This jou nal is © The Royal Socie y o Chemis y 2025 J. Ma e . Chem. C
Ci e his: DOI: 10.1039/d5 c00863h
Gian he mally induced band-gap
eno maliza ion in anha monic
sil e chalcohalide an ipe o ski es†
Pol Benı
´ ez, *
ab
Siyu Chen,
cd
Ruoshi Jiang,
c
Cib a
´nLo
´pez,
ab
Josep-Lluı
´s Tama i ,
ab
Jo ge I
´n
˜iguez-Gonza
´lez,
e
Edga do Saucedo,
bg
Ba omeu Monse a
cd
and Claudio Cazo la *
ab
Sil e chalcohalide an ipe o ski es (CAP), Ag
3
XY (X = S, Se; Y = B , I), a e a amily o highly anha monic
ino ganic compounds wi h g ea po en ial o ene gy applica ions. Howe e , a subs an ial and
un esol ed disc epancy exis s be ween he op oelec onic p ope ies p edic ed by heo e ical i s -
p inciples me hods and hose measu ed expe imen ally a oom empe a u e, hinde ing he undamen al
unde s anding and a ional enginee ing o CAP. In his wo k, we employ densi y unc ional heo y, igh -
binding calcula ions, and anha monic F o
¨hlich heo y o in es iga e he op oelec onic p ope ies o CAP
a ini e empe a u es. Nea oom empe a u e, we obse e a gian band-gap (E
g
) educ ion o
app oxima ely 20–60% ela i e o he alue calcula ed a T= 0 K, b inging he es ima ed E
g
in o excel-
len ag eemen wi h expe imen al measu emen s. This ela i e T-induced band-gap eno maliza ion is
oughly wice he la ges alue p e iously epo ed in he li e a u e o simila empe a u e anges. Low-
ene gy op ical pola phonon modes, which b eak in e sion symme y and enhance he o e lap be ween
sil e and chalcogen s elec onic o bi als in he conduc ion band, a e iden i ied as he p ima y d i e s o
his signi ican E
g
educ ion. Fu he mo e, when empe a u e e ec s a e conside ed, he op ical abso p-
ion coe icien o CAP inc eases by nea ly an o de o magni ude in he isible ligh spec um. These
indings no only b idge a c i ical gap be ween heo y and expe imen bu also pa e he way o u u e
echnologies whe e empe a u e, elec ic ields, and ligh dynamically modula e op oelec onic p ope -
ies, es ablishing CAP as a e sa ile pla o m o ene gy and pho onic applica ions.
In oduc ion
Elec on–phonon coupling (EPC), a ising om he in e ac ions
be ween elec ons and la ice ib a ions, is ubiqui ous in
ma e ials and is esponsible o a wide ange o condensed
ma e physical e ec s.
1–4
Fo example, EPC plays a c ucial ole
in he empe a u e (T) dependence o elec ical esis i i y in
me als, ca ie mobili y in semiconduc o s, op ical abso p ion
in indi ec band gap semiconduc o s, and he onse o con en-
ional supe conduc i i y. Addi ionally, EPC enables he he -
maliza ion o ho ca ie s, in luences he phonon dispe sion in
me als, and de e mines he T-dependence o elec onic ene gy
bands in solids.
5,6
Likewise, he band gap (E
g
) o semiconduc ing and dielec ic
ma e ials can be signi ican ly affec ed by EPC, ypically dec eas-
ing wi h inc easing empe a u e ( he so-called Va shni effec
7
).
This common E
g
beha io can be explained by he Allen–Heine–
Ca dona pe u ba i e heo y, which a ibu es i o a la ge
T-induced ene gy inc ease in he alence band compa ed o
he conduc ion band due o a g ea e sensi i i y o phonon
a
G oup o Cha ac e iza ion o Ma e ials, Depa amen de Fı
´sica, Uni e si a
Poli e
`cnica de Ca alunya, Campus Diagonal Beso
`s, A . Edua d Ma is any 10–14,
08019 Ba celona, Spain. E-mail: pol.b[email p o ec ed], cla[email p o ec ed]
b
Resea ch Cen e in Mul iscale Science and Enginee ing, Uni e si a Poli e
`cnica de
Ca alunya, Campus Diagonal-Beso
`s, A . Edua d Ma is any 10–14,
08019 Ba celona, Spain
c
Depa men o Ma e ials Science and Me allu gy, Uni e si y o Camb idge,
Camb idge, CB30FS, UK
d
Ca endish Labo a o y, Uni e si y o Camb idge, Camb idge, CB30HE, UK
e
Ma e ials Resea ch and Technology Depa men , Luxembou g Ins i u e o Science
and Technology (LIST), A enue des Hau s-Fou neaux 5, L-4362 Esch/Alze e,
Luxembou g
Depa men o Physics and Ma e ials Science, Uni e si y o Luxembou g,
41 Rue du B ill, L-4422 Bel aux, Luxembou g
g
Mic o and Nano echnologies G oup, Eme ging Thin Film Pho o ol aics Lab,
Depa amen dEnginye ia Elec o
`nica, Uni e si a Poli e
`cnica de Ca alunya,
Campus Diagonal Beso
`s, A . Edua d Ma is any 10–14, 08019 Ba celona, Spain
†Elec onic supplemen a y in o ma ion (ESI) a ailable. See DOI: h ps://doi.o g/
10.1039/d5 c00863h
Recei ed 27 h Feb ua y 2025,
Accep ed 11 h Ap il 2025
DOI: 10.1039/d5 c00863h
sc.li/ma e ials-c
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Ma e ials Chemis y C
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popula ion a ia ions (i.e., la ge second-o de elec on–pho-
non coupling cons an s).
8–10
Rep esen a i e examples o his
he mal E
g
dependence include diamond, which exhibi s a
B5% band-gap educ ion a a ound 1000 K;
11,12
an imony
sul ide (Sb
2
S
3
), which shows a E
g
educ ion o 200 meV in he
empe a u e ange o 10 T 300 K;
13
MgO, which displays a
band-gap educ ion o B15% in he empe a u e in e al 0
T 1500 K;
14
S TiO
3
, which exhibi s a B15% band-gap
educ ion om 300 o 1000 K;
15
and molecula c ys als, which
display eco d band-gap educ ions o 15–20% a low
empe a u es.
16
Anomalous band-gap he mal beha iou , in
which E
g
inc eases wi h inc easing empe a u e, has also been
obse ed in a a ie y o ma e ials such as black phospho us,
17
halide pe o ski es,
18
and chalcopy i e
19
and hyd ide
12
compounds.
Highly anha monic sil e chalcohalide an ipe o ski es
(CAP)
20
wi h chemical o mula Ag
3
XY (X = S, Se; Y = B , I) a e
s uc u ally simila o lead halide pe o ski es (e.g., CsPbI
3
),
wi h he ‘‘an i’’ designa ion indica ing he exchange o anions
and ca ions compa ed o he ypical ionic pe o ski e a ange-
men . Analogous o lead halide pe o ski es, CAP a e highly
p omising ma e ials o ene gy and op oelec onic
applica ions,
21–26
o e ing low oxici y due o hei lead- ee
composi ion.
27,28
The wo mos ex ensi ely s udied CAP com-
pounds, Ag
3
SB and Ag
3
SI, possess expe imen ally de e mined
band gaps o app oxima ely 1.0 eV,
29,30
making hem a o able
o pho o ol aic applica ions. These ma e ials ha e also been
ecognized as high empe a u e supe ionic conduc o s.
21,22
Addi ionally, CAP ha e been in es iga ed as po en ial he mo-
elec ic ma e ials
25,26
owing o hei subs an ial ib a ional
anha monici y and unique cha ge anspo p ope ies.
23,24
In iguingly, o bo h Ag
3
SB and Ag
3
SI, he e is an eno -
mous disag eemen be ween he E
g
p edic ed by i s -p inciples
me hods (a T= 0 K, unde s a ic la ice condi ions) and hose
measu ed expe imen ally a oom empe a u e. In pa icula ,
high-le el densi y unc ional heo y (DFT) calcula ions employ-
ing hyb id unc ionals and including spin–o bi coupling (SOC)
e ec s es ima e he band gap o hese wo a che ypal CAP o be
1.8 and 1.4 eV, espec i ely.
29–31
The E
g
disc epancies be ween
heo y and measu emen s amoun o 60–80% o he expe i-
men al alues (i.e., di e ences o 0.5–0.8 eV), which a e unu-
sually la ge and call o a ca e ul inspec ion o he ac o s
causing hem.
In his s udy, we assessed he in luence o EPC effec s on he
E
g
and op ical abso p ion spec a o CAP using i s -p inciples
DFT me hods, igh -binding calcula ions, and anha monic
F o
¨hlich heo y. Nea oom empe a u e, ou compu a ional
in es iga ions e ealed a gian E
g
educ ion o 20–60% ela i e
o he alue calcula ed a T= 0 K, b inging he es ima ed band
gap in o excellen ag eemen wi h he expe imen al alues.
Low-ene gy op ical pola phonons, which cause la ge
symme y-b eaking s uc u al dis o ions and p omo e he
o e lap be ween sil e and chalcogen s elec onic o bi als in
he conduc ion band, we e iden i ied o be he p ima y mecha-
nism d i ing his subs an ial T-induced band-gap educ ion.
Fu he mo e, a ini e empe a u es he op ical abso p ion
spec a o CAP we e signi ican ly enhanced, in some cases by
nea ly an o de o magni ude. The pola na u e o he phonons
causing hese e ec s opens up new echnological possibili ies,
whe e he op oelec onic p ope ies o ma e ials could be
e ec i ely manipula ed by ex e nal elec ic ields and ligh .
Resul s
The oom- empe a u e phase o bo h Ag
3
SI and Ag
3
SB ha e
been expe imen ally iden i ied as cubic wi h he space g oup
Pm%
3m.
30,32–34
This phase is cha ac e ized by a i e-a om uni
cell: a chalcogen a om a he cen e o he cube, halide a oms in
each e ex, and sil e a oms a he cen e o each ace (Fig. 1a
and b). Phonon calcula ions o his phase wi hin he ha monic
app oxima ion (T= 0 K) e eal imagina y phonon b anches,
hus indica ing dynamical ins abili y. Howe e , when phonons
a e calcula ed ully accoun ing o anha monic e ec s a ini e-
Tcondi ions, he esul ing phonon spec um is well-beha ed
wi h no signs o ins abili y (Fig. 1c).
20
Thus, he cubic Pm%
3m
phase was conside ed h oughou his wo k o all CAP.
As discussed in he In oduc ion, he disc epancies be ween
he expe imen ally measu ed (a T= 300 K) and heo e ically
de e mined (a T= 0 K) band gaps o Ag
3
SB and Ag
3
SI a e
emendously la ge (i.e., 60–80% o he expe imen al alues).
The e o e, we in es iga ed hei po en ial causes by assessing
he impac o elec on–phonon coupling (EPC) and empe a-
u e on he band gap o CAP.
Band gap eno maliza ion due o elec on–phonon in e ac-
ions is ypically es ima ed using wo main app oaches: densi y
unc ional pe u ba ion heo y (DFPT)
1
and ini e-di e ences.
2
In DFPT, elec on–phonon in e ac ions a e ea ed as a pe u -
ba ion, wi h band ene gy a ia ions (and consequen ly, band
gap shi s) de i ed om he Fan-Migdal and Debye–Walle sel -
ene gies, as desc ibed by Allen and Heine.
36
A key ad an age o
DFPT is i s compu a ional e iciency, as i does no equi e he
use o supe cells. This me hod has been widely employed in
band gap eno maliza ion s udies
37
and is implemen ed in
popula ab ini io codes such as EPW.
38
Fini e-diffe ences app oaches, on he o he hand, a e com-
pu a ionally mo e demanding, equi ing supe cells o ep o-
duce phonons in eal space and dynamical simula ions o
co ec ly sample he elec onic esponse o ionic luc ua ions.
Howe e , hey offe dis inc ad an ages o e DFPT. One key
bene i is hei lexibili y, as hey can be applied wi h any
unde lying elec onic s uc u e me hod. Addi ionally, ini e-
diffe ences me hods na u ally inco po a e e ms beyond he
lowes o de in he elec on–phonon in e ac ion, making hem
pa icula ly use ul o cap u ing highe -o de e ec s.
18
Reade s
seeking a mo e comp ehensi e discussion o DFPT and ini e-
di e ences me hods a e e e ed o he e iew a icles,
1,2
which
ex ensi ely co e hese echniques.
In his s udy, we employ he ini e-diffe ences app oach o
es ima e empe a u e- eno malized band gaps, as CAP ma e i-
als exhibi s ong anha monici y.
20
Consequen ly, eno maliz-
ing hei elec onic band ene gies a he ha monic le el would
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be inadequa e. Mo eo e , since accu a e CAP band gap p edic-
ions equi e he use o hyb id unc ionals and spin–o bi
coupling, he ini e-di e ences app oach eme ges as he mos
p ac ical and eliable choice.
In pa icula , we pe o med i s -p inciples calcula ions and
ab ini io molecula dynamics (AIMD) simula ions based on DFT
(Me hods). Addi ionally, o cap u e long- ange EPC effec s, we
employed anha monic F o
¨hlich heo y
39–42
conside ing long-
ange dipole–dipole in e ac ions and T- eno malized phonons
(Me hods). Fu he mo e, he op ical abso p ion spec a o all
CAP we e assessed a Ta0 K condi ions and he main EPC
mechanisms unde lying he E
g
disc epancies we e iden i ied
wi h he help o a igh -binding model.
EPC band-gap eno maliza ion in CAP
The T- eno malized band gap o CAP was calcula ed as ollows:
E
g
(T)=E
g
(0) + DE
g
(T), (1)
whe e E
g
(0) ep esen s he s a ic band gap and he co ec ion
e m DE
g
can be exp essed as he sum o sho -(S) and long-
wa eleng h (L) phonon con ibu ions:
40
DE
g
(T)=DE
S
g
(T)+DE
L
g
(T). (2)
The sho -wa eleng h phonon co ec ion was es ima ed
h ough AIMD simula ions using a supe cell (Me hods), whe e
he band-gap alue was a e aged o e mul iple gene a ed
con igu a ions, as desc ibed in e . 39:
DES
gðTÞ¼ 1
NX
N
k¼1
Egð RkðTÞgÞ Egð0Þ;(3)
whe e N ep esen s he o al numbe o conside ed con igu a-
ions and {R
k
} he a omic posi ion o he k- h con igu a ion.
To achie e accu a e band-gap co ec ions, i was essen ial o
use hyb id unc ionals and inco po a e spin–o bi coupling
e ec s (Supplemen a y me hods, ESI†), which signi ican ly
inc ease he compu a ional e o . Ex ensi e nume ical es s
we e conduc ed o op imize he supe cell size, k-poin mesh,
and alue o N, ensu ing ha hese c i ical e ec s we e accu-
a ely cap u ed in he calcula ions (Me hods and Supplemen-
a y me hods, ESI†).
In pola ma e ials, he e is an addi ional con ibu ion o he
band-gap eno maliza ion s emming om long- ange F o
¨hlich
coupling ha is no ully cap u ed by he ini e size o he
supe cells employed in he AIMD simula ions.
39–43
This long-
wa eleng h phonon band-gap co ec ion can be exp essed as
ollows:
DEL
gTðÞ¼DEL
CB TðÞDEL
VB TðÞ;(4)
whe e CB and VB e e o he bo om conduc ion and op
alence band le els, espec i ely.
Fig. 1 Gene al physical p ope ies o he a che ypal CAP Ag
3
SB . (a) The cubic Pm
%
3mphase expe imen ally obse ed a oom empe a u e.
(b) Expe imen al diff ac og am o Ag
3
SB
30
compa ed wi h he heo e ical one es ima ed o he cubic Pm
%
3mphase. (c) Vib a ional phonon spec um
(le ) and phonon densi y o s a es ( igh ) calcula ed wi hin he ha monic app oxima ion o he cubic Pm
%
3mphase o Ag
3
SB a T= 0 K (black lines) and a
T= 200 K ( ed lines) ully conside ing anha monic effec s. (d) Elec onic band s uc u e (le ) and densi y o s a es ( igh ) o Ag
3
SB calcula ed wi h he
hyb id unc ional HSEsol.
35
Red and blue lines (do s) ep esen alence ( op o he alence) and conduc ion (bo om o he conduc ion) bands,
espec i ely.
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Fo a 3D pola ma e ial, he T-induced ene gy le el shi s
appea ing in eqn (4) can be compu ed as ollows:
39
DEL
iðTÞ¼2aP
p
hoLO an1qF
qLO;i
2nTþ1
½
;(5)
whe e a
P
ep esen s he pola on cons an , o
LO
he phonon
equency a e aged o e he h ee longi udinal op ical Gpho-
non modes,
43
and q
F
a unca ion ac o . The unca ion ac o
q
F
can be app oxima ed as he Debye sphe e adius and q
LO,i
is
de ined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2moLO þoi
ðÞ=h
p,m* being he cha ge ca ie
effec i e mass and ho
i
he s a e ene gy. The e m n
T
is he
Bose–Eins ein occupa ion numbe co esponding o he a e -
age LO ib a ional equency, and he pola on cons an can be
compu ed as ollows:
39
aP¼e2
4pE0h
1
e1
1
e0
m
2hoLO
1=2
;(6)
whe e e
N
is he high- equency dielec ic cons an and e
0
he
s a ic pe mi i i y o he sys em. Quan um nuclea e ec s ha e
been dis ega ded h oughou his wo k, hence he T-induced
ene gy le el shi s in eqn (5) we e o se by hei ze o-
empe a u e alues De
L
i
(0).
Fig. 2a p esen s he anha monic phonon spec um calcu-
la ed o Ag
3
SB unde ini e- empe a u e condi ions, accoun ing
o long- ange dipole–dipole in e ac ions (i.e., including non-
analy ical co ec ions), which esul in LO–TO spli ing nea he
ecip ocal space poin G. Fig. 2b shows he co esponding sho -
and long-wa eleng h phonon band-gap co ec ions exp essed as a
unc ion o empe a u e, which a e always nega i e. Since in his
s udy he DE
L
g
co ec ion e m has been calcula ed using he
ma e ial’s anha monic phonon spec um, we e e o his me hod
as anha monic F o
¨hlich heo y (Me hods).
In Fig. 2b, i is obse ed ha nea oom empe a u e he DE
S
g
co ec ion is dominan and signi ican ly la ge han DE
L
g
,
app oxima ely six imes g ea e in he absolu e alue. No ably,
a T= 400 K, he o al band-gap co ec ion o Ag
3
SB amoun s
o 0.7 eV, which is o gian p opo ions, ep esen ing oughly
40% o he E
g
alue calcula ed a ze o empe a u e (excluding
quan um nuclea effec s).
Fig. 3 shows he ela i e band-gap a ia ion, e e enced o
he alue calcula ed a ze o empe a u e and exp essed as a
unc ion o empe a u e, o he ou CAP compounds Ag
3
SB ,
Ag
3
SI, Ag
3
SeB and Ag
3
SeI. In all cases, he band gap signi i-
can ly dec eases as he empe a u e inc eases (Table 1). The
ela i e T-induced E
g
educ ion is la ges o Ag
3
SeB and
smalles o Ag
3
SI. In pa icula , nea oom empe a u e, he
band gap o Ag
3
SB and Ag
3
SI is educed by 39% and 29% while
hose o Ag
3
SeB and Ag
3
SeI dec ease by 56% and 38%,
espec i ely (Fig. 3). As shown in Table 1, he ag eemen
be ween he expe imen al and heo e ical E
g
alues o Ag
3
SB
and Ag
3
SI imp o es as he empe a u e inc eases. In Ag
3
SeB
and Ag
3
SeI, he liquid phase is s abilized o e he c ys al phase
a mode a e empe a u es (Fig. 3c and d); hus no band gaps
we e es ima ed o hese wo compounds unde T4400 K
condi ions.
No ably, ou heo e ical E
g
esul s ob ained a T= 400 K a e
ully consis en wi h he a ailable expe imen al da a ob ained
a oom empe a u e. This excellen ag eemen nea ambien
condi ions s ongly sugges s ha he neglec o EPC effec s is
he main eason o he huge heo e ical–expe imen al E
g
disc epancies discussed in he In oduc ion. The T-induced
ela i e band-gap eno maliza ion ound in CAP a e o gian
p opo ions, anging om 20 o 60% nea oom empe a u e,
se ing a new eco d p e iously held by molecula c ys als,
which exhibi ed a 15 o 20% band-gap eno maliza ion o
simila empe a u e anges.
16
Table 1 also p esen s he alue o he DE
S
g
and DE
L
g
co ec ion
e ms es ima ed o each CAP a h ee diffe en empe a u es.
In all cases, bo h he sho - and long-wa eleng h phonon
Fig. 2 Anha monic phonon spec um and he mal band-gap co ec ions es ima ed o he a che ypal CAP Ag
3
SB . (a) Anha monic phonon spec um
ob ained a T= 200 K neglec ing (black solid lines) and conside ing ( ed dashed lines) non-analy ical co ec ions (NAC). (b) Sho - and long-wa eleng h
phonon band-gap co ec ions, DE
S
g
and DE
L
g
, espec i ely, exp essed as a unc ion o empe a u e (excluding quan um nuclea effec s). The sho - ange
co ec ion e m was e alua ed a se e al empe a u e poin s (blue ci cles and e o ba s); as a guide o he eye, he DE
S
g
da a poin s we e i ed o an
a bi a y polynomial unc ion (blue dashed line). Calcula ions we e pe o med a he HSEsol + SOC le el.
35
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co ec ions a e nega i e, wi h he o me e m conside ably
su passing he la e in absolu e alue. Fo example, a T=
400 K, he sho - ange band-gap co ec ions a e se en and ou
imes la ge han he long- ange ones calcula ed o Ag
3
SeB
and Ag
3
SI, espec i ely. As he empe a u e is aised, he size
o he wo band-gap co ec ion e ms inc eases in absolu e
alue, wi h |DE
L
g
| exhibi ing he la ges ela i e enhancemen
(e.g., app oxima ely a 320% ela i e inc ease o Ag
3
SB om
200 o 600 K).
To p o ide u he insigh s in o he impac o he mal
effec s on he elec onic band s uc u e o CAP, we also exam-
ined how band mo phology e ol es wi h empe a u e. Since
band-gap eno maliza ion in his s udy is assessed using he
ini e-diffe ences app oach based on AIMD simula ions, i is
con enien o un old he ene gy bands calcula ed o he supe -
cell in o he ecip ocal space o he p imi i e uni cell.
44
This
was done using he Easyun old so wa e,
45
ocusing on he
ep esen a i e CAP compound Ag
3
SB (Fig. S1, ESI†).
Speci ically, we analyzed i e unco ela ed supe cell snap-
sho s ex ac ed om a long AIMD ajec o y (B100 ps) a T=
200 K. Fo his pa icula case, he PBEsol exchange–co ela ion
unc ional was employed due o he e y high compu a ional
cos o pe o ming hyb id unc ional calcula ions on supe cells.
Reassu ingly, we e i ied ha he band s uc u es ob ained
using semilocal and hyb id unc ionals a e p ac ically equi-
alen in mo phology (Fig. S2, ESI†).
Table 1 Theo e ical band gaps o CAP as a unc ion o empe a u e. E
g
alues we e ob ained a ze o empe a u e (excluding quan um nuclea effec s) a
T= 200, 400 and 600 K. Calcula ions we e pe o med a he HSEsol + SOC le el.
35
Nume ical unce ain ies a e p o ided, which mainly esul om he
DE
S
g
co ec ion e m. Sho - and long-wa eleng h phonon band-gap co ec ions, DE
S
g
and DE
L
g
, espec i ely, a e p o ided a each empe a u e (excluding
quan um nuclea effec s). The expe imen al band gaps measu ed a oom empe a u e o Ag
3
SB and Ag
3
SI
30
a e shown o compa ison
CAP E
0K
g
[eV] E
200K
g
[eV] DE
S
g
[meV] DE
L
g
[meV] E
400K
g
[eV] DE
S
g
[meV] DE
L
g
[meV] E
600K
g
[eV] DE
S
g
[meV] DE
L
g
[meV] E
exp
g
[eV]
Ag
3
SB 1.8 1.3 0.1 440 42 1.1 0.1 570 108 0.9 0.2 680 175 1.0
Ag
3
SI 1.4 1.1 0.1 260 29 1.0 0.1 290 75 0.8 0.1 490 122 0.9
Ag
3
SeB 1.6 0.9 0.1 630 42 0.7 0.1 770 105 Liquid — — —
Ag
3
SeI 1.3 0.9 0.1 370 37 0.8 0.2 400 90 Liquid — — —
Fig. 3 Tempe a u e-induced ela i e band-gap a ia ion in CAP. Pe cen ages a e e e enced o he band gap calcula ed a T= 0 K condi ions (excluding
quan um nuclea effec s), namely, DE
g
(T)=E
g
(T)E
g
(0), o (a) Ag
3
SB , (b) Ag
3
SI, (c) Ag
3
SeB , and (d) Ag
3
SeI. E o ba s indica e nume ical unce ain ies
and dashed lines a e a guide o he eye. Shaded a eas indica e egions o he modynamic s abili y o he liquid phase ( heo y). Calcula ions we e
pe o med a he HSEsol + SOC le el.
35
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Ou esul s indica e ha he mal effec s lead o a downwa d
shi o he conduc ion band minimum, which emains loca ed
a he Gpoin , which is consis en wi h he band-gap ends
obse ed in he co esponding elec onic densi y o s a es.
Howe e , he ionic diso de induced by la ice ib a ions causes
a no iceable la ening o he alence band nea i s op. As a
esul , he alence band maximum becomes poo ly de ined,
unlike in he s a ic case (Fig. S1, ESI†). The consis ency
o esul s ac oss he i e ionically diso de ed con igu a ions
sugges s ha his small sampling is sufficien o cap u e he key
empe a u e-induced changes in band mo phology.
The mal effec s on he op ical abso p ion coefficien o CAP
Following a simila app oach o ha used o he calcula ion o
T- eno malized band gaps (i.e., pe o ming AIMD simula ions
wi h a supe cell and a e aging he quan i y o in e es o e se e al
o he gene a ed con igu a ions), we de e mined he equency-
dependen complex dielec ic enso o CAP unde Ta0K
condi ions (Me hods and Supplemen a y me hods, ESI†), employ-
ing linea esponse heo y. F om he a e age dielec ic enso , we
compu ed se e al mac oscopic op ical p ope ies like he op ical
abso p ion coefficien , a(o), e ac i e index, and e lec i i y.
46
Fig. 4 shows he op ical abso p ion spec a es ima ed o
CAP as a unc ion o inciden ligh wa eleng h and empe a u e.
I is ound ha ais signi ican ly enhanced unde inc easing
empe a u e,insomecasesbyasmuchasano de o magni ude.
Simila ly o he band gap, he T-induced op ical abso p ion
a ia ions a e he la ges o Ag
3
SeB ( o which aB10
3
–
10
5
cm
1
a ze o empe a u e and B10
4
–10
6
cm
1
a 200 K)
and smalles o Ag
3
SI ( o which aB10
3
–10
5
cm
1
a any
empe a u e). I is also no ed ha he mos signi ican op ical
abso p ion changes gene ally occu a low empe a u es, ha is,
wi hin he 0 T 200 K in e al. These T-induced a ends align
well wi h he ema kably la ge in luence o he EPC on he band
gap, unde sco ing he c i ical ole o he mal eno maliza ion
e ec s on he op oelec onic p ope ies o CAP.
Un o una ely, we canno di ec ly compa e ou heo e ical
a(o) esul s wi h expe imen al da a, as such da a a e no
a ailable in he li e a u e. No ably, Can
˜oe al.
30
measu ed he
op ical abso p ion coefficien o CAP ilms scaled by hei laye
hickness, d, speci ically,
aad. Howe e , since he hickness
o he syn hesized CAP ilms was no de e mined in wo k,
30
we
canno access he physical quan i y o in e es . In his ega d,
pe o ming new op oelec onic expe imen s on CAP ilms
ac oss a b oad ange o empe a u es, including he low-T
egime, would be highly desi able.
EPC mechanisms in CAP
We ha e al eady shown ha empe a u e and EPC effec s
a e essen ial o unde s anding he he mal e olu ion o he
Fig. 4 Op ical abso p ion coefficien (a) o CAP calcula ed a diffe en empe a u es as a unc ion o pho on ene gy. (a) Ag
3
SB , (b) Ag
3
SI, (c) Ag
3
SeB ,
and (d) Ag
3
SeI. Solid lines ep esen he es ima ed a e age alues and s a is ical e o s a e indica ed wi h shaded hick cu es. The ainbow-colo ed
egion deno es pho ons wi h ene gy in he isible spec um. Calcula ions we e pe o med a he HSEsol + SOC le el.
35
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op oelec onic p ope ies o CAP and o achie ing a consis en
ag eemen be ween i s -p inciples calcula ions and oom-
empe a u e expe imen s. Nex , we ocus on un a eling he
p ima y ionic–elec onic mechanisms unde lying hese key
empe a u e and EPC e ec s.
As shown in Fig. 5a, he in luence o each o he i een G
phonon modes on he band gap o Ag
3
SB was analysed by
moni o ing he change in E
g
d i en by ozen-phonon eigen-
mode dis o ions o inc easing ampli ude, u. The Gphonons
we e classi ied in o acous ic (A), op ical pola (P) and op ical
nonpola (NP), whe e he P phonons b eak he in e sion
symme y o he cen osymme ic cubic Pm%
3mphase. I was
ound ha E
g
is un esponsi e o acous ic phonon dis o ions,
as expec ed, while op ical P phonons p oduce he la ges band-
gap a ia ions. As he ampli ude o he op ical phonon dis o -
ions inc eases, E
g
sys ema ically dec eases in bo h he P and
NP cases.
Fig. 5b shows he alue o he de i a i e o he band gap wi h
espec o he phonon dis o ion ampli ude, u, exp essed as a
unc ion o he phonon eigenmode ene gy (as ob ained om
T- eno malised phonon calcula ions, Me hods). We ound ha
low-ene gy pola phonon modes (B10 meV) cause he mos
signi ican band-gap educ ions, ollowed by high-ene gy la ice
ib a ions o he same ype (B200 meV). A oom empe a u e,
phonon exci a ions wi h he lowes ene gy hos he highes
popula ions and, consequen ly, ep esen he mos cha ac e -
is ic la ice ib a ions in he c ys al. The e o e, based on he
esul s shown in Fig. 3 and 5, we conclude ha low-ene gy pola
phonon modes a e p ima ily esponsible o he subs an ial
empe a u e-induced E
g
educ ion epo ed in his s udy o
CAP compounds.
The eigenmode o he op ical P phonon wi h he lowes
ene gy is ep esen ed in Fig. 5b. As obse ed he ein, his
ozen-phonon la ice dis o ion educes he dis ance be ween
he cen al sul u a om and one adjacen sil e a om (Ag2),
while inc easing he o he wo S–Ag1 and S–Ag3 bond leng hs,
compa ed o he undis o ed cubic uni cell. Fig. 5c summa izes
he ela i e bond leng h a ia ion, in absolu e alue, o all
pai s o a oms esul ing om each o he i een Gphonon
modes calcula ed o he cubic Pm%
3mphase. As shown he ein,
he op ical P phonons p oduce he la ges S–Ag dis ance
changes (up o 20%), while he op ical NP phonons cause he
la ges B –Ag bond leng h a ia ions (up o 12%). The B –Ag
bond leng hs a e also app eciably impac ed by he op ical P
phonons (5–10%). This gene al beha iou is eminiscen o ha
obse ed o op ical pola phonons in model pe o ski e oxides
like BaTiO
3
(wi h a omic subs i u ions Ag 2O, S 2Ti and
B 2Ba).
47,48
A e iden i ying he phonon modes ha unde pin he gian
T-induced band-gap educ ion epo ed in his s udy o CAP,
speci ically low-ene gy op ical P modes, we u he analyse he
induced changes in he elec onic band s uc u e. Fig. 6a shows
he elec onic densi y o s a es calcula ed o he a che ypal
compound Ag
3
SB (equilib ium geome y). I is obse ed ha
he op o he alence band (VB) is domina ed by highly
hyb idized sil e d and chalcohalide p elec onic o bi als, while
Fig. 5 Phonon-induced band-gap a ia ion es ima ed o he a che ypal CAP Ag
3
SB . (a) Band gap as a unc ion o he la ice dis o ion ampli ude u o
acous ic, pola op ical (P) and non-pola op ical (NP) Gphonons. (b) De i a i e o he band gap wi h espec o he phonon dis o ion ampli ude
calcula ed a u
0
= 0.4 Å and exp essed as a unc ion o he phonon ene gy. The eigenmode o he op ical pola Gphonon ende ing he la ges band-gap
de i a i e in absolu e alue is ske ched: Ag, S and B a oms a e ep esen ed wi h g ey, yellow and b own sphe es, espec i ely. Calcula ions we e pe o med a
he HSEsol + SOC le el.
35
(c) Gphonon-induced ela i e bond leng h dis o ions in he cubic Pm
%
3mphase o a dis o ion ampli ude o 0.4 Å.
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he bo om o he conduc ion band (CB) is domina ed by
iso opic and mo e delocalized S and Ag s o bi als. The elec o-
nic band s uc u e in Fig. 6b shows ha he VB co esponds o
he high-symme y ecip ocal space poin M (1/2,1/2,0), while
he CB o he cen e o he B illouin zone, G(0,0,0); hus he
band gap o Ag
3
SB is indi ec (we ha e checked ha he same
conclusion applies o he es o CAP compounds analyzed in
his s udy).
The effec s on he elec onic band s uc u e esul ing om a
ozen-phonon la ice dis o ion co esponding o he lowes -
ene gy op ical P eigenmode (u
0
= 0.4 Å) a e wo old (Fig. 6b).
Fi s , due o he b eaking o phonon-induced in e sion sym-
me y, he ene gy band degene acy a he ecip ocal space poin
M is li ed. Howe e , he band gap o he sys em is unaffec ed
by his ene gy degene acy li ing effec since he VB emains
p ac ically in a ian . Second, he CB edge expe iences a signi-
ican dec ease in ene gy and, as a consequence, he band
gap o he sys em is educed by app oxima ely 30%. The e o e,
we may conclude ha he gian T-induced E
g
educ ion
epo ed in his s udy o CAP is p ima ily caused by low-
ene gy pola phonon modes ha induce a p onounced CB
ene gy dec ease.
To be e unde s and he elec onic o igins o he op ical P
phonon-induced CB ene gy lowe ing, we cons uc ed a igh -
binding (TB) model based on Wannie unc ions ha accu a ely
ep oduces ou DFT band s uc u e esul s (Me hods and
Fig. S3, ESI†). Speci ically, he TB model consis s o s, p, and
d o bi als o he i e a oms in he uni cell, esul ing in a o al
o 45 dis inc Wannie o bi als. Consis en ly, he TB model
ep oduces he dominan Ag and S s cha ac e o he CB and i s
Fig. 6 Elec onic band s uc u e p ope ies o he a che ypal CAP Ag
3
SB . (a) Elec onic densi y o s a es calcula ed o he equilib ium cubic Pm
%
3m
phase. Calcula ions we e pe o med a he HSEsol + SOC le el.
35
(b) Elec onic band s uc u e calcula ed o he equilib ium and phonon dis o ed
(i.e., conside ing he lowes -ene gy Gop ical P mode wi h ampli ude u
0
= 0.4 Å) cubic Pm
%
3mphase; in bo h cases, he ene gy bands a e e e ed o a
same ene gy o igin, a deep co e elec onic le el ha emains unaffec ed by he dis o ion (Me hods). (c) and (d) The conduc ion band nea i s minimum a
Gcompu ed wi h a TB model o sil e and sul u a oms (Me hods); solid and dashed lines co espond o he equilib ium and dis o ed s uc u es,
espec i ely. (e) Ske ch illus a ing he mechanism o band-gap closu e induced by low-ene gy pola so phonon modes in CAP. Upon phonon
dis o ion, he hyb idiza ion o sil e and sul u s elec ons in he conduc ion band is enhanced, lowe ing (inc easing) he ene gy o he co esponding
bonding s(an ibonding s*) s a e.
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ene gy lowe ing unde he pola la ice dis o ion o in e es
(Fig. 6c and d).
Acco ding o his TB model, he impac o he ozen-
phonon dis o ion on he Ag and S s conduc ion o bi als is
wo old. Fi s , he diffe ence in hei kine ic ene gies, co es-
ponding o he diagonal Hamil onian ma ix elemen s diffe -
ence |hAg s|H|Ag sihSs|H|S si|, dec eases (Fig. S3, ESI†). And
second, he hopping s e m in ol ing he Ag2 and S a oms,
ep esen ed by he off-diagonal Hamil onian ma ix elemen
hAg2 s|H|S si, inc eases (Fig. S3, ESI†). The gene al physical
in e p e a ion ha ollows om hese TB esul s is ha he
pola ozen-phonon dis o ion enhances he hyb idiza ion o
Ag2 and S s conduc ion o bi als, which lowe s and inc eases he
ene gy o he co esponding bonding (s) and an ibonding (s*)
s a es, espec i ely. Consequen ly, he CB, which is domina ed
by he Ag2-S sin e ac ion, is lowe ed. The e ealed EPC
mechanism, which o e all p oduces a band-gap educ ion, is
schema ically ep esen ed in Fig. 6e.
Discussion
The mal expansion is ano he physical mechanism ha can
in luence he band gap o ma e ials,
40
bu i was no explici ly
conside ed in his s udy due o i s high compu a ional cos .
Howe e , we pe o med a se ies o es s in which we a bi a ily
inc eased and dec eased he uni cell olume o Ag
3
SB and
es ima ed he co esponding ela i e E
g
a ia ion (Fig. S4 and
Table S1, ESI†). We ound ha , when he olume o he sys em
inc eases by a easonable B1%, as migh occu due o he mal
expansion nea oom empe a u e, he band gap dec eases by
only B40 meV (i.e., an o de o magni ude smalle change han
sho -wa eleng h phonon co ec ions, Table 1). Addi ionally,
he mal expansion likely con ibu es o he so ening o op ical
P modes, inc easing hei popula ion and consequen ly enhan-
cing he E
g
educ ion due o EPC. The e o e, while he mal
expansion effec s we e omi ed in ou calcula ions, he epo ed
esul s likely ep esen a lowe bound o he ac ual effec , and ou
conclusions o CAP compounds emain obus and accu a e.
One may wonde whe he , in addi ion o sil e chalcohalide
an ipe o ski es, he e exis o he amilies o ma e ials exhibi -
ing simila ly la ge T- eno maliza ion effec s on he band gap
and op ical abso p ion coefficien . As discussed in p e ious
sec ions, he pola na u e o low-ene gy op ical phonons
appea s o be essen ial in his ega d. Consequen ly, a en a i e
se o necessa y condi ions o iden i ying po en ial ma e ials
ha display simila T-induced effec s on he op oelec onic
p ope ies may include dielec ic ma e ials exhibi ing (1) cen-
osymme ic c ys alline phases, (2) low-ene gy o e en imagin-
a y op ical pola phonons, and (3) highly hyb idized and
delocalized elec onic o bi als nea he Fe mi ene gy le el.
The a ailabili y o la ge DFT calcula ions and phonon da a-
bases may enable high- h oughpu ma e ial sc eening o such a
kind.
49,50
Fe oelec ic oxide pe o ski es, exempli ied by he a che y-
pal compounds S TiO
3
(STO) and BaTiO
3
(BTO), appea o
sa is y he se o necessa y condi ions ou lined abo e. No ably,
a signi ican band-gap modula ion has been epo ed o STO
unde biaxial s ain condi ions, al hough his phenomenon
a ises om diffe en physical mechanisms han hose iden i-
ied in his s udy o CAP compounds (i.e., ene gy degene acy
li ing due o symme y b eaking).
51
Mo eo e , he expe i-
men al oom- empe a u e band gap o BTO (E3.2 eV
52
) shows
subs an ial disag eemen wi h ze o- empe a u e heo e ical
es ima es ob ained wi h hyb id unc ionals (E4.0 eV
53
), high-
ligh ing an expe imen – heo y inconsis ency simila o ha
desc ibed o Ag
3
SB and Ag
3
SI in he In oduc ion. Addi ionally,
hebandgapo hemul i e oicoxidepe o ski eBiFeO
3
exhibi s
a ema kable empe a u e-dependen sh inkage, dec easing by
app oxima ely 50% wi hin he empe a u e ange 300 T
1200 K,
54
likely in luenced by he magne ic deg ees o
eedom.
55
These indings sugges ha he empe a u e e ec s
and EPC mechanisms iden i ied in his s udy o CAP com-
pounds may ha e b oade ele ance, po en ially ex ending o
o he well-known amilies o unc ional ma e ials. Theo e ical
in es iga ions explo ing his possibili y a e cu en ly unde way.
The pola na u e o he op ical phonon modes, which cause
he signi ican T-induced educ ion in he E
g
o CAP, opens up
exci ing echnological possibili ies. Simila o how an elec ic
ield can s abilize a pola phase wi h e oelec ic pola iza ion
o e a pa aelec ic s a e a cons an empe a u e h ough a
phase ans o ma ion,
56
i is likely ha pola op ical phonons
in CAP can also be s imula ed using ex e nal elec ic ields.
This possibili y implies ha he op oelec onic p ope ies o
CAP could be effec i ely uned by applying an elec ic ield
a he han al e ing he empe a u e, p o iding a mo e p ac-
ical app oach o he de elopmen o ad anced op ical de ices
and o he echnological applica ions. Expe imen al alida ion
o his hypo hesis would be highly aluable.
Finally, ad ances in ligh sou ces and ime- esol ed spec o-
scopy ha e made i possible o exci e speci ic a omic ib a ions
in solids and o obse e he esul ing changes in hei elec onic
and elec on–phonon coupling p ope ies.
57–59
These de elop-
men s also sugges he possibili y o uning he op oelec onic
p ope ies o CAP, as well as o simila ma e ials like oxide
pe o ski es,
60–62
h ough speci ic phonon exci a ions using
op ical means such as lase s. This app oach may simpli y he
design and manu ac u e o p ac ical se ups by elimina ing he
need o elec ode deposi ion. The e o e, he esul s p esen ed
in his wo k a e signi ican no only om a undamen al
pe spec i e bu also o en isioning po en ial echnological
applica ions in which he op ical and elec onic p ope ies
o ma e ials could be e ec i ely uned by ex e nal ields and
pho oexci a ion.
Conclusions
In his s udy, we ha e explo ed he empe a u e effec s on he
band gap o sil e chalcohalide an ipe o ski es (CAP), speci i-
cally Ag
3
XY compounds (X = S, Se; Y = B , I), which a e
p omising o ene gy and pho onic applica ions due o hei
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