Quantum structural fluxion in superconducting lanthanum polyhydride

A icle h ps://doi.o g/10.1038/s41467-023-37295-1
Quan um s uc u al fluxion in
supe conduc ing lan hanum polyhyd ide
Hui Wang
1,2
,PascalT.Salzb enne
3
, Ion E ea
4,5,6
, Feng Peng
7
,
Ziheng Lu
3
,HanyuLiu
2,8
, Li Zhu
9
,Ch isJ.Picka d
3,10
&YansunYao
11
The disco e y o 250-kel in supe conduc ing lan hanum polyhyd ide unde
high p essu e ma ked a significan ad ance owa d he ealiza ion o a oom‐
empe a u e supe conduc o . X- ay di ac ion (XRD) s udies e eal a non-
s oichiome ic LaH
9.6
o LaH
10±δ
polyhyd ide esponsible o he supe -
conduc i i y, which in he li e a u e is commonly ea ed as LaH
10
wi hou
accoun ing o s oichiome ic de ec s. He e, we disco e significan nuclea
quan um e ec s (NQE) in his polyhyd ide, and demons a e ha a mino
amoun o s oichiome ic de ec s will cause quan um p o on di usion in he
o he wise igid lan hanum la ice in he g ound s a e. The di usion coe ficien
eaches ~10−7cm2/s in LaH
9.63
a 150 gigapascals and 240 kel in, app oaching
he uppe bound alue o in e s i ial hyd ides a compa able empe a u es. A
puzzling phenomenon obse ed in p e ious expe imen s, he posi i e p es-
su e dependence o he supe conduc ing c i ical empe a u e T
c
below 150
gigapascals, is explained by a modula ion o he elec onic s uc u e due o a
p ema u edis o iono hehyd ogen la ice in his quan um fluxional s uc-
u e upon decomp ession, and esul ing changes o he elec on-phonon
coupling. This finding sugges s he coexis ence o he quan um p o on fluxion
and hyd ogen-induced supe conduc i i y in his lan hanum polyhyd ide, and
leads o an unde s anding o he s uc u al na u e and supe conduc i i y o
nons oichiomec ic hyd ogen- ich ma e ials.
Supe conduc i i y a nea oom empe a u e has been disco e ed in
cla h a e polyhyd ides a megaba p essu es1–7. De e mining he
c ys al s uc u e esponsible o he supe conduc i i y is o c i ical
impo ance, ye a g ea challenge8,9. Expe imen ally, di ficul ies in
p obing ligh e elemen s a ise in XRD, impeding a di ec
de e mina ion o he comple e la ice symme y. The measu ed T
c
and
i s p essu e dependence (dT
c
/dp) igh en cons ain s on s uc u al
models; howe e , calcula ions on he candida e s uc u es based on
Ba deen–Coope –Sch ie e (BCS) heo y10 unequi ocally sugges a
nega i e dT
c
/dp11–20, which con adic s he expe imen ally obse ed
Recei ed: 29 No embe 2022
Accep ed: 9 Ma ch 2023
Check o upda es
1
Key Labo a o y o Pho onic and Elec onic Bandgap Ma e ials (Minis y o Educa ion), School o Physics and Elec onic Enginee ing, Ha bin No mal
Uni e si y, 150025 Ha bin, China.
2
In e na ional Cen e o Compu a ional Me hod & So wa e, College o Physics, Jilin Uni e si y, 130012 Changchun, China.
3
Depa men o Ma e ials Science & Me allu gy, Uni e si y o Camb idge, 27 Cha les Babbage Road, Camb idge CB3 0FS, UK.
4
Fisika Aplika ua Saila,
Gipuzkoako Ingenia i za Eskola, Uni e si y o he Basque Coun y (UPV/EHU), Eu opa Plaza 1, 20018 Donos ia/San Sebas ián, Spain.
5
Cen o de Física de
Ma e iales (CSIC-UPV/EHU), Manuel de La dizabal Pasealekua 5, 20018 Donos ia/San Sebas ián, Spain.
6
Donos ia In e na ional Physics Cen e (DIPC), Manuel
de La dizabal Pasealekua 4, 20018 Donos ia/San Sebas ián, Spain.
7
College o Physics and Elec onic In o ma ion, Luoyang No mal Uni e si y, 471022
Luoyang, P. R. China.
8
S a e Key Labo a o y o Supe ha d Ma e ials and In e na ional Cen e o Fu u e Science, Jilin Uni e si y, 130012 Changchun, China.
9
Depa men o Physics, Ru ge s Uni e si y, Newa k, NJ 07102, USA.
10
Ad anced Ins i u e o Ma e ials Resea ch, Tohoku Uni e si y 2-1-1 Ka ahi a, Aoba,
Sendai 980-8577, Japan.
11
Depa men o Physics and Enginee ing Physics, Uni e si y o Saska chewan, Saska oon, Saska chewan S7N 5E2, Canada.
e-mail: wh@ ysik.cn
Na u e Communica ions | (2023) 14:1674 1
1234567890():,;
1234567890():,;
posi i e dT
c
/dpin some p essu e egimes (Supplemen a y Fig. 1). This
phenomenon was ini ially obse ed in he 250-kel in supe conduc ing
lan hanum polyhyd ide1, and la e in polyhyd ides o y ium6,7and
calcium21 wi h T
c
eaching 257 K and 212 K, espec i ely. This ‘posi i e
dT
c
/dpcon adic ion’p esen s a majo obs acle o ou ull unde -
s anding o he c ys al s uc u es o hese supe conduc ing
polyhyd ides.
As he fi s supe conduc o wi h a T
c
abo e 250K, lan hanum
polyhyd ide has been s udied by se e al g oups independen ly1–5.In
addi ion o he high T
c
, a posi i e dT
c
/dpwas obse ed be ween 137
and 150GPa by measu emen s on ou samples (samples 1–4inRe .1)
syn hesized a di e en p essu es wi h lase hea ing. XRD measu e-
men sde e mine ha he lan hanuma oms o ma ace-cen e ed cubic
( cc) la ice a p essu es o 137–218 GPa, while he hyd ogen a oms
ha eunde e minedloca ionswi hin hela ice.Basedon hemeasu ed
c ys al olume, he hyd ogen- o-lan hanum (H/La) a io a ound he
maximum T
c
was es ima ed o be 9.6 (150GPa)1o 9–11
(180–200 GPa)2,3in he wo s udies, espec i ely. This end was
ecen ly confi med by new expe imen s on he cc →C2/mphase
ans o ma ion a p
c
= 135 GPa, which shows an e en s eepe dec ease
o T
c
below p
c
5.
Wi h a mode a e syn he ic p essu e and high symme y, ‘ cc’
lan hanum polyhyd ide p o ides a model sys em o explo ing he
s uc u al na u e o high-T
c
hyd ides in bo h expe imen and heo y. In
ac , ab ini io calcula ions ha e guided he expe imen al disco e y o
lan hanum polyhyd ides, and p edic ed he appea ance o high T
c
supe conduc i i y, i.e., an es ima ed T
c
o 280-kel in o cc-LaH
10
a
210 GPa11,12. This p edic ion u ned ou o be e y close o he cc
lan hanum polyhyd ide la e syn hesized, wi h some di e ences in T
c
,
syn he ic p essu e, and hyd ogen con en . Recen ly, i has been he-
o ized ha he inclusion o quan um a omic fluc ua ion is essen ial o
a co ec calcula ion o T
c
and he p essu e bounda y o cc-LaH
10
,
which is dic a ed by he quan um na u e o lan hanum polyhyd ide
s uc u es16.Howe e ,anega i edT
c
/dpp edic ed a 137–150 GPa16
emains in con adic ion o he expe imen al obse a ions (Supple-
men a y Fig. 1). This disag eemen sugges s ha c i ical ac o s ha e
been o e looked in p e ious calcula ions o supe conduc i i y. In
pa icula , he s oichiome ic de ec obse ed by expe imen al s udies
may play an impo an ole in de e mining he s uc u e and unde -
lying supe conduc i i y.
Resul s and discussion
Al hough mino in amoun , de ec s in solid ma e ials can s ongly
a ec hei p ope ies. The he modynamic s abili y o a de ec in
high-p essu e solids as well as he ela i e s abili ies o di e en
de ec s uc u es can be e alua ed by he de ec o ma ion en halpy
(H ). As shown in Fig. 1a, he c ys al s uc u e o cc-LaH
10
is ep e-
sen ed as he inse ion o an ‘Hcube’in o oc ahed al in e s ices o
he fluo i e-LaH
2
s uc u e. In his model, we calcula ed he H o a
acancy de ec ei he a a co ne o he H cube (V
C
) o a a e ahed al
in e s ice o he La la ice (V
T
), and ound ha he configu a ionally
a e aged H becomes nega i e below 158 GPa. This sugges s ha cc-
LaH
10
is p one o acancy de ec s, which ag ees well wi h he
expe imen al finding o hypos oichiome ic LaH
10-δ
a 150 GPa spe-
cifically LaH
9.6
1. No ably, he calcula ed ‘ acancy occu ing egion’
coincides he egion whe e he T
c
o he cc lan hanum polyhyd ide
depends posi i ely on he p essu e (which p e ious calcula ions ha e
ailed o p edic ), indica ing ha he acancy s uc u e has a key ole
o play in he supe conduc i i y.
A low concen a ion o acancies in hyd ogen si es does no
change he XRD pa e n o he cc lan hanum polyhyd ide, which is
de e mined p ima ily by he La subla ice. Howe e , hese acancies
significan ly a ec he dynamics o hyd ogen in he c ys al (Supple-
men a y Fig. 2). The dis o ion in he hyd ogen subla ice sp eads ou
away om he acancy si es in bo h he V
C
and V
T
models o LaH
10
wi h
an H/La a io o 9.97, and e en esul s in a ‘liquid-like’H amewo kin
135 140 145 150 155 160 165
-60
-50
-40
-30
-20
-10
0
10
20
135 150 165 180
31
32
33
34
35 b
VT
VC
A e age
H (meV)
P essu e (GPa)
158 GPa
a
Exp . D ozdo
LaH9.6
LaD10
Cal. This wo k
LaH10
LaH9
LaD10
VCMD
Volume ( / .u.)
P essu e (GPa)
VT
VC
H
La
Å3
Fig. 1 | Vacancy o ma ion en halpy and p essu e- olume ela ion. a The o -
ma ion en halpy (H
) o a single acancy in cc-LaH
10
a wo inequi alen la ice si es
in he cla h a e hyd ogen amewo k, V
T
and V
C
,and hei configu a ional a e age
calcula ed a di e en p essu es using 2 × 2 × 2 ex ension o con en ional uni cell
o cc-LaH
10
. The loca ions o V
T
and V
C
a e illus a ed in a con en ional uni cell o
he cc-LaH
10
s uc u e by ed and black balls, espec i ely. bThe expe imen al
p essu e- olume ela ions o cc-LaH
9.6
and cc-LaD
10
measu ed by D ozdo e al.1
compa ed o he heo e ical alues o cc-LaH
10
, cc-LaH
9
and cc-LaD
10
de i ed
om quan um simula ions a cons an p essu e and empe a u e o 300K. The
olumes selec ed o subsequen quan um simula ion (V
CMD
) o cc-LaH
9.63
a e
he eby linked o he ‘quan um’p essu e. Fo mula uni is abb e ia ed as .u.
A icle h ps://doi.o g/10.1038/s41467-023-37295-1
Na u e Communica ions | (2023) 14:1674 2
he o me a 150GPa. Fo LaH
10-δ
, he p esence o acancies changes
he po en ial ene gy su ace, while quan um nuclea fluc ua ion is
expec ed o a ec he anspo beha io o acancies. The calcula ed
ze o-poin ene gy (ZPE) o ~150 meV/LaH
10
is compa able o he
acancy mig a ion ene gy o 160/200 meV o he mig a ion pa h o
V
C
→V
C’
/V
C
→V
T
a 150GPa. This sugges s ha nuclea ze o-poin
mo ion can p omo e p o on hopping o neighbo ing acancies and
cause he LaH
10-δ
g ound s a e o no be a single s uc u e a ound
which he a oms ib a e, bu ins ead one whe e he p o ons dynami-
cally explo e di e en acancies in a fixed cc-La amewo k.
We e e o his beha io as being ‘fluxional’in he sense used by
Goncha o e al. o phase IV o solid hyd ogen22. I is well-es ablished
by expe imen and heo y ha he mal e ec s domina e he fluxion-
ali y in his s uc u e: A classically o de ed phase III eplaces he flux-
ional phase IV a low empe a u e23–25. He e, we demons a e ha , in
he p esence o acancies, hyd ogen NQEs a e s ong enough—and
indeed essen ial— o s abilize a fluxional amewo k in he cc lan ha-
num polyhyd ide. The e o e, we a e dealing wi h a quan um fluxional
s uc u e (QFS).
Taking LaH
9.63
as an example, we in es iga e he NQE and he mal
e ec s on he s uc u alfluxion a 150 GPa and low empe a u es up o
240 K. The s oichiome y is selec ed acco ding o he expe imen al
es ima ion (LaH
9.6
). Ou heo e ical simula ions sugges a H/La a io o
9.54 (o 9.71) o he expe imen al samples syn hesized a 150 GPa (o
137 GPa), in good ag eemen wi h expe imen s1(Fig. 1b). We adop ab
ini io cen oid molecula dynamics (CMD)26 o ea he nuclei quan-
um mechanically. As shown by he mean squa e displacemen (MSD)
cu es in Fig. 2a, he simula ions e eal app eciable p o on mig a ion,
wi h equi alen mean mig a ion dis ances o acancies eaching 1.0 o
2.0 Å be ween 60 and 240 K (Supplemen a y Fig. 3). These dis ances
a ealmos asla geaso la ge han he sho es H-Hsepa a ion a ound
1.2Å.Thisisconsis en wi h he‘ne wo k’-like densi y dis ibu ion
pa e ns ob ained, o ins ance, a 60 K (Fig. 2b). In con as , he di -
usi i y is much smalle in ab ini io molecula dynamics (MD), whe e
he nuclei a e ea ed classically (Fig. 2a, c). This confi ms ha he
s uc u al fluxion in LaH
9.63
is domina ed by he NQE, a he han
he mal e ec s. The c ucial ole o acancies in his in iguing
quan um phenomenon is demons a ed by a compa ison o he
dynamics o s oichiome ic cc-LaH
10
a he same p essu e, which
exhibi s no di usion below 700 K27. Expe imen ally, he occu ence o
‘quenched-in’ acancies in polyhyd ides is ine i able, as a esul o he
he mal ea men o he samples. This finding he e o e sugges s ha
acancy e ec s in e ac wi h he s ong NQE o esul in s ong
quan um s uc u al fluxion in polyhyd ides.
Weapp oxima e he p o on di usion coe ficien DinLaH
9.63
om
he slope o he MSD in simula ions a 240K and be ween 137 o
176 GPa. The empe a u e is chosen o be close o he measu ed T
c
o
he cc lan hanum polyhyd ide. As shown in Fig. 3a, he D alue alls in
he ange o 10−6 o 10−7cm2/s in ou quan um simula ion, whe eas i is
significan lylowe in he classical simula ion. The D alue is 2–3o de s
o magni ude below he h eshold c i e ion o classical supe ionici y
(~10−4cm2/s) o eely di using p o ons28. This no wi hs anding, i
eaches o indeed exceeds he uppe bound di usi i ies obse ed in
in e s i ial hyd ides a oom empe a u e, such as 3.8 × 10−7cm2/s in
cc-Cu
2
H29, and app oaches ha o phase IV hyd ogen30.Wi h
inc easing p essu e, Ddec eases in in e s i ial hyd ides (e.g. Cu
2
H29
and FeH31), due o he con ac ion o he me alla ice and he esul ing
inc ease o he ac i a ion ene gy o p o on hopping. In phase IV
hyd ogen, on he o he hand, Dinc eases wi h p essu e30, p obably
owing o he d as ic inc ease o he p o on hopping a e wi h he
sho ened o he dis ance. Ou p elimina y esul s sugges ha he
unique ‘hos -gues ’s uc u e o LaH
9.63
likely induces a significan
compe i ion be ween he wo mechanisms men ioned, esul ing in a
nonmono onic p essu e end o he coe ficien D.
In LaH
9.63
, p o on di usion b eaks he balance o in e nal Coulomb
epulsions in he ‘Hcube’loca ed in oc ahed al in e s ices o La sub-
la ice, which he e o e dis o s he H subla ice. La ice expansion
p omo es he dis o ion, as indica ed by he heigh ening o he second
coo dina ion shell (peaked a ~1.85 Å) in he a e age adial dis ibu ion
unc ion [RDF, g( ), Supplemen a y Fig. 4], which is co ela ed o a
smea ing o he in e - and inne -cube H-H sepa a ions. Based on a
configu a ional dis ance (deno ed as ξ)o c ys alfinge p in ma ices32,
which is sensi i e o local s uc u al changes, we pa ame ized he
s uc u al di e ence o he QFS ela i e o he c ys al la ices o s a ic
cc and C2/mphases, and a iclinic P1 s uc u e mimicking he QFS, as
showninFig.3bbyξ
q
,ξ
qc
and ξ
qp
espec i ely. Fo he H subs uc u e,
we find ξ
q
>ξ
qc
>ξ
qp
,wi h heξ
q
much beyond he o he wo a la ge
olumes, e ealing an inc easing dis o ion o he H subla ice in QFS
om cc o lowe symme y upon decomp ession. Fo he La sub-
s uc u e, ξ
qc
>ξ
q
≈ξ
qp
,wi hξ
qc
much la ge han ξ
q
and ξ
qp
.This
ag ees wi h he XRD measu emen s ha de e mine an cc subla ice o
La a oms abo e 137 GPa1. The iny dis o ions ela ed o ξ
q
and ξ
qp
a e
bo h wi hin he unce ain y o efinemen s o he cc phase in XRD
s udies (Supplemen a y Fig. 4). The esul s sugges a p ema u e low-
symme y dis o ion o he H subs uc u e upon olume expansion
ela i e o he La subs uc u e in he quan um fluxional LaH
9.63
.
Using he cen oid configu a ions o CMD simula ions a 240 K, i.e.
he QFS, we calcula ed he elec onic densi y o s a es a he Fe mi le el
NϵF

in LaH
9.63
, which exhibi s opposi e p essu e ends a p essu e
below and abo e 150 GPa (Fig. 4a). The cc-LaH
10
(quan um) c ys al has a
mono onously nega i e p essu e dependence o NϵF

abo e 100
GPa12,16, and his p edic ion can be ex ended o LaH
9.63
using he igid-
band model o he elec onic s uc u e by i ue o an a ificial shi o
ϵF33. The p essu e dependences o NϵF

in he C2/mphase o LaH
10
and
LaH
9.63
(wi hin he igid-band app oxima ion) a e bo h non-mono onic,
simila o quan um fluxional LaH
9.63
, sugges ing ha p essu e e ec s on
dNϵF

/dpa emuchmo esignifican in dis o ed s uc u es compa ed
o he high-symme y cc phase. The non-mono onic p essu e end o
NϵF

in LaH
9.63
is a s a is ical consequence o he QFS ha ing an
a e age cc-La subla ice wi h di e sely dis o ed H subs uc u es. This
end canno be a ibu ed o a single classical configu a ion, bu can
oughly be ep oduced by he P1 s uc u e mimicking he QFS
01234
0.00
0.04
0.08
0.12
0.16
0.20
0.24
Classical Quan um
Nuclea densi y dis ibu ions
Quan um:
240 K
120 K
60 K
Classical:
240 K
120 K
60 K
MSD ( )
Time (ps)
Å2
ab
c
Fig. 2 | S uc u al fluxion in LaH
9.63
a 150 GPa. a The p o on MSD de i ed om
cen oid ajec o ies o he CMD simula ions and hose o MD simula ions. bThe
[100] iew o he quan um nuclea densi y dis ibu ion a 60 K ex ac ed om a
CMD simula ion, wi h ull conside a ion o he 16 beads. Neighbo ing p o ons a e
illus a ed in di e en colo s. c he [100] iew o he classical nuclea densi y dis-
ibu ion in 16 MD simula ion uns o 4 ps dis inguished by a ious p o on colo s.
The 16 uns we e ini ialized om di e en cen oid configu a ions o he
4-picosecond CMD ajec o y wi h a sampling in e al o 0.25 ps.
A icle h ps://doi.o g/10.1038/s41467-023-37295-1
Na u e Communica ions | (2023) 14:1674 3
(Supplemen a y Fig. 5), which implies ha he p ema u e dis o ion o
he H subs uc u e (e.g. cc →P1 symme y) accoun s o he sign change
o dNϵF

/dpin he quan um fluxional LaH
9.63
.
The analy ic McMillan34 and Allen-Dynes (AD)35 o mulas (see de ails
in Supplemen al In o ma ion) ha e played an impo an ole in analyz-
ing he mechanism o p essu e-dependen supe conduc i i y in high-T
c
hyd ides. In hese o mulas, he elec on-phonon coupling (EPC) con-
s an λdepends explici ly on cha ac e is ic pa ame e s o bo h elec ons
and phonons, in addi ion o he a e age o he elec on-phonon ma ix
elemen s, hI2i.Specifically, λcan be exp essed as he p oduc o hI2i=M
(deno ed as β,wi hMbeinga omicmass)andNϵF

=
ω2
2

(deno ed as
ζ,wi h
ω2being he second equency momen o he Eliashbe g unc-
ion). ζdi ec ly desc ibes he compe i ion be ween elec on and phonon
con ibu ions. Analysis o li e a u e da a sugges ha ζdec eases mo e
han wo imes as e han βinc eases upon comp ession (Supplemen-
a y Table 1). Fu he mo e, i was ound ha dλ/dpplays a domina ing
ole in de e mining he dT
c
/dp,despi e he ac ha heT
c
also depends
explici ly on ωlog and 
ω2/ωlog (wi h ωlog being he loga i hmic equency
momen o he Eliashbe g unc ion). These findings sugges wo ules
o single c ys al phases o high-T
c
hyd ides: (I) phonons a ec dT
c
/dp
mainly h ough he pa ame e λ; and (II) dλ/dpis p ima ily de e mined
by dζ/dp, a he handβ/dp.
In iew o ζbeing p opo ional o NϵF

, hep essu e endo
NϵF

in LaH
9.63
(Fig. 4a) sugges s a sign change o dT
c
/dpa s udied
p essu es. Since he quan um fluxional na u e o LaH
9.63
o bids a
di ec calcula ion o he Eliashbe g unc ion αωðÞ
2FωðÞby densi y
unc ional me hods, a quan i a i e confi ma ion o his is unachie able
ye . Howe e , conside ing ha he change is so significan ha e en
quali a i e es ima ions could p o ide insigh s, we s udy he sign o
dT
c
/dpslope in LaH
9.63
unde wo app oxima ions, as he fi s s ep o
access he p oblem: (I) calcula ing ωlog and 
ω2based on he phonon
spec um FωðÞo LaH
9.63
, combined wi h αωðÞ
2app oxima ed by ha
o quan um cc-LaH
10
; (II) app oxima ing he βin LaH
9.63
by ha o he
quan um cc-LaH
10
. The a ionale o such app oxima ion is based on
empi ical ules, and he ac ha quan um fluxional LaH
9.63
la gely
e ains he same local a omic en i onmen as quan um cc-LaH
10
(Supplemen a y Fig. 6). The FωðÞde i ed om he Fou ie ans o m
o he eloci y au oco ela ion unc ions in he CMD simula ions a
240 K, as well as he app oxima ed αωðÞ
2and αωðÞ
2FωðÞa e shown in
Supplemen a y Fig. 7.
Wi h hisapp oach,wee alua ed hep essu e endo T
c
in LaH
9.63
using he AD o mula (wi h pa ame e s lis ed in Supplemen a y
Table 2). I does indeed exhibi a posi i e dT
c
/dpslope a 137–163 GPa,
in quali a i e ag eemen wi h expe imen al obse a ion1(Fig. 4b).
Mo eo e , a nega i e dT
c
/dpslope is ob ained a highe p essu es in line
wi h bo h expe imen al obse a ions and heo e ical esul s o cc-
LaH
10
(Supplemen a y Fig. 1). The s anda d de ia ion o NϵF

(Sup-
plemen a y Fig. 5) e eals fluc ua ions o he elec on s uc u e nea he
Fe mi le el, which may a ec he T
c
h ough pa ame e s ζand λ.
Howe e , conside ing ha supe conduc i i y o LaH
9.6
is measu ed a a
ime scale a beyond ha o he simula ion, i is easible o calcula e he
T
c
by a a e aged NϵF

. Recen ly, he C2/mphase has been success ully
p epa ed a 120 GPa by ab up decomp ession o he cc phase5.A
posi i e dT
c
/dpslope measu ed in he subsequen comp ession was
a ibu ed o a boos in he T
c
due o phonon so ening. As illus a ed in
he li e a u e12,16, he so ening o phonons esul s in a nega i e dT
c
/dp
slope o cc-LaH
10
, in con as o he expe imen al obse a ions below
150 GPa1. Howe e , he concep o ‘ahighe T
c
nea s uc u al
ins abili y’5ag ees well wi h ou finding o a p ema u e dis o ion o
hyd ogen subs uc u e ela i e o an a e age cc-La subs uc u e in
quan um fluxional LaH
9.63
, upon decomp ession o 137 GPa.
In summa y, he p esen wo k illus a es in lan hanum polyhyd ide
he coexis ence o quan um p o on fluxion and hyd ogen-induced high-
T
c
supe conduc i i y. A p ema u e dis o ion o hyd ogen subs uc u e
upon olume expansion and i s impac on he supe conduc i i y
h ough modula ing he elec onic s uc u e is e ealed. The findings
34.2 33.3 32.4 31.5
10-9
10-8
10-7
10-6
10-5
34.2 33.3 32.4 31.5
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
Hyd ogen phase IV, 250 GPa
Hyd ogen phase IV, 350 GPa
240 K un1
240 K un2
120 K
60 K
240 K
FeH, 202 GPa
Cu2H, 96 GPa
FeH, 33 GPa
D (cm2/s)
Volume ( )
Quan um:
Classical:
Cu2H, 15 GPa
LaH9.63
137 143 150 163 176
P essu e (GPa) P essu e (GPa)
XRD
H | La
|QF: QFS s. FCC
|QC: QFS s. C2/m
|QP: QFS s. P1
(a b. uni s)
Volume ( )
Decomp ession
XRD
137 143 150 163 176
Å3Å3
ξ
ba
Fig. 3 | P essu e e ec s on he di usi i y and s uc u al dis o ions. a The
p o on di usion coe ficien Dde i ed om he MSD in CMD simula ions ( un1) o
4 ps, and om he a e age MSD o ou MD simula ions o 24 ps. The Din h ee
addi ional CMD simula ions o 4 ps a 150GPa a e also shown. These a e wo
simula ions a 120K and 60 K, and a simula ion using highe o al-ene gy con-
e gence c i e ia ( un2). The MSD in he CMD simula ions is de i ed om he
cen oid ajec o ies. The measu ed Din Cu
2
H29 and FeH31, and calcula ed Din
phase IV hyd ogen30 a e shown o compa ison. bThe configu a ional dis ance ξ
(wi h e o ba indica ing he s anda d de ia ion) o QFS om he c ys alla ices o
s a ic cc and C2mphases, and a iclinic P1 s uc u e o H and La subs uc u e. The
P1 s uc u e is buil by scaling he la ice pa ame e a o a QFS sampled om he
CMD simula ion o LaH
9.63
a 176 GPa.
A icle h ps://doi.o g/10.1038/s41467-023-37295-1
Na u e Communica ions | (2023) 14:1674 4
suppo he expe imen al a gumen conce ning he hypos oichiome ic
na u e o he supe conduc ing samples, e eal he c ucial ole o
acancy de ec s in he s uc u e, and p o ide a heo e ical explana ion
o he posi i e dT
c
/dpslope o lan hanum polyhyd ide, wi h implica-
ions o unde s anding he s uc u e and supe conduc i i y o o he
hyd ogen- ich ma e ials. Wi h significan p og ess o nuclea magne ic
esonance echniques in diamond an il cells, he measu emen o p o-
on mobili y in me al hyd ides is cu en ly accessible a megaba
p essu es29,31.Wean icipa e ha con inuous expe imen al and heo e-
ical s udies o quan um fluxional s uc u e o high-T
c
hyd ides will s i-
mula e new heo e ical ools o unde s anding he supe conduc ing
beha io o quan um fluxional ma e ials ha canno be p ecisely
desc ibed by a single unde lying s a ic s uc u e.
Me hods
The ab ini io calcula ions we e pe o med using he plane-wa e
pseudopo en ial me hod, as implemen ed in he Vienna ab ini io
simula ion p og am (VASP)36,wi h hePe dew–Bu ke–E nze ho (PBE)
Gene alized G adien App oxima ion (GGA) densi y unc ional37,and
he ba e ion Coulomb po en ial ea ed in he p ojec o augmen ed
wa e (PAW) amewo k38. The acancy o ma ion en halpy o cc-LaH
10
wascalcula ed om acancy s uc u es o cc-LaH
10
and B
2
/n-H
2
39,wi h
en halpies calcula ed using he hi d-o de Bi ch–Mu naghan iso-
he mal equa ion o s a e (EOS)40 om ab ini io ene gy- olume ela-
ions, asimplemen edin he EOS code41. The acancymig a ionene gy
wascalcula ed by he climbing image nudged elas ic band me hod (CI-
NEB)42. The quan um nuclea dynamics we e s udied by pa h-in eg al
molecula dynamics (PIMD) and cen oid molecula dynamics
(CMD)30, as implemen in he PIMD code43. The supe conduc i i y was
analyzed using McMillan’s heo y
34 wi h he T
c
es ima ed by he Allen-
Dynes equa ions35. The c ys al s uc u es and MD ajec o ies we e
isualized by he VESTA44 and OVITO so wa es45 espec i ely. Mo e
de ails a e shown in he Supplemen a y In o ma ion.
Da a a ailabili y
All da a a e a ailable in he pape and om he au ho upon eques .
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p essu e end o he a e aged elec onic densi y o s a es a he Fe mi le el NϵF

in quan um fluxional LaH
9.63
(wi h dis ibu ions and s anda d de ia ion o s a ics
shown in Supplemen a y Fig. 5), and he p essu e end o NϵF

in s a ic P1-LaH
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,
cc-LaH
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c
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smea ings (σ) o he phonon spec um FωðÞ, oge he wi h he alues measu ed o
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9.6
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A icle h ps://doi.o g/10.1038/s41467-023-37295-1
Na u e Communica ions | (2023) 14:1674 5

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Acknowledgemen s
H.W. is hank ul o Y. Ma and M. Shiga o aluable discussions, o V.
Minko and M. E eme s o he XRD da a, and o C.Wang o he EPC
da a. The p ojec is suppo ed by he Na ional Na u al Science Foun-
da ion o China (G an No. 11974135, 11874176, 12174170, and 12074138),
he Na u al Sciences and Enginee ing Resea ch Council o Canada, he
EPSRC h ough g an s EP/P022596/1, and EP/S021981/1, and he s a up
unds o he o fice o he Dean o SASN o Ru ge s Uni e si y-Newa k. P.
T. S. hanks he Depa men o Ma e ials Science and Me allu gy a he
Uni e si y o Camb idge o gene ous unding. The wo k o P. T. S. is
u he suppo ed h ough a T ini y Hall esea ch s uden ship. I. E.
acknowledges financial suppo by he Eu opean Resea ch Council
(ERC) unde he Eu opean Union’s Ho izon 2020 esea ch and inno a ion
p og am (g an ag eemen no. 802533). We used he compu ing acil-
i ies a Beijing Supe Cloud Compu ing Cen e .
Au ho con ibu ions
H.W., P.S., I.E., F.P., Z.L., H.L., and L.Z. pe o med he calcula ions. H.W.,
Y.Y., P.S., and C.P. w o e he manusc ip , wi h inpu om all co-au ho s.
Compe ing in e es s
The au ho s decla e no compe ing in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains
supplemen a y ma e ial a ailable a
h ps://doi.o g/10.1038/s41467-023-37295-1.
Co espondence and eques s o ma e ials should be add essed o Hui
Wang.
Pee e iew in o ma ion Na u e Communica ions hanks he anon-
ymous e iewe s o hei con ibu ion o he pee e iew o his
wo k. Pee e iewe epo s a e a ailable.
Rep in s and pe missions in o ma ion is a ailable a
h p://www.na u e.com/ ep in s
Publishe ’s no e Sp inge Na u e emains neu al wi h ega d o ju -
isdic ional claims in published maps and ins i u ional a filia ions.
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Na u e Communica ions | (2023) 14:1674 7
Supplemen a y In o ma ion o
“Quan um s uc u al luxion in supe conduc ing lan hanum polyhyd ide”
Hui Wang1,2*, Pascal T. Salzb enne 3, Ion E ea4,5,6, Feng Peng7, Ziheng Lu3, Hanyu Liu2,8, Li Zhu9, Ch is J. Picka d3,10
and Yansun Yao11
1Key Labo a o y o Pho onic and Elec onic Bandgap Ma e ials (Minis y o Educa ion), School o Physics and Elec onic Enginee ing,
Ha bin No mal Uni e si y, Ha bin 150025, China
2In e na ional Cen e o Compu a ional Me hod & So wa e, College o Physics, Jilin Uni e si y, Changchun 130012, China
3Depa men o Ma e ials Science & Me allu gy, Uni e si y o Camb idge, 27 Cha les Babbage Road, Camb idge CB3 0FS, Uni ed Kingdom
4Fisika Aplika ua Saila, Gipuzkoako Ingenia i za Eskola, Uni e si y o he Basque Coun y (UPV/EHU), Eu opa Plaza 1, 20018 Donos ia/San
Sebas ián, Spain
5Cen o de Física de Ma e iales (CSIC-UPV/EHU), Manuel de La dizabal Pasealekua 5, 20018 Donos ia/San Sebas ián, Spain
6Donos ia In e na ional Physics Cen e (DIPC), Manuel de La dizabal Pasealekua 4, 20018 Donos ia/San Sebas ián, Spain
7College o Physics and Elec onic In o ma ion, Luoyang No mal Uni e si y, Luoyang 471022, P. R. China
8S a e Key Labo a o y o Supe ha d Ma e ials and In e na ional Cen e o Fu u e Science, Jilin Uni e si y, Changchun 130012, China
9Depa men o Physics, Ru ge s Uni e si y, Newa k, NJ 07102, USA
10Ad anced Ins i u e o Ma e ials Resea ch, Tohoku Uni e si y 2-1-1 Ka ahi a, Aoba, Sendai, 980-8577, Japan
11Depa men o Physics and Enginee ing Physics, Uni e si y o Saska chewan, Saska oon, Saska chewan S7N 5E2, Canada
*e-mail: [email p o ec ed]
Table o con en s
1. Ex ended Da a Figu es; Page 1 ~ 7
Supplemen a y Figu e 1 ~ Supplemen a y Figu e 7
2. Ex ended Da a Tables; Page 8 ~ 9
Supplemen a y Table 1 ~ Supplemen a y Table 2
3. Compu a ional De ails; Page 10 ~ 15
4. Supplemen a y Re e ences; Page 16 ~ 17
Supplemen a y Figu e 1
Supplemen a y Figu e 1| The ‘posi i e dTc/dp con adic ion’.
Recen expe imen al s udies by D ozdo e . al.1, Somayazulu e . al.2, S uzhkin e . al.3, Kong e . al.4, Snide
e . al.5 and Ma e . al.6 disco e ed se e al nea - oom empe a u e supe conduc ing polyhyd ides, and he
measu ed p essu e end o Tc usually e eals a posi i e dTc/dp a some p essu e egimes, in con adic ion wi h
he nega i e slope p edic ed by ea lie o subsequen calcula ions by Peng e . al.7, Liu H. e . al.8, Liu L. e .
al.9, Wang C. e . al.10, Quan e . al.11, E ea e . al.12, K uglo e . al.13, Wang H. e . al.14 and Duan e . al.15, 16 on
he same hyd ide based on BCS heo y. The con adic ion is clea ly seen om he da a. The p esen wo k
ocuses on he 250-kel in lan hanum polyhyd ide, which has a posi i e dTc/dp measu ed by D ozdo e . al.
on LaH9.61, and a nega i e dTc/dp calcula ed by E ea e . al.12 on cc-LaH10 a p essu es o 137-150 GPa.
1／17
Supplemen a y Table 1
Supplemen a y Table 1| Pa ame e s impac ing he p essu e end o Tc.
Based on e ie able Ω(p) (whe e Ω ep esen s 𝜔𝑙𝑜𝑔 o 𝜔2) (Re 11) o 𝛼(𝜔)2𝐹(𝜔) om he li e a u e
(Re 12, 8, 10 and 14), we de i ed dΩ/dp and analyzed he ole played by se e al pa ame e s in he p essu e
end o Tc using AD equa ions, wi h he Coulomb coupling cons an μ* se o 0.1. The esul ing p essu e
ends o Tc o cubic LaH10 and CaH6 a e in good ag eemen wi h he li e a u e (Supplemen a y Figu e 1).
The analysis demons a es ha 𝜁 dec eases mo e han wo imes as e han 𝛽 inc eases upon comp ession,
which leads o a nega i e sign o he dλ/dp slope. This sugges s ha d𝜁/dp domina es he p essu e end o λ.
λ has a mono onously nega i e p essu e dependence, bu dec eases less signi ican ly han 𝜔𝑙𝑜𝑔 inc eases.
Howe e , due o he nonlinea dependence o Tc on λ, ei he h ough he well-known exponen ial unc ion o
he ‘s ong-coupling co ec ion’ and ‘shape co ec ion’ ac o s (see de ails in page 14), dλ/dp plays a dominan
ole in de e mining he dTc/dp slope.
Ou esul s o LaH9.63 a e also shown in o de o enable a di ec compa ison. Ou d𝜔𝑙𝑜𝑔 /dp is
compa able o hose o cubic LaH10 ( cc) and CaH6 (bcc), bu d𝜁/dp dec eases signi ican ly a lowe p essu es,
which e e ses he sign o he dTc/dp slope h ough a conside able educ ion o dλ/dp below 163 GPa. We
u he es ima ed he limi o s abili y o he +dTc/dp slope agains a ia ions o
𝛽
and d
𝛽
/dp: I) a a ixed
d
𝛽
/dp o 0.01235 eV3/megaba , dec easing (inc easing)
𝛽
by 50 % esul s in a dec ease (an inc ease) o dTc/dp
by 59% (41%), bu does no change he posi i e sign; II) a a ixed
𝛽
o 0.04352 eV3 a 176 GPa, an abou
127 % dec ease o he d
𝛽
/dp slope is needed o in e he sign o dTc/dp. Li e a u e da a gene ally epo a
posi i e d
𝛽
/dp slope o cubic hyd ides co e ing a ange o H/M a ios (M= La and Ca) om 10 o 6. The e o e,
LaH9.63 is e y unlikely o exhibi a -d
𝛽
/dp slop in quali a i e con adic ion wi h mos o he hyd ides. The
analysis indica es ha he +dTc/dp slope is obus o e o s induced by he app oxima ion o
𝛽
and d
𝛽
/dp in
LaH9.63 by hose in quan um cc-LaH10.
8／17

Supplemen a y Table 2
Supplemen a y Table 2| Pa ame e s o es ima ing he p essu e end o Tc.
Pa ame e s o calcula ing he p essu e end o Tc shown in Fig 4b wi h smea ing 𝜎 = 0.0025 eV o F(ω) a e
lis ed he e o illus a ion. Fu he mo e, we compa e he p essu e ends o Tc calcula ed o a ious 𝜎 o
𝛼(𝜔)2 (i.e. 0, 0.01 and 0.05). I is ound ha he end is de e mined by he o e all shape o 𝛼(𝜔)2. Due o
he inexis ence o a me hod ha can di ec ly calcula e he 𝛼2𝐹(𝜔) o a quan um luxional s uc u e, we
es ima ed equency momen s Ω o 𝛼2𝐹(𝜔) in LaH9.63 based on he calcula ed 𝐹(𝜔) o LaH9.63 and he
𝛼(𝜔)2 app oxima ed by ha o quan um cc-LaH1012. To e alua e he po en ial e o s associa ed wi h he
app oxima ion, we calcula ed he Tc using eshaped 𝛼2𝐹(𝜔) ob ained by a i icial scaling o 𝛥𝜔. Empi ically,
EPC weigh s he phonon spec um o lowe equencies in high-Tc hyd ides. This weigh ing e ec changes
smoo hly wi h p essu e, as obse ed om he shape o 𝛼(𝜔)2 in quan um cc-LaH10 (Supplemen a y Figu e
7). Conside ing his, and ha quan um LaH9.63 and cc-LaH10 exhibi a simila phonon ha dening end wi h
inc easing p essu e as well as simila shapes o 𝐹(𝜔), ou esul s sugges ha he p essu e end o Tc is obus
o a ia ions o Ω. This indica es ha i is p ac icable o es ima e quali a i ely he sign o dTc/dp slop in LaH9.63
unde his app oxima ion.
9／17
Compu a ional De ails
Molecula dynamics calcula ions
Quan um nuclea dynamics we e s udied using pa h-in eg al molecula dynamics (PIMD) and cen oid molecula
dynamics (CMD), wi h he massi e Nosé-Hoo e chain (NHC) he mos a s on NVT and NVE ensembles (N-numbe o
pa icles; V- olume; T- empe a u e, and E-ene gy), espec i ely, as implemen ed in he PIMD code17. NpT-PIMD (p-
p essu e) simula ions we e pe o med o es ima e he quan um p essu e- olume ela ions. Beads numbe was se o 16
o he cases wi hou a speci ica ion, and ime s ep was se o 0.5 and 0.05 s o PIMD and CMD, espec i ely. A sho
PIMD simula ion was always ca ied ou on he ini ial s uc u e o p epa e he p e-equilib ium s a e in o de o p omo e
he e iciency and s abili y o he CMD simula ion. The pa ame e s o he s anda d ab ini io MD simula ions a e he
same as hose o he PIMD simula ions, excep o a educed bead numbe o 1. The MD uns o LaH9.63 we e ini ialized
om di e en cen oid con igu a ions o he 4-picosecond CMD ajec o y wi h a sampling in e al o 0.25 ps o 16
sho uns o 12000 s eps and o 1 ps o 4 long uns o 52000 s eps.
The unde lying ab ini io o al ene gy and o ces we e calcula ed based on he plane-wa e pseudopo en ial me hod, as
implemen ed in he Vienna ab ini io simula ion p og am (VASP)18. The con e gence c i e ion o he o al ene gy (Ecc)
was chosen o be 3×10−6 eV/a om wi h a plane-wa e cu o o 325 eV and Γ-poin sampling o he i s B illouin zone.
Dec easing Ecc o 3×10−7 eV/a om ( un 2) esul s in a nea ly negligible change o he p o on di usion coe icien D in
LaH9.63 a 150 GPa (see Fig. 3a in he main ex ). The olume o he cubic simula ion cells o LaH9.63 we e ixed a
1097.70, 1079.98, 1062.39, 1038.16, and 1014.35 Å3. The esul ing empe a u e luc ua ions a equilib ium a e wi hin
± 15 K, ± 10 K and ± 25 K wi h espec o he a ge in he PIMD, CMD and MD simula ions, espec i ely. O he de ails
a e summa ized in Supplemen a y Table 3.
Supplemen a y Table 3. De ails o simula ions.
Phase
Numbe o
A oms,
H : La
P essu es,
GPa
Tempe a u e,
K
Me hod
S eps
uns
LaH9.63
308:32
150
60, 120, 240
MD
12000
1-16
137, 143, 150, 163,176
240
MD
52000
1-4
150
240
PIMD
2000
1,2
CMD
82000
1,2
137, 143, 163,176
240
PIMD
2000
1
CMD
82000
1
150
60
PIMD
8000
3§
60, 120
PIMD
2000
1
CMD
82000
1
LaH9.0
288:32
130, 137, 150, 170, 180
300
PIMD†
2000
1
LaH10
320:32
130, 137, 150, 170, 180
300
PIMD†
2000
1
137, 150
240
PIMD
2000
1
CMD
42000
1
§
NVT simula ions wi h 16, 32 and 64 beads we e pe o med o check he impac o bead numbe on MSD; †NpT simula ions.
10／17
The quan um p essu e- olume ela ion o cc-LaH10 and cc-LaH9.
The quan um p essu e- olume ela ion o cc-LaH10 and cc-LaH9 a 300 K was es ima ed om he olume a e aged
o e 1 ps a equilib ium ob ained in NpT-PIMD simula ions using a cubic box and Ma yna’s equa ion o mo ion, as
implemen ed in he PIMD code17. Ou esul s e eal a H/La a io o 9.54 (o 9.71) o he expe imen al samples
syn hesized a 150 GPa (o 137 GPa), in good ag eemen wi h expe imen al es ima ion o a LaH9.61. To ensu e ou
p essu e end o Tc can di ec ly compa e wi h he expe imen al one, we buil s uc u al models a 137 and 150 GPa
using expe imen al la ice pa ame e s. This gua an ees ha he coincidence o his p essu e ange (i.e. ‘137-150 GPa’)
be ween calcula ion and expe imen s does no dependen on he expe imen al o heo e ical p essu e scales. No ably,
his is indeed he p essu e ange ha he posi i e dTc/dp slope we e measu ed expe imen ally and calcula ed in his wo k.
The PBE unc ionals may lead o po en ial shi o he calcula ed Tc .s. p essu e cu e in ela i e o he expe imen al
measu ed one a p essu e abo e 150 GPa, howe e , which does no a ec ou conclusion. The quan um p essu e o 143,
163 and 176 GPa we e es ima ed based on in e pola ion.
The mean squa e displacemen , 〈∆𝒓𝟐〉
〈∆𝑟(𝑡)𝟐〉 = 〈|𝑟𝑖(𝑡)−𝑟0(𝑡)−[𝑟𝑐𝑚(𝑡)−𝑟𝑐𝑚(0)]|2〉𝐼,
whe e 𝑟𝑖(𝑡) is he posi ion o he di usi e p o on i, which exhibi s a leas one jump o e a h eshold alue o 0.7 Å
o e he cou se o he simula ion, 𝑟𝑐𝑚(𝑡) ep esen s he posi ion o he cen e o mass o he sys em a he ime , and
he 〈 〉 ep esen an a e age o e 10 ime s eps and he o al numbe o ‘pa icle’ I, which equals 308 o p o on and 12
o acancy in LaH9.63. In he simula ions, some o di usi e p o ons hop back o he p e iously occupied posi ion
occasionally due o ‘ a ic jam’, which esul s in a dec ease o 〈∆𝒓𝟐〉 . The p o on a ic jam and local s uc u al
elaxa ion esul in pla eaus in 〈∆𝒓𝟐〉 cu es in bo h he CMD and MD simula ions. Howe e , 〈∆𝒓𝟐〉 gene ally
inc eases wi h inc easing o , which is clea ly e ealed by long- ime MD simula ions o 24 ps.
Supplemen a y Figu e 8. MSD o PIMD simula ion wi h a ious bead numbe s a 60 K.
MSDs de i ed om he cen oid ajec o y o PIMD simula ions a 150 GPa and an equilib ium empe a u e o 60 K
wi h 16, 32 and 64 beads a e shown in Supplemen a y Figu e 8. All cu es sugges quan um p o on di usion a his
empe a u e. Conside ing ha he s eng h o NQE inc eases wi h dec easing empe a u e, la ge bead numbe gi es
be e MSD. The di e ence o 〈∆𝒓𝟐〉 alues a he end o he simula ion (i.e. 2 ps) is no signi ican , which may be
a ibu ed o he ‘ a ic jam’ o p o ons in LaH9.63 associa ed wi h he small amoun o s oichiome ic de ec .
11／17
The di usion coe icien , D
We app oxima e he p o on di usion coe icien D in LaH9.63 om he slope o 〈∆𝑟𝟐〉 ob ained om a linea i , as
shown in Supplemen a y Figu e 9 o CMD simula ions un1 and un2 a 150 GPa and 240 K. D is compu ed om 𝐷 =
𝑠𝑙𝑜𝑝𝑒(〈∆𝑟𝟐〉) 6
⁄.
Supplemen a y Figu e 9. Linea i o MSD.
Vacancy o ma ion en halpy, H
We ca ied ou a ull a iable-cell op imiza ion (using ISIF=3 in VASP) o cc-LaH107, 8 (in a 2×2×2 supe cell o he
con en ional uni cell con aining 352 a oms) and B2/n-H2 (in he con en ional uni cell o 48 a oms)19 a p essu es
be ween 120 and 220 GPa wi h an in e al o 5 GPa. We employed k-mesh g ids o sizes 3×3×3 and 15×9×9, espec i ely,
o ca y ou he equi ed DFT calcula ions.
The VT and VC LaH9.97 acancy s uc u es we e hen cons uc ed by emo ing he app op ia e H a om om he elaxed
cc-LaH10 s uc u e a each p essu e, as desc ibed in he main ex . Va iable-cell op imiza ion o hese s uc u es induced
p o on di usion, esul ing in a dis o ion o he s uc u es. In o de o isola e he e ec o he acancies, he e o e, we
only elaxed he a omic posi ions, keeping he cell shape and olume ixed (ISIF=2 in VASP). We used a k-mesh g id
equi alen o ha o cc-LaH10. A cu -o ene gy o 325 eV and a con e gence c i e ion o he o al ene gy o 3×10−7
eV/a om we e employed in all calcula ions.
Using he esul ing V-E ela ionship o he elaxed s uc u es, he p essu e-en halpy cu es we e hen calcula ed using
he hi d-o de Bi ch–Mu naghan iso he mal equa ion o s a e20. F om he en halpies o each p essu e, he H o he VC
and VT s uc u es we e calcula ed as ollows,
H = H(LaH9.97) – H( cc-LaH10) + H(B2/n-H2)/48,
wi h he con igu a ionally a e aged H exp essed as,
H (a e age) = 0.2H (VT) + 0.8H (VC).
The calcula ed p essu e- olume cu e gi es a p essu e shi o ~5 GPa o LaH9.97 wi h espec o he expe imen al
measu emen s on cc-LaH9.6 a 120-220 GPa, which does no a ec ou discussions.
I should be no ed ha H is calcula ed classically in he s a ic la ice app oxima ion on he Bo n–Oppenheime ene gy
su ace, which di e s om he con igu a ional ene gy su ace o he quan um c ys al (e.g., cc-LaH10 in Re . 12) o he
ee ene gy su ace o he classical o quan um c ys al a ini e empe a u e. The s abili y and p essu e bounda y o he
12／17
acancy s uc u e may be a ec ed by mo e complex physical e ec s exis ing in he supe conduc ing samples, o
ins ance nuclea quan um e ec s (including ze o-poin mo ion o nuclei and p o on unnelling), o empe a u e e ec s.
Indeed, empe a u e e ec s only enhance he o ma ion o acancies. In his sense, he conclusion om he H cu es
shown in Fig.1a o he main ex , namely ha acancy o ma ion is a o ed below 158 GPa in he classical c ys al
pic u e, while being based on a model calcula ion subjec o many app oxima ions, indica es ha lan hanum
supe hyd ide should be a he p one o acancy o ma ion.
The con igu a ional dis ance, ξ
The c ys al inge p in echnique u ilized in his s udy is based on Gaussian o e lap ma ices, which ep esen he local
en i onmen s o all a oms in a uni cell and can e icien ly de e mine con igu a ional dis ances ξ be ween c ys alline
s uc u es sa is ying he ma hema ical equi emen s o a me ic21. The ξ o he QFS o LaH9.63 s. a ious s a ic s uc u es
(e.g. cc, C2/m and P1) we e ob ained by s a is ical analysis o e he ξ o 400 cen oid con igu a ions sampled om 16-
bead CMD ajec o ies o 4 ps wi h a ime in e al o 10 s a each p essu e.
Elec onic densi y o s a es a he Fe mi le el, 𝑵(𝝐𝑭)
The 𝑁(𝜖𝐹) is o cen al impo ance o he unde s anding o con en ional supe conduc i i y o hyd ogen- ich ma e ials.
I is qui e gene al ha he p essu e dependence o 𝑁(𝜖𝐹) di ec ly co ela es o he p essu e end o he supe conduc ing
Tc. The 𝑁(𝜖𝐹) o he quan um s uc u es o LaH9.63 was calcula ed by ca ying ou a s a is ical a e age o e he 𝑁(𝜖𝐹)
o 400 cen oid con igu a ions sampled om 16-bead CMD ajec o ies o 4 ps wi h a ime in e al o 10 s a each
p essu e. The 𝑁(𝜖𝐹) o each cen oid con igu a ion was calcula ed using he e ahed on me hod wi h Blöchl
co ec ions22, wi h a con e gence c i e ion o he o al ene gy o 3×10−6 eV/a om o a cu -o ene gy o 325 eV and a
5×5×5 k-mesh g id. The 𝑁(𝜖𝐹) o s a ic cc-LaH10, C2/m-LaH10, and P1-LaH10 we e calcula ed based on p imi i e
cells by he e ahed on me hod wi h Blöchl co ec ions, wi h a con e gence c i e ion o he o al ene gy o 3×10−6
eV/a om o a cu -o ene gy o 325 eV and 21×21×21, 27×27×15 and 5×5×5 k-mesh g ids, espec i ely. Fo he cc and
C2/m phases, he 𝑁(𝜖𝐹) o s a ic LaH9.63 we e es ima ed based on he igid-band model23 o he elec onic s uc u e o
LaH10 by i ue o an a i icial shi o 𝜖𝐹.
Supplemen a y Figu e 10. The 𝑁(𝜖𝐹) and 𝜉 (o H subs uc u e) in LaH9.63 compa ed o hose o LaH10. The .u. is he
abb e ia ion o o mula uni , h oughou he a icle.
13／17

Compa ison on 𝑵(𝝐𝑭) and ξ be ween LaH9.6 and LaH10
As shown in Supplemen a y Figu e 10, we compa ed he con igu a ionally a e aged 𝜉 and 𝑁(𝜖𝐹) in quan um LaH9.63
and LaH10 in he p essu e ange o 137–150 GPa. The esul s sugges ha he H subla ice dis o ion is supp essed in he
acancy- ee case. Mo eo e , a nega i e d𝑁(𝜖𝐹)/dp slop is obse ed in LaH10, in con as o he posi i e d𝑁(𝜖𝐹)/dp
slop o LaH9.63.
The equency momen s o 𝜶𝟐𝑭(𝝎) in LaH9.63, 𝝎𝒍𝒐𝒈 and 𝝎
𝟐
The 𝛼2𝐹(𝜔) o LaH9.63 is es ima ed based on he 𝐹(𝜔) o LaH9.63 de i ed by Fou ie ans o ming he eloci y
au oco ela ion unc ions (VACF) in he CMD simula ions a 240 K in combina ion wi h coupling unc ions 𝛼(𝜔)2
app oxima ed by ha o quan um cc-LaH10:
1. De i ing he 𝐹(𝜔) o LaH9.63 by Fou ie ans o ming he VACF in he CMD simula ions a 240 K and olume (V)
o 34.30, 33.75, 33.20, 32.44, and 31.70 Å3/ .u.;
2. Calcula ing he 𝛼(𝜔)2 om 𝛼2𝐹(𝜔) and 𝐹(𝜔) in quan um cc-LaH10: 𝛼(𝜔)2= 𝛼2𝐹(𝜔)/ 𝐹(𝜔) a olume
(V′) o 35.23, 32.97 and 30.36 Å3/ .u., and de i ing he 𝛼(𝜔)2 a V by in e se olume weigh in e pola ion o
𝛼(𝜔)2 o neighbo ing V′;
3. Calcula ing he 𝛼(𝜔)2𝐹(𝜔) o LaH9.63 based on 𝐹(𝜔) o LaH9.63 and 𝛼(𝜔)2 o quan um cc-LaH10:
𝛼2𝐹(𝜔)_LaH9.63 = 𝛼(𝜔)2_LaH10 × 𝐹(𝜔)_LaH9.63 a a ious V alues;
4. Calcula ing he 𝜔𝑙𝑜𝑔 and 𝜔2 ollowing hei de ini ions in Re . 24.
The equency s ep 𝛥𝜔 is se o 0.0004 eV in he calcula ions. To include he ib a ional modes in bo h o low- and
high- equency egime, he 𝐹(𝜔) a e de i ed om he VACFs o he La and H subla ices sepa a ely. Since smea ed
𝐹(𝜔) is gene ally adop ed o con e ge EPC pa ame e s, we smea 𝐹(𝜔) wi h Gaussian p o iles wi h wid hs (𝜎) o
0.002 ~ 0.003 eV. 𝜔𝑙𝑜𝑔 and 𝜔2 a e he e o e ob ained ollowing hei de ini ions in Re . 24. In addi ion, we eshaped
he 𝛼2𝐹(𝜔) o LaH9.63 by a i icial scaling o 𝛥𝜔 o 0.0003 and 0.0005 eV o e alua e he po en ial e o s associa ed
wi h he app oxima ion in s ep 3, see he cap ion o Supplemen a y Table 2.
The elec on-phonon coupling cons an , λ
We es ima e λ in LaH9.63 using McMillan’s o mula25 (𝜆 = 𝑁(𝜖𝐹)⟨𝛪2⟩/𝑀𝜔2
2) as ollows:
1. Combining a iables as 𝜆 = 𝛽 ⋅𝜁 wi h 𝛽 = ⟨𝛪2⟩/𝑀 and 𝜁 = 𝑁(𝜖𝐹)/𝜔2
2;
2. Es ima ing 𝜁(𝑉) based on 𝑁(𝜖𝐹) and 𝜔2 o LaH9.63 a V o 34.30, 33.75, 33.20, 32.44, and 31.70 Å3/ .u.;
3. Calcula ing 𝛽(V′) om 𝜆, 𝑁(𝜖𝐹) and 𝜔2o quan um cc-LaH10 in Re . 12 a V′ o 35.23, 32.97 and 30.36 Å3/ .u.
and de i ing he 𝛽(V) by in e se olume weigh in e pola ion o neighbo ing 𝛽(V′);
4. Calcula ing 𝜆(𝑉) in LaH9.63 using he es ima ed 𝜁(𝑉) o LaH9.63 and he 𝛽(V) o quan um cc-LaH10.
C i ical empe a u e o supe conduc i i y, Tc
The Tc was e alua ed using he Allen–Dynes-modi ied McMillan equa ion (AD)24,
𝑇𝑐=𝑓1𝑓2𝜔𝑙𝑜𝑔
1.2 𝑒𝑥𝑝[−1.04(1+𝜆)
𝜆−𝜇∗(1+0.62𝜆)],
whe e 𝑓1 and 𝑓2 a e he ‘s ong-coupling co ec ion’ and ‘shape co ec ion’ ac o s o s ong-coupling sys ems:
𝑓1= [1+(𝜆/

1)3/2]1/3 wi h

1= 2.46(1+3.8

∗);
𝑓2= 1+(𝜔2𝜔𝑙𝑜𝑔
⁄−1)𝜆2(𝜆2+

2
2)
⁄ wi h

2= 1.82(1+6.3

∗)(𝜔2𝜔𝑙𝑜𝑔
⁄).
μ* is he Coulomb coupling cons an , which is se o a ypical alue o 0.1 in his wok.
14／17
S uc u al in o ma ion o s a ic LaH10-𝛿.
The simula ion cells o cc-LaH9 (320 a oms) and cc-LaH10 (352 a oms) a e ini ialized by a 2×2×2 ex ension o he uni
cell o F-43m-LaH913 and Fm-3m-LaH1012, espec i ely. We ini ialize he LaH9.63 s uc u e (340 a oms) by adding
andomly 20 H a oms o he simula ion cell o cc-LaH9, cons aining neighbo ing acancies o be no less han 4.4 Å
apa o ensu e an app oxima ely uni o m acancy dis ibu ion, as shown in Supplemen a y Figu e 11. A PIMD
simula ion (2000 s eps) was always ca ied ou on he ini ial s uc u e o p epa e he p e-equilib ium s a e o he CMD
simula ion. Vacancy di usion gene ally esul s in a andom dis ibu ion o acancies on he 8c and 32 Wycko posi ion
in Fm-3m-LaH10 wi hin se e al hund ed PIMD s eps. The s a ic P1-LaH9.63, mimicking he quan um luxional LaH9.63,
is buil by scaling he la ice pa ame e a o a QFS sampled om he CMD simula ion o LaH9.63 a 176 GPa. The ini ial
acancy s uc u e o LaH9.63 and s a ic P1-LaH9.63 a 150 GPa can be ound a he link—
h ps://doi.o g/10.6084/m9. igsha e.20484420. 1.
Supplemen a y Figu e 11. Ini ial acancy s uc u e o LaH9.63.
15／17
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