axis, a 0◦and 90◦ he spec a o he wo molecules o e lap in
a single eigh -line Hype ine s uc u e. This con i ms ha , wi hin he expe imen al
esolu ion o CW-EPR, hei magne ic enso s appea o sha e he same o ien a ion
wi h espec o he applied ield (see ig. 3.12). As we will see in he ollowing, his
aspec will be e ined by he NMR analysis which ea u es a ine sensi i i y on he
hype ine componen s and enables o obse e a sligh misalignmen o he magne ic
enso s, no app eciable in CW-EPR.
Figu e 3.16: B oadband NMR spec a measu ed a T= 4 K a ixed applied s a ic
ield B0= 0.2 T, o he ield di ec ions BS,BM,BLschema ized in inse wi h
espec o he c ys al aces. In black, he Gaussian spec al lines i ing. Rep in ed
wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y.
Fo in es iga ing he nuclea ine-s uc u e componen s o he spin Hamil onian 3.3,
51V (I= 7/2, γ/2π= 11.21 MHz) b oadband NMR spec a we e measu ed a T=4
K in he ield ange B0=0 o 0.4 T, exploi ing he ”HyReSpec ” spec ome e [11]
a he Uni e si y o Pa ma. The [V(Cp)2Cl2] single c ys al 3b was ins alled on
65 Simone Chicco
Chap e 3
a speci ically p in ed sample holde , in o de o apply he s a ic magne ic ield as
much as possible along he c ys al aces labeled as L, M and S. The ield o ien a ion
was subsequen ly u he e ined as a i ing pa ame e o he spec a. Figu e 3.16
shows an example o spec a collec ed in each o he ield con igu a ions (BS,BM
and BL). Despi e he esonance o e c owding (14 nuclea ansi ions mul iplied o
he wo molecules), hanks o he ine expe imen al esolu ion, all he ansi ions
a e dis inguishable and can be i ed by a ibu ing a Gaussian line-shape o each
esonan equency. The e olu ion o he NMR peak equencies as a unc ion o he
applied ield has been i ed wi h he model spin Hamil onian in o de o e ine he
pa ame e ob ained by CW-EPR and o e alua e also he nuclea quad upola and
Hype ine con ibu ions. Only he elec onic spec oscopic g enso was kep ixed
om he EPR analysis, which is mo e sensi i e o i s de e mina ion. The i ing o
he spec al componen s a di e en applied ield is shown in igu e 3.17.
The achie ed ag eemen be ween he da a and he modeled elec o-nuclea spin
Hamil onian is excellen e en when conside ing bo h he molecules in he uni cell
( ed and black lines). All he spin Hamil onian pa ame e s ob ained om his dou-
ble echnique e ining app oach a e shown in able 3.2 below.
[V(Cp2)Cl2] spin Hamil onian
A(MHz) p(MHz) gS(MHz)
x−55 ±2 0.09 ±0.01 2.0010 ±5×10−4
y−216 ±2 0.09 ±0.01) 1.9834 ±5×10−4
z−315 ±2−0.18 ±0.01 1.9721 ±5×10−4
Table 3.2: Bes i pa ame e o he [V(Cp)2Cl2] spin Hamil onian 3.3. [12]
Bo h he spec oscopic gand Hype ine A enso s show a hombic symme y, while
a modeled nuclea quad upola con ibu ion, axial along he Cp-V-Cp double-decke
axis, was essen ial o he ep oduc ion o NMR spec a. Apa om he nuclea ine
s uc u e componen s o 3.3, b oadband NMR also enables an accu a e e inemen
o he Hype ine enso alues and absolu e sign. The la e is in ac no accessible
in CW-EPR.
By diagonalizing he spin Hamil onian 3.3, we calcula e he sys em ene gy le els o
he h ee ex e nal ield con igu a ions, shown in igu e 3.18. While he elec onic
mul iple s spli ing is no widely a ec ed by he o ien a ion o he ield, he nuclea
mul i-le el s uc u e is s ongly modi ied. Indeed, he ema kable di e ences in he
hype ine componen s causes he o e all nuclea spli ing o a y signi ican ly be-
ween he h ee con igu a ions. Mo eo e , a he han he small nuclea quad upo-
la coupling, i is he hype ine e ms ans e se o he applied ield esponsible o
he second-o de pseudo-quad upola e ec ha play a key ole he e in he in a-
mul iple gaps aniso opy. The calcula ed ene gy le els ul ill he equi emen s o
a p omising qudi candida e. Indeed, he nuclea dynamics is condi ioned by he
elec onic spin s a e, since he wo mSmul iple s a e well dis inguished in ene gy.
Mo eo e , he aniso opy in he nuclea ansi ions exci a ion ene gies is su icien
o enable hei selec i e exci a ion wi h s a e o he a esona o s.
The ene gy le els o he wo inequi alen molecules a e iden ical, wi h he excep ion
o small di e ences due o he di ec ion o he applied ield in he e e ence ame
o he wo molecules. These small di e ences a e howe e su icien o dis inguish
he exci a ion spec al equencies o each molecule, in mos o he applied ield
66 Simone Chicco
Chap e 3
Figu e 3.17: The measu ed NMR equencies o a single c ys al o 3b (yellow and
cyan do s) a e compa ed wi h he spin Hamil onian i ing model (black and ed
do s o each molecule, espec i ely). The applied ield di ec ions BS,BM,BLa e
schema ized in inse . Cyan expe imen al poin s label he ansi ions whose nT2
was measu ed. Shaded a eas we e no explo ed expe imen ally. Rep in ed wi h
pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y.
67 Simone Chicco
Chap e 3
Figu e 3.18: Calcula ed ene gy le els o he expe imen al ield con igu a ions BS,
BM,BLwi h espec o he c ys al aces. Black and ed lines ela es o he wo
inequi alen molecules. Fo he nuclea ansi ions labeled by e ical colo ed lines
i was also measu ed he nuclea phase memo y ime nT2( ig. 3.19). Rep in ed wi h
pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y.
68 Simone Chicco

Chap e 3
condi ions explo ed expe imen ally (see ig. 3.17).
The mul i echnique app oach exploi ed he e has p o en o be also essen ial in he
s udy o he elaxa ion and cohe ence imes o he elec o-nuclea coupled sys em.
An echo de ec ed ield sweep (EDFS) expe imen pe o med by ou collabo a o a
he Uni e si y o Tu in (see he Lis o Collabo a o s) wi h he pulsed EPR ech-
nique in he X band ( ig. 3.15-(a)) con i ms he p esence o elec onic cohe ence
also a empe a u es abo e 20 K. The elec onic spin-la ice elaxa ion ime eT1
was measu ed as a unc ion o empe a u e on powde s, wi h an in e sion- eco e y
sequence. The applied ield was ixed a 0.334 T in o de o ma ch he highe in-
ensi y ansi ion in he spec a, co esponding o ∆mS=±1 and ∆mI= 0 om
mI=−1/2. The eco e y a es a ixed empe a u e we e i ed wi h a s e ched
single exponen ial sa u a ion model I=I0+C1exp(−(τ/eT1)β). The ex ac ed
elaxa ion imes eT1a e shown in igu e 3.19. We obse e a maximum and hen la
elaxa ion a es below 10 K, wi h alues inc easing wi h dilu ion. A highe em-
pe a u es he elaxa ion ime dec eases ab up ly and becomes independen om he
dilu ion pe cen age. This anomalous empe a u e dependence o eT1has been mod-
eled by he sum o a di ec and a Raman-like con ibu ions o elaxa ion, domina ing
a lowe and highe empe a u e, espec i ely:
eT−1
1=aT +bTn.(3.4)
The e iden di e ence wi h espec o o he Vanadium based molecula sys em ha
display spin cohe ence app oaching oom empe a u e,[5,6] can be linked o an
unexpec edly e icien Raman-like pa h o elaxa ion, whose exponen app oach
n= 5 in he phenomenological model used he e. The dilu ion-dependen di e ences
a e ins ead ela ed o he enhanced dipola in e ac ion. A ine model o elaxa ion
in he high empe a u e egime can be se up by making explici he coupling o he
spin wi h local ib a ional modes:[116]
T−1
1=cT +dexp(¯hω/kT)
(exp(¯hω/kT)−1)2.(3.5)
This mo e de ailed model indeed e ined he i in igu e 3.19 and sugges s ha a
po en ial impo an ole in he ab up dec ease o he elaxa ion ime a 30-40 K is
played by an op ical phonon wi h ene gy ¯hω ≈14 meV, oge he wi h low ene gy
phonons (¯hω ≈2−6 meV) commonly in ol ed in hese p ocesses. Spaces o a ine
and in-dep h analysis o he elaxa ion p ocesses we e le o a u u e dedica ed
wo k.
Elec onic quan um cohe ence eTmas a unc ion o empe a u e was also in es i-
ga ed by ou collabo a o s a he Uni e si y o Tu in (see he Lis o Collabo a o s)
by pulsed X-band EPR on he same spec al line (0.334 T), wi h a Hahn-echo de-
cay expe imen . The echo decays we e i ed wi h a s e ched exponen ial decay
I=I0+C2exp[−(2τ/eTm)β] and he ex apola ed cohe ence imes a e shown in
igu e 3.19-(a). The a es a e empe a u e independen below 40 K and in he o de
o magni ude o he µs (2.3 µs o he highe dilu ion). Thus, he low empe a u e
cohe ence imes a e compa able wi h o he pa en s compounds. [6,35] The con-
cen a ion dependence also below 1% sugges s ha in his empe a u e ange he
elaxa ion is s ill due o he elec onic spin-spin in e ac ion, ha domina es o e
he less e ec i e elec on-nuclea in e ac ion, despi e he abundance o spin ac i e
nuclei (1H). A highe empe a u e he spin cohe ence ime eTmsha es wi h he
69 Simone Chicco
Chap e 3
spin la ice eT1a simila ab up d op. E en i eTmis no limi ed by eT1in his
empe a u e ange, i is clea ha he wo phenomena a e co ela ed. A possible
explana ion is ha he as e spin-la ice elaxa ion eT1causes as e p onounced
elec onic luc ua ions ha di ec ly impac he spin cohe ence eTm. Thus, in he
de ini ion o he elec onic cohe ence ime 1/eTm= 1/eT2+ 1/2eT1 he spin-la ice
eT1canno be neglec ed, e en i he spin-spin elaxa ion eT2is s ill he dominan
con ibu ion.
Figu e 3.19: (a) Tempe a u e e olu ion o he elec onic spin-la ice eT1and spin
cohe ence eTm elaxa ion imes, o each dilu ion pe cen age o [V(Cp)2Cl2]. Solid
and dashed lines ep esen espec i ely he models 3.4 and 3.5 i ing esul s. (b,c,d)
Nuclea phase memo y ime nT2measu ed on dilu ion 3b o he ansi ions labeled
in igu e 3.18 a T= 4 K as a unc ion o he applied ield, o BS,BM,BL
con igu a ion, espec i ely (ske ch in inse ). Rep in ed wi h pe mission om Re .
[12]. Copy igh 2021 Ame ican Chemical Socie y.
Nuclea phase memo y imes we e also in es iga ed o a single c ys al o sample
3b (1 % dilu ion). The measu emen s ha e been pe o med a he Uni e si y o
Pa ma on he ”HyReSpec ” [11] b oadband NMR spec ome e (see 2.1.1). Hahn-
echo decays we e measu ed as a unc ion o he applied ield, o all he ansi ions
labeled in he ene gy le els scheme o igu e 3.18. The echo decays as a unc ion o
he delay τbe ween he exci ing and e ocusing 2π/3 -pulses, was modeled wi h
a single exponen ial M(τ) = M0exp(−2τ/nT2) ( ig. 3.20). The ex ac ed phase
memo y imes nT2≈20 −30 µs a e compe i i e wi h espec o pa en compounds
as he [VO(TPP)] epo ed in sec ion 3.2 and in Re .[4].
70 Simone Chicco
Chap e 3
Figu e 3.20: T ans e se magne iza ion decay om NMR echo elaxa ion expe i-
men s, as a unc ion o he delay τbe ween exci ing and e ocusing pulses, a T= 4
K and ixed s a ic ield B0, along BS(a), BL(b), BM(c) o ansi ions S1,L2,
M2. Lines shows he single exponen ial decay i ing. Rep in ed wi h pe mission
om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y.
The nuclea cohe ence inc eases wi h he applied ield, because o he simul aneous
educ ion o he elec o-nuclea mixing o he wa e unc ions and inc emen o he
elec onic pola iza ion. Indeed, he dominan phenomenon ha imes he nuclea
decohe ence is he indi ec dipola in e ac ion wi h nea by nuclea and elec onic
spins. The la e is se e ely educed by dilu ion in diamagne ic analogues ( i a-
nium(IV) in his case).
Conside ing he abo e esul s o he elec onic and nuclea spin cohe ence imes,
his sys ems can be unequi ocally conside ed a p omising coupled qubi -qudi sys-
em. The u he s ep o con i m his claim and p o e ha he cohe ence imes allows
a bi a y and selec i e manipula ions o he elec onic and nuclea mul ile el s uc-
u e, is o ealize expe imen ally elec onic and nuclea spin nu a ion expe imen s.
Low empe a u e elec onic Rabi oscilla ions we e ealized on c ys al powde s a
X-band equency by ou collabo a o s a he Uni e si y o Tu in (see he Lis o
Collabo a o s). The sys em is a ge ed by a i s esonan pulse o a iable leng h
θ( ) ha exci es he chosen ansi ion, ollowed by he canonical π−τ−π/2 Hahn-
echo e ocusing sequence o s imula e he spin-echo. The Rabi equency eωR( ig.
3.21) scales linea ly wi h he manipula ion pulse ampli ude. To u he suppo he
71 Simone Chicco
Chap e 3
selec i i y o he exci a ion, he calcula ed equency domain spec a ( ig. 3.21-(b))
a e basically monoch oma ic, showing only he genuine Rabi eωRcon ibu ion and a
small con amina ion in co espondence o he p o on (1H) nuclea La mo equency
eω
H, which a e ine i ably coupled wi h 51V.
Figu e 3.21: (a) Elec onic Rabi nu a ion expe imen s (EPR X-band) on 3c a
T= 20 K a di e en exci a ion powe s. (b) F equency domain pic u e o highligh
he monoch oma ici y o he oscilla ions. (c) Linea dependence o Rabi equency
(ΩR=ωR/2π) on he exci ing mic owa e pulse ampli ude B1. Rep in ed wi h
pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y.
Nuclea cohe en manipula ions we e also ealized a he Uni e si y o Pa ma o
a ious ield di ec ions wi h espec o he 3b single c ys al (see ig. 3.22). A i s
exci ing pulse o a iable leng h θ( ) esonan wi h a speci ic nuclea ansi ion
and wi h a na ow bandwid h is used o selec i ely induce he ansi ion o in e es .
Then, he s imula ed spin exci a ion is e ocused by a subsequen πpulse o he
de ec ion o he echo in ensi y. The induced Rabi oscilla ions a e damped by a ac-
o λ ha combines he e ec s o igina ing om he sys em nuclea cohe ence imes
and he inomogenei ies o he manipula ing - ield. The la e gene ally domina es
o e he long phase memo y imes o hese sys ems. The oscilla ions a e hus mod-
eled wi h an exponen ially damped oscilla o unc ion I∝exp(−λ ) sin(nωR ). Fo
all he ield di ec ions and pulse powe s p obed, he damping a e is conside ably
longe han he ime equi ed o induce a single in e sion (πpulse) o he nuclea
spin sys em. Combining his esul s wi h he ac ha he damping is mainly due o
72 Simone Chicco
Chap e 4
Un eiling phonons in a
p o o ypical molecula qubi by
IXS
In o de o push u he he pe o mances o molecula qubi s owa d hei use in
quan um de ices, i is undamen al o each a ho ough unde s anding o hei e-
laxa ion dynamics and decohe ence.
In his wo k we exploi ed o he i s ime high- esolu ion Inelas ic X- ays Sca e ing
(IXS) oge he wi h cu ing-edge ab ini io calcula ions o s udy phonon dispe sions
in he p o o ypical molecula qudi [VO(TPP)] [4,14] (see sec . 3.2 o i s cha ac-
e iza ion wi h b oadband NMR). Wi h his syne ge ic expe imen al and heo e ical
app oach, we alida ed he ab ini io phonon calcula ions by compa ing he measu ed
spec a wi h he simula ed sca e ing c oss sec ion. IXS da a pe mi o in es iga e
phonon dispe sions o e a wide sec ion o he ecip ocal space and o un a el he ex-
is ence o ex emely low ene gy op ical phonons. Finally, we de e mined he c i ical
ole o hose ex emely low ene gy modes in he spin elaxa ion, by calcula ing hei
spin-phonon coupling cons an s. Iden i ying he ib a ions which mos ly con ibu es
o elaxa ion indeed p o ides a iable op ion o syn he ically imp o ing he pe o -
mances o molecula magne s. This new and powe ul syne gis ic app oach ga e us
an unp eceden ed insigh s on he spin dynamics o he [VO(TPP)] molecule, pa ing
he way o he applica ion o his echnique o he cha ac e iza ion o phonons in
MNMs.
4.1 S a e o a
Reaching a p o ound unde s anding o Molecula qubi s elaxa ion dynamics and
decohe ence mechanisms is an aspec o pa amoun impo ance in o de o unleash
he po en iali ies o hese sys ems o quan um in o ma ion applica ions and com-
pu a ion. The cohe ence imes o hese sys ems mus be long and empe a u e
esilien in o de o implemen quan um algo i hms.[70] A key ole in de e mining
hose cohe ence imes and hei empe a u e dependence is played by he in e ac ion
o molecula spin wi h in e - and in a-molecula ib a ions.[29,86]
Recen ly, Inelas ic Neu on Sca e ing expe imen s (INS) and ab ini io spin dynam-
ics simula ion ha e been used o he in es iga ion o phonons in molecula qubi s
79

Chap e 4
and single-molecule magne s. Fo ins ance, phonons ha e been ecen ly in es iga ed
in he VO-based compound [VO(acac)2] wi h he 4-dimensional inelas ic neu on
sca e ing echnique. [30,118]. This s udy un a eled he p esence low ene gy op i-
cal phonons, in ol ed in an i-c ossing wi h he acous ic b anches, causing a mu ual
spin-phonon coupling s eng h ans e . [29,30] This pic u e was in clea con as
wi h he canonical ea men o phonons wi h he Debye model, in which acous ic
phonons a e p edic o be well sepa a ed in ene gy om he op ical modes, being
also he main playe s in he spin elaxa ion o MNMs.
An expe imen al echnique allowing us o access bo h phonons ene gies and pola iza-
ion ec o s is undamen al in o de o cons uc a sound model o phonon-induced
elaxa ion mechanisms. Indeed, only wi h such a solid s a ing poin i is possible
o de e mine he ole o speci ic phonon b anches in di e en elaxa ion p ocesses.
[28]. In ac , be o e he ecen INS expe imen s on phonons, we only had in o ma-
ion on phonon ene gy spec a a he Γ poin ob ained wi h Raman, IR o THZ
spec oscopies, which can only be in e p e ed in he con ex o a Debye-like pic u e,
which is iola ed in ou molecula complexes. Howe e , INS echnique poses s ong
limi a ions (be e de ailed in he ollowing sec ion) in e ms, e.g., o samples di-
mensions and chemical composi ions. The pu pose o his wo k is hen o exploi
o he i s ime inelas ic X- ays sca e ing o a molecula qubi , o expe imen ally
in es iga e he ene gies and he pola iza ion ec o s o phonons in a p o o ypical
molecula qubi [VO(TPP)]. A he same ime he implemen a ion o ab ini io s a e-
o -a spin dynamic simula ions enables o unde s and he c i ical ole o di e en
phonon b anches in he elaxa ion dynamics.
The [VO(TPP)] complex is chosen no only as a che ypes o he new gene a ion o
molecula qubi s wi h p omising pe o mances (long cohe ence imes e en a high
empe a u es), [4] bu also because, among he amily o anadium-based sys ems,
i is conside ed o be one o he mos obus o high ene gy i adia ion. The expe -
imen al obse a ion o ul a-low ene gy op ical modes makes his sys em an ideal
es -bed o s udy he ole o hese non-Debye modes in he spin dynamics.
4.2 Inelas ic X- ay sca e ing expe imen
The expe imen al s udy o phonon dispe sion and pola iza ion ec o s on a molecula
c ys al is ealized o he i s ime by exploi ing high- esolu ion Inelas ic X- ay Sca -
e ing (IXS). The ad an ages o his echnique wi h espec o he complemen a y
Inelas ic Neu on Sca e ing (INS) a e se e al. Fi s o all, mode n high- esolu ion
IXS spec ome e s as ID28 [119] a he Eu opean Sinch o on Radia ion Facili y
(ESRF) allows o measu e acous ics and op ical phonons also o e y small samples
(o de o 1 mm3), while he small dimensions o molecula c ys als s ill ep esen an
hu dle e en o mode n high- lux neu on spec ome e , ypically p e en ing he use
o INS o s udying phonon dispe sion. Mo eo e , ano he downside o neu ons is
ep esen ed by he la ge incohe en c oss sec ion o ligh elemen s, like 1H, which a e
abundan in MNMs, causing he blu ing o he phonons cohe en sca e ing signal
and hus o phonon dispe sions. This d awback can be o e come by subs i u ing
he elemen s wi h la ge incohe en c oss sec ion wi h chemical analogues, such as
deu e ium in place o Hyd ogen. Howe e , deu e a ion has signi ican e ec s on he
INS signal only i high pe cen age can be ob ained, which, in he case o MNMs,
80 Simone Chicco
Chap e 4
i is e y di icul . On he con a y, he la ge ene gy ans e in ol ed in he X- ay
incohe en sca e ing p ocesses (∼eV ) is o de o magni ude la ge han he ene gy
scale o in e es when measu ing phonons in MNMs (∼meV ), hus i does no con-
ibu e o he measu ed c oss sec ion. This, combined wi h he nea ly absence o
mul iple sca e ing p ocesses, makes IXS basically a backg ound- ee echnique in
he meV ange. Fu he mo e, he employmen o ha d X- ays wi h high inciden
ene gies (∼keV ), se e al o de o magni ude la ge han he ene gy scale o MMNs
molecula and la ice ib a ions, causes he spec al esolu ion o be independen
om he ene gy and momen um ans e o he sca e ing e en . This is an im-
po an ea u e o he cha ac e iza ion o op ical phonons. Indeed, in p eceden
s udies pe o med wi h INS, acous ic modes dispe sions we e success ully obse ed,
while he ene gy-dependen esolu ion was no su icien o esol e op ical modes
wi h ene gies abo e 10-12 meV. [30] The ene gy and momen um ans e also esul
o be comple ely decoupled.
All he lis ed IXS ad an ages and ID28 [119] unique capabili ies allowed us o suc-
ceed a measu ing o he i s ime acous ic and op ical phononic exci a ion in a
molecula c ys al o a p o o ypical qubi . A [VO(TPP)] single c ys al (c ys allog a ic
cell in ig 4.1) o dimensions 1 ×1×0.5 mm3was syn hesized by ou collabo a o s
a he Uni e si y o Flo ence (see he Lis o Collabo a o s). The c ys al was o i-
en ed by means o di use sca e ing on he ID28 beamline side s a ion. We ocused
on he (h0l) sca e ing plane, and in pa icula on B agg e lec ions 006 and 600,
which o e ed he bes comp omise in e ms o in ensi y o he inelas ic signal and
supp ession o he elas ic one. Wi h his choice o B illouin zone (BZ) we could
explo e he Γ −Kz, Γ −Kxand Γ −Npa h o he [VO(TPP)] ecip ocal space,
co esponding o di ec ions (00l), (h00) and (h0h) espec i ely, in longi udinal and
ans e se con igu a ion.
Figu e 4.1: (a-b) iew o he [VO(TPP)] c ys al s uc u e along he a and c axis
espec i ely. (c) B illouin zone pic u e, in which he symme y di ec ion p obed in
ou expe imen a e highligh ed ( ed a ows).
81 Simone Chicco
Chap e 4
IXS measu emen we e collec ed a 300 K, wi h he sample glued on a s anda d
sample holde . The ID28 spec ome e was used in backsca e ing con igu a ion,
exploi ing wo di e en e lec ions o he Si monoch oma o s: he Si(9 9 9), which
selec s an incoming ene gy E= 17.794 KeV and ea u es a esolu ion o δE = 3.0
meV, and he Si(12 12 12), whose selec ed ene gy is E= 23.725 KeV wi h a esolu-
ion δE = 1.5 meV. We e e o his wo con igu a ions as Low and High- esolu ion
con igu a ions. By exploi ing he iple-axis geome y o he spec ome e in low-
esolu ion con igu a ion we pe o med an explo a ion scan o he phonon modes.
The p incipal symme y di ec ions in he ecip ocal space Γ−N, Γ−Kxand Γ−Kz
(see he i s B illouin Zone o he c ys al in ig. 4.1) we e sampled spanning he
ene gy om -25 meV o 25 meV a cons an sca e ing ec o Q alues. In igu e 4.2
examples o low esolu ion spec a a e shown, o longi udinal and ans e se scans
a di e en poin s in he ecip ocal space, o all he explo ed symme y di ec ions1.
Figu e 4.2: Longi udinal (a,b,c) and ans e se (d,e, ) IXS scans along he
di ec ions Γ −N, Γ −Kxand Γ −Kz, collec ed in low- esolu ion con igu a ion.
F om hese scans is al eady e iden he p esence o low ene gy phonons, oge he
wi h highe ene gy op ical ones a a ound 8 and 15 meV ( ig. 4.2-(a-c)). To be -
e in es iga e low-ene gy phonon modes, we measu ed he di ec ion in which hese
exci a ions a e expec ed wi h he high- esolu ion con igu a ion o he spec ome e
(see ig. 4.3).
High esolu ion measu emen s indeed con i med he ul a-low ene gy o [VO(TPP)]
op ical phonons (≤2 meV) in mos o he p obed BZ di ec ions. The su p isingly-
low ene gy o hese modes is he lowe e e measu ed in a molecula c ys al.
The ID28 beamline has a se o nine analyze s, each o hem co esponding o a
de ec o in he de ec o desk. This ea u e enables he simul aneous collec ion o
phonon spec a o nine di e en Q alues. This is an impo an esou ce, since i
1I wo h no icing ha he de ini ion o longi udinal and ans e se scans along a diagonal
ecip ocal space di ec ion as Γ −Nis essen ially meaningless. The eason o which i is done is
simply a no a ion con en ion o uni o m he language.
82 Simone Chicco
Chap e 4
Figu e 4.3: Longi udinal (a) and ans e se (c) IXS scans along he di ec ions Γ−Kx,
Γ−N, and Γ −Kz, collec ed in high- esolu ion con igu a ion, o esol e he low
ene gy phonons con ibu ion close o he elas ic line.
allowed us o de ec a seconda y se o exci a ions, e y close o he p obed sym-
me y di ec ions and wi h sligh ly di e en |q|. Combining he wo se o da a we
inc ease se e ely he sampling o he BZ.
4.2.1 Resul s
The measu ed spec a ha e been i ed using he beamline so wa e @ i 28, by in-
cluding an elas ic line cen e ed a ze o exchanged ene gy and a easonable numbe o
inelas ic peaks, depending on he exci a ion clea ly isible in each scan. The spec al
unc ion used was a damped ha monic oscilla o (DHO) dho =I0(cδE)2/(E2+cδE),
ep oducing he lineshape o he ins umen , wi h a FWHM co esponding o he
measu emen esolu ion (δE = 3 meV o δE = 1.5). The ene gies o he esul ing
phonon exci a ions as a unc ion o he BZ sampling ep esen he phonon dis-
pe sions along he p obed symme y di ec ions. The e o e, hey can be di ec ly
compa ed wi h hose ob ained wi h pe iodic densi y unc ional heo y (DFT) cal-
cula ions. Howe e , no e e y phonon b anch has he same IXS c oss-sec ion. The
expe imen al da a mus he e o e be compa ed wi h he simula ed IXS c oss sec ion
∂2σ
∂E∂Ω. The sca e ing in ensi y o each phonon b anch can be simula ed s a ing
om calcula ed phonon eigen alues and eigen ec o s, as desc ibed in sec ion 2.2.3.
These DFT calcula ions on [VO(TPP)] phonons we e ealized by ou collabo a o s
om he T ini y College Dublin and om he Uni e si y o Flo ence (see he Lis
o Collabo a o s).
The mos isible exci a ions in he measu ed spec a a e hus expec ed o be he
one wi h he s onge sca e ing c oss sec ion. The compa ison be ween da a and
DFT calcula ions is done by supe imposing he phonon ene gies ex ac ed om he
da a i ing o he 2D colo map o he simula ed c oss sec ion in he ene gy/phonon
wa e ec o plane. The esul is shown in igu e 4.4 o longi udinal and ans e se
modes along he h ee sampled symme y di ec ions Γ−N, Γ−Kxand Γ−Kz. The
ag eemen is excellen and he majo i y o he op ical o acous ics phonons wi h a
signi ican c oss sec ion a e de ec ed expe imen ally. F om igu e 4.4 he p esence o
he ex emely low ene gy op ical phonons obse ed in high- esolu ion con igu a ion
is also con i med by DFT esul s. These modes a e pe sis en in he whole BZ and
cha ac e ized by a ema kable sca e ing in ensi y.
A comple e desc ip ion o phonon modes is undamen al o calcula ing he spin-
83 Simone Chicco
Chap e 4
Figu e 4.4: The 2D colo map shows he inelas ic c oss-sec ion calcula ed s a ing
om phonon ene gies and pola isazion ec o s ex ac ed om ab ini io calcula ions,
along he h ee sampled symme y di ec ions in he eci pocal space in bo h lon-
gi udinal and ans e se con igu a ion: (a,e) Γ −N, (b, ) Γ −Kx, (c,d) Γ −Kz.
The line-wid h associa ed o each phonon modes is chosen small enough o enable
he dis inc ion o he con ibu ion a ising om any b anch. Cyan do s and squa es
(δE = 3 / 1.5 meV espec i ely) ep esen he exci a ion ene gies ex ac ed om he
spec a in igu es 4.2 and 4.3. Whi e do s/squa es ep esen he ene gies ex ac ed
om he inspec ion o seconda y analyze s. Shaded a eas highligh he egions non
accessible expe imen ally due o he dominan elas ic con ibu ion.
phonon coupling coe icien s. The e o e, he alida ion o DFT calcula ions mus be
suppo ed by he analysis o bo h phonon ene gies ωj(q) and pola iza ion ec o s
σd
j(q). Bo h hese quan i ies, as seen in sec ion 2.2.3, con ibu e o he inelas ic sca -
e ing c oss sec ion. Thus, a compa ison be ween he expe imen al spec a and he
IXS c oss sec ion simula ed wi h he DFT calcula ed ωj(q) and σd
j(q) gi es a di ec
eedback on he accu acy o calcula ions. In o de o pu o he es he pola iza ion
ec o o each phonon mode ob ained wi h DFT, he mos e ec i e app oach is o
implemen a con olu ion o he c oss sec ion o each mode wi h he expe imen al
esolu ion and di ec ly compa e i wi h he expe imen al spec a as a unc ion o
he ene gy. The compa ison is shown in igu e 4.5 o some ep esen a i e Q alues
and is excellen o he whole ene gy ange o bo h low and high- esolu ion con ig-
u a ions. This es u he con i ms he eliabili y o he calcula ed phonon model.
A ypical small escaling o ∼10% is applied [30] o accoun o an de Waals in-
e ac ion and empe a u e e ec s. Mo eo e , small disc epancies in he low ene gy
egime exci a ion in ensi y a e mainly due o he Bose ac o nj(q)=(exp(βωj(q))−
1)−1which exponen ially depends on he exac phonon ene gy and hus enhance
small di e ences.
84 Simone Chicco

Chap e 4
Figu e 4.5: Expe imen al IXS spec a (black do s) a e compa ed wi h he simula ed
c oss-sec ions ( ed line). The la e a e con olu ed wi h he expe imen al esolu ion
((a-e,g): δE = 3 meV; ( ,h,i): δE = 1.5 meV). A ep esen a i e spec a is shown
o each p obed symme y di ec ion. Blue lines ep esen he inelas ic c oss-sec ion,
calcula ed wi h a line-wid h ha enables o disc imina e he con ibu ion o each
phonon mode. The elas ic line is omi ed o cla i y.
85 Simone Chicco
Chap e 4
4.3 The c i ical ole o ul a-low ene gy ib a-
ions
This ex ensi e expe imen al IXS cha ac e iza ion o he [VO(TPP)] phonon dis-
pe sions enabled he alida ion o he ab ini io phonon calcula ions. F om his
alida ed esul s i has been possible o mo e a s ep u he and simula e also he
sys em spin elaxa ion. The ab ini io calcula ion o spin-phonon coupling pa ame-
e s equi es o simula e he e ec o phonons on he pa ame e s o he sys em spin
Hamil onian. As in sec ion 3.2, he spin Hamil onian o [VO(TPP)] is w i en as
H0=ˆ
I·A·ˆ
S+pˆ
I2
z+µBˆ
S·gS·B0+µNgNˆ
I·B0and he pa ame e s a e de i ed
bo h expe imen ally in [4] and con i med by DFT calcula ions.
The modula ions induced on he Hype ine Aand spec oscopic spli ing g en-
so s a e calcula ed om he i s and second pa ial-de i a i e o hose enso s wi h
espec o he phonon wa e ec o o each phonon b anch, Vα= (∂A/∂qα) and
Vαβ = (∂2A/∂qα∂qβ), espec i ely. E en o a mode n compu a ional ha dwa e,
his huge calcula ion is e y demanding in e ms o ime and compu a ional e-
sou ces. This hu dle is o e come, by ou collabo a o s om Dublin T ini y College
and om Flo ence Uni e si y (see he Lis o Collabo a o s), exploi ing a ained
neu al ne wo k o in e pola e ab ini io esul s and compu e he di e en ia ion e i-
cien ly. [120] The calcula ed spin-phonon coupling coe icien s, ep esen ed by he
modulus |Vα|, a e shown in igu e 4.6 oge he wi h he calcula ed phonon densi y
o s a e.
Figu e 4.6: Phonon densi y o s a e (black line) and modulus |Vα|o he spin-phonon
coupling coe icien ( ed line, nega i e axis) compu ed ab ini io, as a unc ion o
ene gy. In inse he compa ison be ween he low ene gy pDOS o [VO(TPP)] (black)
and he pa en molecula qubi [VO(acac)2] (g een). [13]
Two impo an conclusions can be ex ac ed he e: he i s is ha he ex emely low
ene gy op ical modes possess a ema kable coupling wi h he spin and hus a e ex-
86 Simone Chicco
Chap e 4
pec ed o play a key ole in spin elaxa ion p ocesses. Secondly, om he calcula ed
low ene gy phonon densi y o s a es (pDOS), we obse e a comple e de ia ion om
he Debye model. Indeed, he [VO(TPP)] pDOS g ows linea ly wi h he phonons
ene gy below 5 meV, while a ∼ω2beha io would be expec ed o he canonical
ep esen a ion o he Debye model, in which op ical and acous ics phonons a e well
sepa a ed in ene gy. This second poin is u he suppo ed in he inse o igu e
4.6 in which he low ene gy pDOS p o ile o [VO(TPP)] is compa ed o he one o
he pa en compound [VO(acac)2], ecen ly s udied by 4D-INS [30] which does no
ea u e op ical phonons a ene gy lowe han ∼6 meV a Γ-poin . [13]
I has been demons a ed ha , o molecula sys ems, he spin li e ime a empe -
a u e highe han ∼10 K is s ongly in luenced by he coupling wi h phonons. [13]
In molecula qubi s like [VO(TPP)] elaxa ion dynamics is mos ly igge ed by non-
esonan wo-phonon Raman p ocesses. [116] The spin elaxa ion induced by hese
phonon abso p ion and e-emission phenomena can be simula ed by including, in he
secula Red ield equa ion, he spin-phonon coupling coe icien s |Vα|and |Vαβ|. A
mode a e applied ield (0.3 T) he leading e m o he elaxa ion a e he modula ion
o he hype ine Aand spec oscopic spli ing g enso s. The la e con ibu ion is
he e neglec ed in he calcula ion wi hou loss o gene ali y, since he con ibu ion
o he wo o he spin-phonon coupling is analogous and ollows a simila end o
he conside ed ields. The ex apola ed spin la ice elaxa ion ime T1as a unc ion
o empe a u e is compa ed in igu e 4.7 wi h he da a measu ed in [14] by X-band
EPR wi h an in e sion and eco e y expe imen (see sec . 2.1.2). The ag eemen
is ema kable abo e ∼10 K and bo h he simula ion and he expe imen display a
T−2dependence, ypical o phonon-induced Raman elaxa ion p ocesses.
Figu e 4.7: Calcula ed spin elaxa ion ime T1 o [VO(TPP)] ( ed squa es) com-
pa ed o he T1measu ed in [14] wi h pulsed X-band EPR (black do s). These da a
a e hen compa ed o he simula ed T1a e he emo al o phonons wi h ene gy
lowe hen 6 meV (blue iangles).
87 Simone Chicco
Chap e 4
We no e ha he low empe a u e expe imen al T1de ia es om he end isible a
highe empe a u e. This can be ela ed o he e ec o spin-spin dipola in e ac ion,
which is only pa ially emo ed by dilu ion a %2 in he isos uc u al [TiO(TPP)]
diamagne ic analogue. The Raman-like end in he spin la ice elaxa ion a es
down o low empe a u es suppo s he leading ole o he ul a-low ene gy op ical
modes which a e demons a ed o be s ongly coupled wi h he molecula spin ( ig.
4.6). Indeed, by emo ing low-lying phonons (¯hω < 6 meV) s ongly coupled wi h
he spin, he simula ed T1inc eases by wo o de o magni ude in he whole p obed
empe a u e ange ( ig. 4.7). This es u he s ess he pa amoun impo ance o
low-ene gy op ical phonons in supp essing spin li e ime.
The ib a ions associa ed wi h low ene gy op ical phonons (∼2meV ) co espond, a
he molecula si e, o dis o ion o he molecula s uc u e. In his pa icula case i
has been obse ed ha he ib a ions in ol ed cause dis o ion o he phenyl g oups
o he molecule, and he bending o he po phi in plana ing (see ig. 4.8). Being
able o associa e he c i ically coupled phonon modes wi h a speci ic molecula dis-
o ion ep esen s an e ec i e app oach o imp o ing he spin li e ime pe o mances
by chemically ailo ing he molecule s uc u e and hus he ib a ional spec a.
Figu e 4.8: Ske ch o he dis o ion o he molecula s uc u e, om he equilib ium
s uc u e (yellow) o an ex ensi ely dis o ed one ( ed). These dis o ions consis
in a bending o he po phi in plana ing and a wis o he phenyl g oups, and a e
associa ed wi h he lowes ene gy op ical mode a Γ-poin .
I is wo h men ioning ha e en hough he expe imen s a e ypically pe o med on
single molecula c ys als, he applica ion o hese molecules in quan um de ices e-
qui es he exploi a ion o single molecules. Thus, he in es iga ion o single molecule
ib a ions is also an impo an piece o in o ma ion, when compa ed wi h he sin-
gle c ys al esul s. The pe sis ence o he low ene gy in a-molecula dis o ions in
[VO(TPP)] was con i med by gas phase calcula ions on a single molecule. I was also
e i ied ha by emo ing he phenyl g oups om he molecule, he lowes ene gy
op ical phonons a e shi ed up o 6.5 meV. The e o e, he s a egy o chemically
enginee ing he molecule by emo ing he phenyl ings as in Vanadyl po phi in can
po en ially imp o e he elaxa ion imes by emo ing low ene gy ib a ions c i ically
coupled wi h he spin.
88 Simone Chicco
Chap e 5
elaxa ion p ocesses, wi h and an ex ac ed e ec i e ba ie Ue = 1760 K. A
lowe empe a u e ins ead, elaxa ion in 1 esul s o be domina ed by powe law
Raman-like mechanisms, wi h τ−1=CTnand a exponen n= 1.1, hal o he one
epo ed o a. Indeed, o e all, 1displays a elaxa ion 10 o 100 imes as e wi h
espec o a, wi h a 100 seconds blocking empe a u e o 23 K, compa ed o 56
K o a. Since a hese empe a u es he spin dynamics is con ined o he lowes
K ame s double , he elaxa ion is d i en by he coupling wi h low ene gy dispe -
si e phonons ha igge non- esonan Raman mechanisms. The e o e, he as e
elaxa ion can be a ibu ed o he di e ences in he low ene gy phonon spec a.
Insigh s on he low-ene gy phonons spec a a e hen c ucial o unde s and he p o-
cesses go e ning he elaxa ion and o un a el he key ing edien s o each a high
blocking empe a u e.[31] Fo samples a,2and 3 o example, simula ions o he
pDOS e eal wo e y peculia ends in he low and high ene gy spec al egions.
The high ene gy DOS, domina ed by localized ib a ional modes esonan wi h he
c ys al ield gaps and p omo e o O bach elaxa ion o e he e ec i e ene gy ba -
ie Ue , display simila spec al ea u es, especially abo e 100 meV. Con e sely,
he DOS o low-ene gy dispe si e modes, igge ing non- esonan Raman p ocesses
a ies signi ican ly (see ig. 5.6).
Figu e 5.6: Compa ison o he calcula ed pDOS o samples a,2,3. Panel (a)
exhibi s he whole calcula ed ene gy ange, while panel (b) shows he es ic ed
low-ene gy egime, highligh ing he signi ican di e ences in he low ene gy spec a,
compa ed o he less p onounced ones in he in e media e-ene gy spec a.
95 Simone Chicco

Chap e 5
By compa ing hese ab ini io pDOS calcula ions wi h hose measu ed by INS e-
po ed in he ollowing sec ion, we can alida e he DFT esul s o be used as he
s a ing poin o he calcula ion o he spin-phonon coupling coe icien s and he
simula ion o he empe a u e dependence o he elaxa ion a e in igu e 5.4.
I mus be ema ked ha in his app oach we a e neglec ing he e ec o he quan-
um unneling o he magne iza ion (QTM), which is empe a u e-independen and
hus is no igge ed by he in e ac ion wi h phonons. The e ec o QTM can be
in ac quan i ied only by measu ing he empe a u e and ield dependence o he
magne iza ion elaxa ion a es. Howe e , his mechanism was demons a ed o be
non dominan a low empe a u e in his compounds class, due o he s ong axial
ligands en i onmen ha supp ess he e iciency o his elaxa ion p ocess.[17]
5.1.1 Inelas ic neu on sca e ing expe imen
The inelas ic neu on sca e ing (INS) expe imen was pe o med a he ins i u e
Laue-Lange in (ILL) on he indi ec geome y spec ome e IN1-Lag ange (LA ge
GRaphi e ANalyse o Genuine Exci a ions), desc ibed in sec ion 2.2.2.[109] This
beamline, ins alled on he ho neu on sou ce o he eac o , ea u es a combina ion
o wide ene gy- ange and high esolu ion wi h an high neu on lux. Fo his eason,
i is well sui ed o he s udy o molecula ib a ions in complex sys ems, h ough he
e alua ion o he pDOS. In pa icula , a pa om cons an ac o s (see sec . 2.2.2),
he measu ed sca e ing unc ion co esponds o he neu on weigh ed pDOS o he
s udied sys ems (once he elas ic and quasi elas ic con ibu ion a e sub ac ed).
The expe imen was pe o med a he empe a u e o 5 K on ≈2 g o powde o
each o he sample unde in es iga ion (1,2,3). The ene gy ans e was chosen in
o de o co e he 0 o 200 meV ange. To co e his low and in e media e ene gy
ange wi h a cons an ene gy s ep, h ee monoch oma o s we e used, each o hem
ea u ing di e en esolu ions:

Si(1 1 1), a ailable in a 4.5 o 20 meV ange and ea u ing a 0.8 meV esolu ion;

Si(3 1 1), a ailable in a 16.5 o 35 meV ange and ea u ing he same 0.8 meV
esolu ion;

Cu(220), in he wide ange 26 o 200 meV and a p opo ional esolu ion o
2-3 % o he inciden ene gy.
Each da ase was hen no malized by he numbe o inciden neu on, be o e me g-
ing hem oge he in a single cu e. Fu he e inemen s o he collec ed da a equi e
he sub ac ion o he emp y sample holde signal and he spec al spike smoo hing
by compa ison wi h a wa e spec a p e iously collec ed. Elas ic and quasi-elas ic
sca e ing con ibu ions a e hen sub ac ed by i ing he sum o a Gaussian cu e
cen e ed a E=0 and a Lo en zian unc ion cen e ed a non-ze o exchanged ene gy,
keeping he wid h as a i ing pa ame e . The esul s o hese da a educ ion p o-
cedu e (pe o med wi h he so wa e LAMP [121]) a e shown in igu e 5.7 o he
h ee Dy complexes, compa ed wi h he da a published by Chiesa e al. o he
benchma k Dysp osocenium a.[17]
The measu ed pDOSs con i m wha was expec ed om ab ini io calcula ion. The
high ene gy spec al componen s o he h ee samples ea u e analogous peaks, wi h
96 Simone Chicco
Chap e 5
Figu e 5.7: pDOS measu ed on IN1 o sample 1(g een line), 2(ligh blue line) and
3(magen a line), compa ed wi h he same measu emen collec ed o he benchma k
Dysp osocenium on he MERLIN spec ome e a ISIS (do ed black line).[17] The
wo panels (a) and (b) show he whole collec ed ene gy ange and a close up o he
low ene gy pDOS, espec i ely.
97 Simone Chicco
Chap e 5
a compa able ela i e in ensi y. This end is con i med also by compa ison wi h
sample a. Con e sely, a low ene gy he pDOSs display s ong di e ences in bo h
he ene gy o he modes and hei in ensi y. The la ges di e ences a e ound in he
e y low ene gy b anches, om 0 o 30 meV, whe e he sca e ing in ensi y inc ease
se e ly wi h espec o sample a, eaching he maximum in sample 3. This sample
in pa icula display an o e all highe sca e ed in ensi y, which is jus i ied by he
highe numbe o 1H in he coun e ion, yielding a ele an incohe en con ibu ion.
5.1.2 Resul s
The p elimina y conclusions ou lined om he compa ison be ween he expe imen al
pDOS a e in sound ag eemen wi h hose d awn om he ab ini io calcula ed ones.
Bo h expe imen al da a and DFT esul s in ac con i m ha he main di e ences
in he pDOS o he h ee compunds a e in he low-ene gy ange and hus can be
di ec ly linked o he di e en beha iou o he elaxa ion a es in igu e 5.4 in he
Raman empe a u e egime. To con i m hese hesis, i is undamen al o de elop
a sound model o he phonon-induced elaxa ion o he magne iza ion. This can
be done exploi ing he DFT phonon esul s o de e mine he coupling o molecula
and la ice ib a ion wi h he molecula spin. The compa ison be ween he mea-
su ed pDOSs wi h he calcula ed ones allows us also o alida e he exploi ed DFT
me hod. The pe iodic DFT calcula ions o he pDOSs epo ed in igu e 5.6 ha e
been pe o med by ou collabo a o s a he Uni e si y o Manches e (see he Lis o
Collabo a o s) wi h he code VASP [122], exploi ing he PBE exchange-co ela ion
unc ional [123,124] and D2 an dee Waals co ec ions.[125] Phonon densi y o
s a es a e compu ed using he ini e displacemen me hod as implemen ed in he
PHONOPY package.[126,127]
The compa ison ou come is shown in igu e 5.8 o sample a,2,3 espec i ely.
Figu e 5.8: Compa ison o he expe imen al phonon densi y o s a e (5 K) wi h he
neu on weigh ed pDOS calcula ed ab ini io o (a) sample a, (b) sample 2and (c)
sample 3. The ab ini io calcula ions we e pe o med by ou collabo a o s a he
Uni e si y o Manches e (see he Lis o Collabo a o s).
The ag eemen be ween he expe imen al da a and he calcula ed pDOSs is excel-
len . The mos ele an spec al ea u es a e indeed ep oduced by he calcula ions.
As a esul o his success ul compa ison, he phonon model is assumed o be ac-
cu a e and can be used o he de elopmen o a ealis ic model o he elaxa ion
98 Simone Chicco
Chap e 5
o hese compounds. This wo k in pa icula is s ill ongoing, since he ab ini io
calcula ion o he spin phonon coupling coe icien s and he simula ion o he spin
elaxa ion is e y demanding in e ms o compu a ional ime.
5.2 Measu ing phonon modes o a Dy-based SMM
in applied p essu e
Chemical and s uc u al modi ica ions a e no he only s a egy ha can be exploi ed
in o de o igge a ia ions in he phonon spec a o molecula nanomagne s. In-
deed, by applying an ex e nal hyd os a ic p essu e, i is possible o induce g adual
dis o ion o he c ys al and molecula s uc u e o hese sys ems, and consequen ly
o modi y he phonon spec a. By in es iga ing how phonons and hus he phonon-
induced elaxa ion dynamics a y in SMMs as a unc ion o he applied p essu e,
we can in p inciple disen angle he ole played by speci ic la ice and molecula i-
b a ions in he domina ing elaxa ion mechanisms. In pa icula , his in o ma ion
can gi e undamen al insigh in o de o de elop u u e syn he ic s a egies o im-
p o ing he pe o mances o SMMs.
To apply his inno a i e app oach, we decided o cha ac e ize a mononuclea Dys-
p osium based SMMs [Dy(H2O)5(HMPA)2]I3·2HMPA (wi h HMPA = hexame hylphos-
pho amide, [(CH3)2N]3PO), syn hesized by ou collabo a o s a he Uni e si y o
Glasgow (see he Lis o Collabo a o s). The syn hesis o his sys em, oge he
wi h magne ome y da a ha e al eady been published in 2018 by A. Canaj e al.
[18]. The compound p esen s an almos ideal pen agonal bipy amidal geome y,
wi h a p ominen axial c ys al ield p omo ed by he wo HMPA ligands (see ig.
5.9), which o m a ∼182◦axial angle. The sys em shows magne ic hys e esis up
Figu e 5.9: (a) Molecula s uc u e o [Dy(H2O)5(HMPA)2] and i s coun e ion
I3·2HMPA a ambien p essu e. The Dy ion (seag een) and he O ( ed) en i onmen
a e highligh ed o cla i y. (b) Magne iza ion da a om Re .[18] as a unc ion o he
applied ield showing hys e esis loops a di e en empe a u e in ambien p essu e
(Y-dilu ed sample). (c) Magne iza ion elaxa ion a es om Re .[18] as a unc ion
o empe a u e (black do s), ex ac ed om ac suscep ibili y measu emen s. The
O bach egime is highligh ed by he A henius i ( ed line) o an ene gy ba ie
Ue = 600 K and τ0= 1.2×10−11 s, as epo ed in [18]
o T=9 K (10 K i he molecule is dilu ed in o i s Y diamagne ic analogue). The
quan um unneling mechanism is no supp essed comple ely by he axial symme y;
99 Simone Chicco
Chap e 5
he e o e, a a ound ze o applied ield he hys e esis loop becomes na owe , as a
signa u e o he ac i a ion o his elaxa ion p ocess (see ig. 5.9). F om ac suscep-
ibili y measu emen s a elaxa ion a e was also ex ac ed and i ed wi h a single
exponen ial A henius cu e abo e 30 K ( ig. 5.9). This sugges s ha he dominan
elaxa ion p ocess abo e 30 K is an O bach-like mechanisms wi h an ene gy ba ie
Ue = 600 K, induced by he coupling o he spin wi h phonons wi h ene gy compa-
able wi h he c ys al ield gaps. A empe a u e below 30 K ins ead, we do expec
he de ia ion om he O bach end o ep esen he insu gence o non- esonan wo
phonons Raman p ocesses as he leading mechanisms o he elaxa ion dynamics ( ig
5.9).[18]
When an hyd os a ic p essu e is applied o he sys em, c ys allog aphic in es iga-
ions o he molecule wi h X- ay di ac ion ha e e ealed a g adual il ing o he
ligands axial angle, ha in an applied p essu e o 1.1 GPa becomes 190◦deg ee (see
ig. 5.10). This loss o axiali y causes se e al di e ences in he magne iza ion cu es
measu ed unde p essu e. A an applied p essu e o 1.1 GPa he magne ic hys e esis
Figu e 5.10: (a) Molecula s uc u e o [Dy(H2O)5(HMPA)2] and i s coun e ions
I3·2HMPA unde an applied p essu e o 1.1 GPa. Dysp osium ion (seag een) and
he Oxygen ( ed) en i onmen a e highligh ed. (b) Magne iza ion da a as a unc ion
o he applied ield showing hys e esis loops a di e en empe a u e in an applied
hyd os a ic p essu e o 1.1 GPa. These measu emen s we e pe o med by ou col-
labo a o s a he Uni e si y o Glasgow (see he Lis o Collabo a o s)
loop is in ac comple ely closed in ze o applied ield, also a 2 K, due o he elax-
a ion induced by he magne iza ion unneling unde he ba ie . When he ield is
applied he hys e esis cycle opens, bu wi h a educed ela i e a ea i compa ed o
he da a in ambien p essu e and equi alen empe a u e (see ig. 5.10).
In o de o unde s and he modula ions induced in he phonon spec um by he
applied p essu e and he e ec s on he elaxa ion dynamics, we combine he magne-
iza ion measu emen s wi h unde p essu e INS expe imen s, ha allows he mea-
su emen s o phonon exci a ions as a unc ion o he applied p essu e. These esul s
can hen be compa ed wi h DFT calcula ions o he phonon modes. These DFT
esul s, alida ed by compa ison wi h he expe imen al da a, will be exploi ed o
simula e he sys em spin dynamic unde di e en p essu e egimes.
100 Simone Chicco

Chap e 5
5.2.1 Inelas ic neu on sca e ing expe imen
The inelas ic neu on sca e ing expe imen unde applied p essu e was pe o med a
he Ins i u e Laue-Lange in (ILL) on he he mal neu on iple-axis spec ome e
IN8 (mo e de ails on IN8 see sec . 2.2.2).[108] This high inciden lux spec ome e
is op imized o he cha ac e iza ion o la ice ib a ions and magne ic exci a ions
in he ange om ze o o ∼100 meV, in sample o small olume and weak signals.
Mo eo e , IN8 can be equipped wi h a high p essu e Pa is-Edimbu g cell ha can
emula e he ange o p essu e applied p e iously o he magne ome y expe imen s
(0 o 1.1 GPa). Due o he chemical incompa ibili y o he [Dy(H2O)5(HMPA)2] sin-
gle c ys al wi h he liquid ec o s used o apply he hyd os a ic p essu e, we used
lead ilings as a ec o o dis ibu e uni o mly he ex e nal p essu e. The c ys al has
been placed inside he CuBe gaske o he Pa is-Edimbu g cell, he emp y spaces
ha e been closed wi h he lead ilings and hen p essed o a gapless pocke .
A e he alignemen o he single c ys al sample, we in es iga ed he phonon exci-
a ions a Γ-poin in he ecip ocal space o wo di e en 
Q ec o alues: 
Q1=
(0 3 −12) and 
Q2= (0 8 −16). The ene gy scans a e collec ed by ixing he inal
wa e ec o k and a ying he inciden ki, o keep cons an he sca e ing 
Q alue.
We exploi ed wo di e en monoch oma o s, Si (111) and PG (002), ea u ing wo
di e en esolu ions (0.88 meV and 3 meV, espec i ely). The low esolu ion con ig-
u a ion ha e been used o an exhaus i e scan o he ene gy ange 0 −70 meV wi h
a inal wa e ec o k = 4.11 ˚
A−1. Then he low ene gy ange 0 −34 meV has been
e ined in he high esolu ion con igu a ion wi h k = 2.66 ˚
A−1. In his case, he
ac ual accessible ene gy was limi ed below 34 meV due o he p esence o a spu ious
signal om an analyze a monics o ki= 2k .
The Γ-poin ene gy scans shown in igu e 5.11 we e collec ed a he empe a u e o
77 K o wo di e en applied p essu es 0 GPa and 1 GPa. Each da ase has been
no malized by he numbe o sca e ed neu ons a he p e-sample moni o . Then,
he emp y p essu e cell signal, collec ed wi h he same CuBe gaske bu wi hou he
c ys al inside he lead iling, was sub ac ed, oge he wi h a quasi-elas ic lo en ian-
shape signal a e y low ene gies. These da a a e also compa ed wi h he inelas ic
double di e en ial c oss-sec ion o he Dy-complex (see sec.2.2.2) simula ed wi h he
eigen alues and eigen ec o s o pe iodic DFT-calcula ed phonon modes. F om he
compa ison i is e iden ha he majo di e ence induced by he applied p essu e
lies in he low ene gy egion, be ween 0 and 25 meV. In his ange, phonon modes
shi o highe ene gy by inc easing he applied p essu e (wi h a sizeable shi o he
o de o some meV) and change he sca e ed spec al in ensi y. Some changes in
he phonon exci a ions a e also obse able a highe ene gies; howe e , non sizeable
as he e ec s on he low ene gy phonons.
Da a we e also collec ed as a unc ion o empe a u e ( ig. 5.12). The di e en
scans a e no malized o he empe a u e dependen Bose Fac o (sec . 2.2.2). This
app oach can be use ul o de e mine he eliabili y o he spec al de ails, ha a e
expec ed o g adually b oaden by inc easing he empe a u e. F om he compa ison
o di e en scans we obse e ha mos o he spec al ea u es a low empe a u e
su i e a highe empe a u es, bu b oaden signi ican ly and lose in ensi y.
Va ious single c ys als ha e been hen milled in o ew g ams o powde and placed
in a hin Aluminum cylinde , ha is sc ewed on he c yos a p obe. By measu -
ing powde samples we a e age ou he 
Qdependence o he sca e ing in ensi y
o e a wide ange o o ien a ions. Thus, he sca e ing unc ion can be desc ibed
101 Simone Chicco
Chap e 5
Figu e 5.11: INS c oss sec ion measu ed a 77 K o wo di e en applied p essu e
on a [Dy(H2O)5(HMPA)2] single c ys al (do s), a he Γ-poin o 
Q1= (0 3 −12).
He e, he p essu e cell+gaske blank signal is al eady sub ac ed om he sca e ed
in ensi y. The solid line ep esen he INS c oss sec ion calcula ed s a ing om
DFT-calcula ed eigen alues and eigen ec o s and con olu ed wi h expe imen al es-
olu ion o 3 meV (low esolu ion con igu a ion). The low-ene gy di e ences in he
INS c oss sec ion induced by he applied p essu e a e e iden and eme ge conside -
ably om he e o ba s.
Figu e 5.12: Tempe a u e dependence o he high- esolu ion INS c oss sec ion mea-
su ed a 
Q1= (0 3 −12) in a [Dy(H2O)5(HMPA)2] single c ys al, o a cons an
applied p essu e o 1 GPa.
102 Simone Chicco
Chap e 5
in he so-called incohe en app oxima ion (see sec . 2.2.2), whe e i esul s di ec ly
p opo ional o he neu on weigh ed phonon densi y o s a es (nw-pDOS). F om
he powde spec a measu ed a 77 K, we can he e o e ob ain he nw-pDOS o
he sample. In igu e 5.13 we compa e he measu ed nw-pDOS o wo di e en
|
Q| alues. Fo bo h hese alues, he phonon densi y o s a es p esen he same
spec al de ails, con i ming ha he obse ed exci a ions a e pu ely phononics. The
measu ed nw-pDOS can hen be compa ed wi h he one calcula ed wi h pe iodic
DFT.
Figu e 5.13: Phonon densi y o s a es om no malized inelas ic neu on sca e ing
expe imen on [Dy(H2O)5(HMPA)2] powde s. The spec a was collec ed a 77 K o
wo |
Q| alues. The high and low esolu ion scans a e me ged in o de o maximize
he ex en o he high esolu ion egion.
5.2.2 Resul s
The pDOS as a unc ion o he applied p essu e can be calcula ed ab ini io wi h
pe iodic DFT. These calcula ions we e pe o med by ou collabo a o s a he Uni-
e si y o Pa ma, P. Bon `a and I. J. Onuo ah (see he Lis o Collabo a o s),
using he Quan umESPRESSO package [128,129,130] which uses a plane wa e
basis se and pseudopo en ials. The ene gy cu o o he wa e unc ions was se
o 1225 eV while he ecip ocal space in eg a ion was pe o med wi h a 2 ×2×1
Monkhos -Pack [131] g id when conside ing he p imi i e uni cell. The ONCV
no m conse ing pseudopo en ials [132] we e used o all elemen s and he exchange
and co ela ion con ibu ion was desc ibed wi h he Pe dew, Bu ke, and E nze ho
(PBE) unc ional.[123]
The a omic posi ions we e op imized un il a omic o ces we e smalle han 2.57×10−4
103 Simone Chicco
Chap e 5
eV˚
A−1and o al ene gy di e ences among op imiza ion s eps we e smalle han 2
meV while he expe imen al la ice pa ame e s we e kep ixed du ing he elax-
a ion. Phonon equencies we e ob ained wi h he Phonopy package.[133] In his
case a 2 ×2×1 supe cell is used o collec he in e -a omic o ce cons an s by pe -
o ming 0.011 ˚
A posi ion displacemen s. The ecip ocal space is sampled using only
he Γ poin in his la e case. The s a ing c ys allog aphic cell has been adap ed
o each p essu e condi ion and e i ied by compa ison wi h X- ay di ac ion da a
unde he same condi ions. The esul s o he simula ed pDOS ( ig. 5.14) con i ms
he conclusions de i ed om he expe imen al da a, hus ha he la ges di e ence
as a unc ion o he applied p essu e is obse ed o low ene gy phonon modes.
Figu e 5.14: (a) Simula ed pDOS a wo di e en applied p essu es (a mosphe ic
and 1 GPa). The pDOS is con olu ed wi h a sha p esolu ion σin o de o be e
e alua e he con ibu ion o di e en phonon b anches ha would o e lap wi h he
expe imen al one. (b) Zoom o he lowe ene gy egime, whe e he la ge p essu e-
induced di e ences a e obse ed.
In o de o alida e ou DFT esul s, we ha e compa ed he nw-pDOS measu ed
a a mosphe ic p essu e wi h he one calcula ed ab ini io ( ig. 5.15). The spec al
ea u es a e ep oduced by he calcula ion in bo h he low and high ene gy egime.
The only e iden di e ence esides in he high-ene gy expe imen al backg ound ha
esul s enhanced i compa ed o he simula ed pDOS. This e ec is no comple ely
unde s ood up o da e, bu can be a ibu ed o a non negligible cohe en con ibu-
ion o he sca e ed in ensi y, no included in he incohe en app oxima ion, and
o mul i-phonon con ibu ions a high ene gy, excluded by ocusing only on he i s
o de phonon expansion o he sca e ing unc ion (see sec . 2.2.2). Indeed, he
sha p aspec o all he o he spec al ea u es pe mi o exclude a p io i ha his
e ec is ela ed o he c ys al phonon s uc u e.
Once alida ed, he DFT-calcual ed phonon modes will be used o simula e he
elaxa ion dynamics and hus unde s and he e ec s o p essu e-induced phonons
modi ica ions in p omo ing di e en elaxa ion mechanisms.
104 Simone Chicco
Chap e 6
The echnique used o measu ing nuclea nT1is no a simple in e sion- eco e y.
Con e sely, o measu ing he eco e y o he ans e se nuclea magne iza ion we
add essed he sys em wi h a sa u a ing pulses ain, ollowed by a s anda d Hahn-
echo e ocusing sequence π/2−τ−π/2. Wi h his p ocedu e we a e con iden ha
he nuclea abso p ion line is ully i adia ed (maximum s a is ical empe a u e con-
igu a ion).
The sa u a ion eco e y pa h ollows a bi-exponen ial law I( ) = I1(1−exp(− /nT1,1))+
I2(1 −exp(− /nT1,2)), wi h a ime scales di e ence o se e al o de o magni ude;
hus, he as es elaxa ion we a e in e es ed in is easily dis inguishable ( ig. 6.6).
The ex ac ed spin-la ice elaxa ion a es 1/nT1 o he as es elaxing componen
a e shown in igu e 6.7 as a unc ion o empe a u e o each applied ield.
Figu e 6.6: (a) Fas es 1H-NMR no malized sa u a ion eco e y a 10 K o di e -
en applied s a ic ield B0. (b) No malized single exponen ial 1H-NMR echo decay
measu ed a 10 K o wo applied ields.
The 1/nT1 empe a u e dependence shows a peak a 11 K in 0.33 T applied ield ha ,
by inc easing ield, lowe s in in ensi y, b oadens and shi s a highe empe a u es.
A mo e de ailed analysis o his elaxa ion dynamics is ongoing and he i ing o he
da a wi h a de ailed heo e ical amewo k, will gi e us in o ma ion on he phonon
coupling s eng h and, on he numbe o domina ing elaxa ion a es/ equencies
and on hei empe a u e dependence.
In his analysis, an impo an e ec ha mus be aken in o accoun is he wipe
ou e ec .[141] I consis s in a g adual NMR signal loss a low empe a u e, due o
an inc emen in he elaxa ion a es nT−1
2o ce ain nuclei. The as e elaxa ion
causes he signal o exi om he de ec able window o ou expe imen al se up ( he
so-called ”dead- ime”, which is o he o de o ≈10 µs). Thus, o simula ing he
spin-la ice elaxa ion a e we will e ain only he 1Hnuclei wi h a ans e se e-
laxa ion a e nT−1
2smalle hen he iden i ied wipe ou h eshold. To e alua e his
e ec we measu ed he nuclea spin-spin elaxa ion a es 1/nT2o e a wide empe -
a u e ange, wi h a s anda d Hahn-echo e ocusing sequence. The elaxa ion a es
( ig. 6.6) a e i ed wi h a sligh ly s e ched mono-exponen ial unc ion (wi h al-
mos empe a u e-independend s e ching pa ame e ) and he alues ex ac ed a e
shown in igu e 6.8. The ex ac ed elaxa ion a es show a peak a 10 K in 0.33 T
applied ield ha b oadens, lowe s in in ensi y, and shi s a highe empe a u es by
111 Simone Chicco

Chap e 6
Figu e 6.7: P o on nuclea spin-la ice elaxa ion a es 1/nT1ex ac ed om i ing
he sa u a ion eco e y cu es in igu e 6.6 as a unc ion o he sys em empe a u e,
a h ee di e en applied s a ic ield B0.
Figu e 6.8: (a) P o on nuclea spin-spin elaxa ion a es 1/nT2a wo s a ic applied
ields, ex ac ed om i ing he echo ampli ude decay wi h a s e ched single ex-
ponen ial unc ion a ixed s e ching pa ame e β. (b) T ans e se magne iza ion
MH
x,y(0)T om he echo ampli ude ex apola ed a ze o delay in he echo decays.
Mul iplied by he empe a u e, i gi es a hin on he numbe o nuclei wi h a s ill
measu able decohe ence.
112 Simone Chicco
Chap e 6
inc easing he applied ield. A isualiza ion o he wipe ou e ec can be ob ained
by plo ing he s a ing ans e se magne iza ion alues MH
x,y(0) ex apola ed om
he τ= 0 poin in he echo decay cu es, mul iplied by he measu ing empe a u e
( ig. 6.8). This quan i y MH
x,y(0)Tco espond o he numbe o esona ing nuclei. In
igu e 6.8, a he applied ield o 0.33 T, he numbe o esona ing nuclei is sa u a ed
abo e 20 K, and dec eases ab up ly below 15 K, owa ds a low empe a u e ( ew
Kel in deg ees) cons an minimum alue.
6.3 Elec onic spin Manipula ions
To be e in es iga e he hype ine spli ing in he Cu gzcomponen , we milled
se e al polyc ys als (5 % dilu ed in he diamagne ic analogue {Ti7Ga}[Zn(h ac)2])
in o ine powde s and measu ed he echo de ec ed ield sweep in he Q-band EPR
spec ome e ( ig. 6.9). The signal o noise a io is ex emely high, hanks o a
ine uning o he esonan ca i y, and consequen ly all he spec al ea u es a e
well esol ed. Indeed, in his con igu a ion also he ou old Hype ine spli ing o
he gzcomponen is isible and enables he obse a ion o he u he spli ing ( i e
e iden peaks a e ma ked in ig. 6.9) induced by he Jcoupling.
Taking ad an ages o his ine uning o he expe imen al se up, we pe o med a
Rabi nu a ion expe imen in o de o demons a e he abili y o manipula e bo h
he ing and Cu elec onic spins cohe en ly and independen ly om each o he . The
sys em was a ge ed by a manipula ing pulse o a iable leng h θ( ) ollowed by a
Hahn- e ocusing sequence π/2−τ−π. The Rabi oscilla ions o he gx,y componen s
o he {C 7Ni} ing and [Cu(h ac)2] a e shown in igu e 6.10 o di e en powe a -
enua ion.
Figu e 6.9: Q-band EPR-EDFS a 3K on a 5% dilu ed ine powde o {C 7Ni}[Cu
(h ac)2]. Red and black icks ma k he in plane gx,y and axial gz ing and coppe
signal espec i ely.
113 Simone Chicco
Chap e 6
Figu e 6.10: Rabi nu a ion expe imen s a 3 K on ine powde o {C 7Ni}-
[Cu(h ac)2]. The ield was ixed o ma ch espec i ely he wo highe spec al com-
ponen s (a) B ing = 13.597 kG and (b) BCu = 11.704 kG (as ske ched in inse )
co esponding o he gx,y plane. The pulse powe -a enua ion is a ied o highligh
he Rabi equency eωRand damping λRdependencies. Fo each pulse-powe , he
pulse leng h in he Rabi-sequence was modi ied in o de o main ain he e ec i eness
o he π/2−π e ocusing.
The ob ained Rabi oscilla ions esul s cohe en and monoch oma ic. Rabi equency
eωRscales linea ly wi h he pulse powe a enua ion, oge he wi h he oscilla ion
damping, which is hampe ed in highe powe con igu a ions by he pulse inhomo-
genei ies.
6.4 Conclusions
The elec onic and nuclea spins elaxa ion dynamics o a {C 7Ni}[Cu(h ac)2] com-
plex has been ho oughly in es iga ed by exploi ing a combined pulsed EPR-NMR
app oach. The da a analysis and he heo e ical simula ions on he elaxa ion dy-
namics a e s ill ongoing. The heo e ical model desc ibing he sys em dynamics will
be con i med by i ing he elaxa ion a es o bo h he {C 7Ni}and he Cu elec-
onic spins and Cu nuclea one. In e p e ing hese elaxa ion measu emen s will
gi e undamen al insigh s o he de e mina ions o he leading elaxa ion p ocesses,
such as hei dependence on he empe a u e egime and he s eng h o spin-phonon
coupling.[138,139]
Pu ing oge he all his in o ma ion, we will be able o deduc he key ing edi-
en s go e ning he sys em elaxa ion o his p omising sup amolecula assembly.
Mo eo e , in iew o he ema kable esul s achie ed in he elec onic spin manip-
ula ions h ough he Rabi nu a ions expe imen s on powde samples, s udies on
single c ys als a e ongoing. By exploi ing a combined EPR-NMR app oach, we ex-
pec o ealize cohe en manipula ions o he qubi -qudi sys em, demons a ing he
possibili y o implemen o dina y single qubi quan um algo i hms and o ealize
e icien ly he in o ma ion swapping be ween he elec onic spin p ocesso and i s
nuclea memo y. [49]
114 Simone Chicco
Gene al Conclusions
In conclusion, in his hesis we ha e p esen ed he esul s o di e en expe imen-
al cha ac e iza ions o molecula nanomagne s. The spin Hamil onian and he
spin dynamics o hese sys ems ha e been in es iga ed by means o se e al di e en
cu ing-edge expe imen al echniques. We ha e demons a ed ha b oadband NMR
is an elec ion expe imen al ool o ou s udies, since i enabled a p ecise e alua ion
o he pa ame e s o he elec o-nuclea spin Hamil onian o wo V-based molecu-
la qubi s, oge he wi h hei nuclea phase memo y imes (sec . 3). P o en ha
hese sys ems possess he ene gy le els s uc u e and he cohe ence p e equisi es o
an ideal qudi , by means o NMR -pulses we ealized cohe en manipula ions o
hese elec o-nuclea sys ems, inducing selec i ely he desi ed ansi ions be ween
hei nuclea s a es. In he case o he [VO(TPP)] complex, we also succeeded in
simula ing he encoding o a Quan um e o co ec ion algo i hm and we add essed
expe imen ally he same ansi ions in ol ed in he encoding sequence, wi h eally
p omising esul s, such as as manipula ions, negligible cohe ence losses and encod-
ing eliabili y. By exploi ing his echnique and his, o simila complexes, we expec
o succeed in ealizing he i s s p oo -o -p inciples expe imen s on he implemen a-
ion o quan um e o co ec ion algo i hms in molecula qudi s. In pa icula , an
in e es ing pe spec i e-wo k would be he cha ac e iza ion o complexes om he
class o Lan hanide-based monome s, dime s o ime s wi h he NMR expe imen al
se up desc ibed abo e. [47,44,45] Indeed, also hese sys ems ep esen ideal es -
beds o he implemen a ion o quan um algo i hm on he expanded compu a ional
space ha esul s om he coupling o di e en magne ic ions and o he elec onic
and nuclea spins wi hin each ion.
The phonon-induced elaxa ion p ocesses play a c ucial ole in he spin dynamics o
molecula qudi s such as he [VO(TPP)] molecule. The phonon dispe sions o his
complex we e ho oughly in es iga ed by exploi ing, o he i s ime in a molecula
c ys al, inelas ic X- ay sca e ing (sec . 4). The echnique ea u es se e al ad an-
ages wi h espec o he pa en inelas ic neu on sca e ing and he ou s anding
quali y o he esul s achie ed will pa e he way o a sys ema ic usage o his
echnique in pa en compounds cha ac e iza ions. In pa icula , in he [VO(TPP)]
compound we obse ed ex emely low-ene gy op ical phonons and we alida ed he
pe iodic DFT phonon calcula ions by compa ing he measu ed IXS c oss sec ion
wi h he simula ed one. The alida ed phonon model was hen used o simula e he
spin elaxa ion o [VO(TPP)], iden i ying he c ucial ole o he low ene gy modes,
which esul o be c i ically coupled wi h he elec onic spins. The esul s achie ed
in his s udy will be c ucial o he de elopmen o new classes o molecula magne s
wi h op imized cohe ence by ailo ing o phonon modes.
In his hesis we also ocused on he s udy o single molecule magne s wi h high en-
e gy ba ie s. In pa icula , ou scien i ic e o was de o ed o he de e mina ion o
115
Chap e 6
he key ac o s go e ning he phonon-induced elaxa ion p ocesses. Focusing on Dy-
based SMMs, we s udied by inelas ic neu on sca e ing he e ec s on he phonon
densi y o s a es o chemical/s uc u al subs i u ions in he molecule o o an ex e -
nal applied p essu e (sec . 5). By exploi ing INS, we accessed he phonon densi y
o s a es and, in bo h sys ems, we demons a ed ha he main di e ences a e in he
ene gy ange o phonons in ol ed in Raman elaxa ion p ocesses. These elaxa ion
p ocesses a e indeed he c ucial ones o he de e mina ion o SMMs pe o mances.
The compa ison o he expe imen al phonon densi y o s a es wi h he ones simu-
la ed wi h pe iodic DFT was exploi ed o alida e he phonon models, o be used
o simula ing he spin dynamics o hese complexes. This app oach ep esen a
aluable es -bed o he phonon model alida ion, ha needs o be implemen ed
sys ema ically in u u e wo ks.
Finally, we p esen ed a combined EPR-NMR cha ac e iza ion o he spin dynamics
o a sup amolecula complex in which a {C 7Ni} ing qubi is linked o a Cu-based
nuclea qudi wi h long cohe ence imes (sec . 6). By exploi ing pulsed Q-band EPR
we measu ed he elec onic spin-la ice and spin-cohe ence elaxa ion imes, and we
ealized cohe en indi idual manipula ions o bo h he ing and he Cu elec onic
spins. Mo eo e , wi h 1H-NMR we accessed he p o on nuclea spin-la ice and
spin-spin a es. These esul s will be used o elabo a e a model o he elaxa ion
dynamics o he sys em.
To summa ize, in his hesis we exploi ed se e al ad anced expe imen al echniques
which a e essen ial in he s udy o MNMs and enabled o pinpoin he key in-
g edien s o he op imiza ion o hei pe o mances o applica ions in quan um
echnologies. The nex s ep o his pu pose is he ealiza ion o an expe imen al
se up o mul i- equency add essing o hese sys ems, capable o inducing se e al
cohe en ansi ions wi hin he mul ile el s uc u e o hese molecula qudi s in sho
imes. Such a pla o m would enable he implemen a ion o bo h single and mul i-
qubi p oo -o -p inciple quan um algo i hms. A he same ime, i is also pi o al o
discuss and design he i s single molecule expe imen s, which ep esen s he inal
challenge o he implemen a ion o molecula magne s in eal quan um in o ma ion
p ocessing de ices.
116 Simone Chicco

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Re e eed publica ions
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S. Chicco, A. Chiesa, G. Allodi, E. Ga la i, M. A zo i, L. So ace, R. de Renzi,
R. Sessoli, S. Ca e a, ”Con olled cohe en dynamics o [VO(TPP)], a p o-
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12, 12046, 2021.
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M. A zo i, E. Ga la i, G. Allodi, S. Chicco, A. Chiesa, A. Albino, R. de
Renzi, E. Sal ado i, M. Chiesa, S. Ca e a and L. So ace, ”Radio equency o
mic owa e Cohe en Manipula ion o an O ganome allic Elec onic spin Qubi
coupled o a Nuclea Qudi ”, Ino ganic Chemis y, 60, 11273-11286, 2021.
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L. T ombi, F. Cugini, R. Rosa, N. Sa zi Amad`e, S. Chicco, M. Solzi, P.
Ve onesi, ”Rapid mic owa e syn hesis o magne ocalo ic Ni–Mn–Sn Heusle
compounds”, Sc ip a Ma e ialia, 176, 63-66, 2020.
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F. Cugini, S. Chicco, F. O landi, G. Allodi, P. Bon `a, V. Vezzoni, O. Mi oshk-
ina, M. G une , L. Righi, S. Fabb ici, F. Albe ini, R. De Renzi, M. Solzi,
”E ec i e decoupling o e omagne ic subla ices by con olling magne ic in-
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P. Bon `a, S. Chicco, F. Cugini, S. Sha ma, J. K. Dewhu s and G. Allodi,
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S. Chicco, E. Ga la i, G. Allodi, A. Chiesa, M. A zo i, L. So ace, R. De Renzi,
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Publica ions abou o be published, a e mino e ision
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E. Ga la i, A. Albino, S. Chicco, V.H.A. Nguyen, F. San anni, L. Paolasini,
C. Mazzoli, R. Caciu o, F. To i, P. San ini, A. Lunghi, R. Sessoli and S.
Ca e a, ”The c i ical ole o ul a-low ene gy ib a ions in he elaxa ion
dynamics o molecula qubi s”, Na u e Communica ion, 2023.
129
Acknowledgmen s
To conclude his hesis, I would like o since ely hank all he scien is and collabo-
a o s ha made i possible. Fi s and o emos , I would like o hank my supe iso
P o . S e ano Ca e a, o gi ing me he oppo uni y o s a my esea ch ac i -
i y in his an as ic g oup, and o his ex ao dina y kindness. He in oduced me,
du ing my mas e deg ee, o he ascina ing ield o molecula magne ism and his
en husiasm has ne e ceased o inspi e me. E e y discussion we had o ad ice I
ecei ed was p iceless and I am ex emely g a e ul o e e y hing ha he augh
me. I would conside ed mysel sa is ied i I succeeded in lea ning e en a en h o
his boundless knowledge. I am hono ed and p oud o ha e had he oppo uni y o
wo k wi h him and I also hope ha hese h ee yea s we e only he beginning o a
long un.
I will be o e e hank ul o D . Elena Ga la i, who co-supe ised my PhD, o
he ex ao dina y e o she pu in helping me, wi h he kindness o a sis e and
an ex ao dina y compe ence. Wi hou he suppo and supe ision I would ha e
been los . I hope I ha e cap u ed he passion o esea ch and he expe ise. Many
special hanks o P o . Giuseppe Allodi, who supe ised me du ing he whole expe -
imen al NMR ac i i y. Du ing he ime we spen oge he in he labo a o y I ied
my bes o lea n as much as possible om his immense expe imen al a i ude and
compe ence. I would also like o hank D . Alessand o Chiesa o he collabo a ions
we had du ing hese h ee yea s and o being so en husias ic in eaching me. I is
also impo an o me o hank my colleague D . Emilio Macaluso o he p ecious
ime spen discussing all he i ial ques ions I was no able o answe mysel and I
was ashamed o ask o my supe iso s. All he people in he g oup ha e been, and
always will be, an example o me, and I am immensely g a e ul o each o hem.
I would like o hank ou collabo a o s a he Uni e si y o Manches e P o . Richa d
Winpenny and P o . E ic McInnes o gi ing me he oppo uni y o spend h ee
mon hs in hei an as ic labo a o ies, welcoming me in hei esea ch g oup. I am
e y g a e ul o D . Selena F.J. Lockye o he p ecious suppo du ing my s ay
in Manches e , i has been a g ea lea ning and wo king oppo uni y. I hope o
collabo a e again in he nea u u e. I would also like o hank all he ins umen
scien is s o he X- ays, neu ons and EPR acili ies whose help and assis ance was
p iceless: D . Luigi Paolasini, D . Claudio Mazzoli, D . Cai Yong, P o . Ta iana
Guidi, D . And ea Pio ano, D . Alexand e I ano , D . Monica Jimenez Ruiz, M .
Adam B ook ield. Thanks o all o hem I am e y p oud o he expe imen al esul s
achie ed in his hesis. Las bu no he leas , I would like o wa mly hank my high
school Physics eache Gianni Melega i, who plan ed he seed many yea s ago and
ook ca e o i in he ea ly s ages. Finally I would like o hank all he collabo a o s
ci ed a he beginning o his hesis.
Simone Chicco, Janua y 2023
130
FUNDING PROJECTS
The wo k p esen ed in his hesis has ecei ed unding om he Eu opean Union’s
Ho izon 2020 Resea ch and Inno a ion P og amme, FET-OPEN p ojec FATMOLS
(”FAul Tole an MOLecula Spin p ocesso ”) unde g an ag eemen No. 862893,
including a h ee mon hs s ay a he Uni e si y o Manches e o he esea ch
p ojec : “Elec on Spin Resonance measu emen s on molecula complexes o Quan-
um In o ma ion P ocessing”. O he p ojec s in ol ed in his wo k we e he Eu o-
pean P ojec “Scaling Up quan um compu a ion wi h MOlecula spins” (SUMO)
o he call Quan ERA, and he PRIN P ojec 2017CR5WCH Q-chiSS “Quan um
de ec ion o chi al induced spin selec i i y a he molecula le el” o he I alian
Minis y o Uni e si y and Resea ch (MUR).