Excited-state band structure mapping

PHYSICAL REVIEW B 105, 075417 (2022)
Exci ed-s a e band s uc u e mapping
M. Puppin ,1,2,*C. W. Nicholson ,3C. Monney,3Y. Deng,4R. P. Xian ,2J. Feldl ,2S. Dong ,2A. Dominguez,5,6
H. Hübene ,7A. Rubio,7,8,9M. Wol ,2L. Re ig ,2and R. E ns o e 2,10,†
1Labo a oi e de Spec oscopie Ul a apide and Lausanne Cen e o Ul a as Science (LACUS), École Poly echnique Fédé ale de Lausanne,
ISIC, S a ion 6, CH-1015 Lausanne, Swi ze land
2F i z-Habe -Ins i u de Max-Planck-Gesellscha , Fa adayweg 4-6, 14195 Be lin, Ge many
3Depa men o Physics and F ibou g Cen e o Nanoma e ials, Uni e si y o F ibou g, Chemin du Musée 3, CH-1700 Swi ze land
4Paul Sche e Ins i u e, SwissFEL, 5232 Villigen PSI, Swi ze land
5Shenzhen JL Compu a ional Science and Applied Resea ch Ins i u e (CSAR), Shenzhen 518110, China
6Beijing Compu a ional Resea ch Cen e (CSRC), Beijing 100193, China
7Max Planck Ins i u e o he S uc u e and Dynamics o Ma e and Cen e o F ee Elec on Lase Science, Lu upe Chaussee 149,
Geb. 99 (CFEL), 22761 Hambu g
8Cen e o Compu a ional Quan um Physics, Fla i on Ins i u e, 162 5 h A enue, New Yo k, New Yo k 10010, USA
9Nano-Bio Spec oscopy G oup, Uni e sidad del Paìs Vasco UPV/EHU, 20018 San Sebas ián, Spain
10Ins i u ü Op ik und A oma e Physik, Technische Uni e si ä Be lin, S aße des 17, Juni 135, 10632 Be lin, Ge many
(Recei ed 17 Augus 2021; e ised 6 Decembe 2021; accep ed 26 Janua y 2022; published 17 Feb ua y 2022)
Angle- esol ed pho oelec on spec oscopy is an ex emely powe ul p obe o ma e ials o access he occupied
elec onic s uc u e wi h ene gy and momen um esolu ion. Howe e , i emains blind o hose dynamic s a es
abo e he Fe mi le el ha de e mine echnologically ele an anspo p ope ies. In his wo k we ex end band
s uc u e mapping in o he unoccupied s a es and ac oss he en i e B illouin zone by using a s a e-o - he-a
high epe i ion a e, ex eme ul a iole em osecond ligh sou ce o p obe op ically exci ed samples. The wide-
anging applicabili y and powe o his app oach a e demons a ed by measu emen s on he wo-dimensional
semiconduc o WSe2, whe e he ene gy-momen um dispe sion o alence and conduc ion bands a e obse ed in a
single expe imen . This p o ides a di ec momen um- esol ed iew, no only on he comple e ou -o -equilib ium
band gap bu also on i s eno maliza ion induced by elec onic sc eening. Ou wo k es ablishes a benchma k
o measu ing he band s uc u e o ma e ials, wi h di ec access o he ene gy-momen um dispe sion o he
exci ed-s a e spec al unc ion.
DOI: 10.1103/PhysRe B.105.075417
I. INTRODUCTION
Func ionali y in elec onic and op oelec onic de ices is
based on he con ol o he low o cha ge ca ie s unde ou -
o -equilib ium condi ions. A he mic oscopic le el, cha ge
anspo and de ice ope a ion ely upon gene a ing nonequi-
lib ium elec on dis ibu ions con olled by ex e nal ields o
achie e he desi ed elec onic esponse. The p opaga ion o
elec ons in a c ys al and he e olu ion o hei ene gy dis i-
bu ions a e go e ned by he de ails o he elec onic s uc u e,
as well as he e iciency o elas ic and inelas ic sca e ing
p ocesses.
Time- esol ed angle- esol ed pho oemission spec oscopy
( ARPES) add esses his p oblem by obse ing he spec-
*[email p o ec ed]
†[email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI. Open
access publica ion unded by he Max Planck Socie y.
al unc ion o a ma e ial a e exci a ion ia a em osecond
op ical pulse [1]. The momen um- esol ed dis ibu ion o
exci ed s a es combined wi h he dynamical in o ma ion on
s a e li e imes p o ides a powe ul iew in o exci ed solids
[2], ex ending he scope o ARPES and allowing o obse e
ou -o -equilib ium elec onic p ope ies which can be used
o ex ac he elec onic coupling wi h phonons and o he
deg ees o eedom [3,4]. Ul ima ely, unde s anding ma e
ou o equilib ium is manda o y o achie ing op ical con-
ol in complex ma e ials [5]. Time- esol ed pho oelec on
spec oscopy can esol e s a es unoccupied a equilib ium and
has been ex ensi ely used o e eal image po en ial s a es
a su aces [6,7] and exci ons in semiconduc o s and molec-
ula adso ba es [8,9]. Mo e ecen ly, ARPES was used o
e eal he unoccupied band s uc u e o opological ma e i-
als [10], o measu e op ically d essed s a es [11], o obse e
elec on popula ion li e imes and spin- alley pola iza ion in
he conduc ion band o ansi ion-me al dichalcogenide semi-
conduc o s [12–15] and has enabled he di ec obse a ion o
exci ons [2,16,17].
Ene gy-momen um dispe sion o exci ed s a es can be de-
e mined om ARPES da a, bo h abo e and below he Fe mi
le el, p o iding expe imen al access o quan i ies such as he
2469-9950/2022/105(7)/075417(12) 075417-1 Published by he Ame ican Physical Socie y
M. PUPPIN e al. PHYSICAL REVIEW B 105, 075417 (2022)
band-gap [18–20] and conduc ion-band ca ie e ec masses
[21]. An impo an open ques ion is how band p ope ies
ex ac ed om he ARPES spec al unc ion in he exci ed
s a e compa e wi h con en ional s eady-s a e expe imen s,
e.g., op ical spec oscopy o ARPES.
A common expec a ion is ha a compa ison is possible in
he weak exci a ion limi [22] whe e ARPES expe imen s
become e y challenging, pa icula ly when accessing he ull
B illouin zone (BZ) o he in es iga ed ma e ial. This is be-
yond he each o mos ARPES expe imen s, which a e pe -
o med a UV pho on ene gies. Ex ending hese expe imen s
o he XUV pho on ene gy ange and co espondingly, o high
pho oelec on momen a co e ing he whole BZ, while e ain-
ing a compa able signal- o-noise a io and weak exci a ion
densi ies, has been challenging un il he ecen de elopmen
o sui able high- epe i ion- a e XUV sou ces [22–26].
In his wo k we employ a s a e-o - he-a expe imen al
se up [23] o simul aneously de e mine he ene gy o con-
duc ion s a es (unoccupied a equilib ium) and alence s a es.
This allows us o add ess he band gap, one o he undamen-
al op oelec onic p ope ies, by mapping in ecip ocal space
bo h alence and conduc ion bands o 2H −WSe2,a wo-
dimensional ansi ion-me al dichalcogenide (TMD) semi-
conduc o widely s udied o exci onic and spin- alley onic
applica ions [27–29]. The conduc ion-band popula ion is
p obed wi h a 21.7-eV XUV pulse ollowing pho oexci a ion
by a 3.1-eV pulse, wi h a empo al esolu ion be e han
100 s. This enables exci ed-s a e ARPES measu emen s
be o e ene gy elaxa ion o he conduc ion-band minimum,
e ealing he ene gy e sus momen um dispe sion o a-
lence and conduc ion s a es in a single expe imen , including
high-ene gy conduc ion elec onic s a es a om he band
edge, go e ning he high-ene gy op oelec onic p ope ies
o TMD semiconduc o s [30,31]. As compa ed o p e i-
ous expe imen s on simila TMD compounds [12–15,18,20],
exci ed-s a e band mapping is pe o med in he whole i e-
ducible pa o he B illouin zone, a he han along a ini e
numbe o high-symme y di ec ions, and in a weak exci a ion
egime, whe e he exci ed-s a e band gap and i s eno maliza-
ion due o many-body e ec s a e s udied. We demons a e
ha in he low-exci a ion limi he ARPES gap ag ees wi h
he band gap measu ed by o he spec oscopies and p edic ed
by heo y. This alida es exci ed-s a e band s uc u e mapping
as a gene ally applicable me hod o measu e, wi h momen um
esolu ion, he conduc ion s a es o ma e ials.
II. EXCITED-STATE BAND STRUCTURE MAPPING
To be e unde s and he di e ence and simila i ies be-
ween ARPES and ARPES, we sho ly e iew he wo expe -
imen al app oaches. In an ARPES expe imen , a pho on wi h
ene gy hνexci es a single-c ys alline sample, and he kine ic
ene gy Eo pho oelec ons is measu ed along a wa e- ec o
di ec ion k. I pho oioniza ion is ea ed as a sudden p ocess,
he pho oemission in ensi y can be app oxima ed as [32]
I(k,E)=I0(k,E)A−(k,E) μ,T(E).(1)
Equa ion (1), which o simplici y neglec s he expe i-
men ally ini e angula and ene gy esolu ion, as well as
cha ge anspo a he su ace, links he ARPES spec um
I(k,E) o he unde lying elec onic s uc u e ia h ee ac o s.
The one-elec on- emo al spec al unc ion, A−(k,E), con-
ains he in o ma ion abou he quasipa icle band s uc u e
and many-body in e ac ions. The spec al weigh is modu-
la ed by a ma ix elemen e m I0(k,E), which depends on
ini ial- and inal-s a e symme y and wa e ec o s, as well
as pho on ene gy (hν) and pola iza ion, and he expe imen-
al geome y [33,34]. Thi dly, he Fe mi-Di ac dis ibu ion
μ,T(E) imposes ha only s a es popula ed a he empe -
a u e Tcan con ibu e o he measu ed spec um, se ing a
limi o he highes accessible ene gy o ew kBTabo e he
chemical po en ial μ. The ma ix elemen e m is anishing
unless momen um conse a ion pa allel o he sample’s su -
ace is ul illed by he escaping pho oelec on, allowing o
link he measu ed pho oelec on angula dis ibu ion I(k,E)
o he quasipa icle bands in ecip ocal space, as illus a ed
in Fig. 1(a). Pa allel momen um (k) conse a ion, oge he
wi h ene gy conse a ion, imposes ha ypically only ene -
ge ic pho ons in he XUV ange can access he whole BZ
[35]. As an example, pho ons wi h an ene gy o ≈20 eV
a e necessa y o measu e he i s BZ bounda y o WSe2,as
indica ed by he iole dashed line in Fig. 1(a). In ou expe i-
men pho oelec on spec a a e collec ed wi h a hemisphe ical
ene gy analyze (HEA) which measu es kine ic ene gy (EK)
and angle o emission along he en ance sli [Fig. 1(b)], which
co esponds o a line-cu h oughou he unc ion I(k,E) [ ull
g een lines in Fig. 1(a)]. Band mapping is achie ed by an-
gula scanning o he sample [g een a ows in Figs. 1(a) and
1(b)] ac oss he analyze sli . The mul idimensional unc ion
I(k,E) is cons uc ed om di e en images, and da a can
be displayed as cons an ene gy cu s o as ene gy e sus
momen um plo s, as shown in Fig. 1(a), whe e a ho izon al
cons an ene gy cu close o he alence-band maximum and a
e ical ene gy e sus momen um dispe sion ac oss he BZ
a e plo ed. I is wo h no ing he al e na i e app oach o
momen um mic oscopy in which he whole accessible pho-
oemission space is collec ed a he same ime [36]. A de ailed
compa ison be ween he wo me hods e eals ha an HEA
ensu es highe coun ing s a is ics when acqui ing da a along
a speci ic di ec ion [37], whe eas he ixed geome y p o ided
by momen um mic oscopy is sui able o he s udy o he
symme y-dependen ma ix elemen I0(k,E)[34].
A ime- esol ed ARPES expe imen accesses an exci ed
s a e o he ma e ial by pe o ming an ARPES expe imen a a
well-de ined empo al delay ollowing a em osecond op ical
pump pulse [Fig. 1(b)]. The ARPES spec um ˜
I(k,E, )
he eby measu es he (quasi)-elec on- emo al spec um as a
unc ion o his ime delay:
˜
I(k,E, )=˜
I0(k,E, )˜
A−(k,E, )˜
(k,E, ).(2)
He e Eq. (1) is modi ied o include he explici ime de-
pendence o each e m. The op ical exci a ion p oduces no
only an ou -o -equilib ium elec onic dis ibu ion ˜
bu also
pe u bs he many-body in e ac ions in he spec al e m ˜
A−.
The ma ix elemen e m ˜
I0can become a ime-dependen
quan i y i he symme y o he ini ial o inal s a es is mod-
i ied [38]. We ollow he con en ion ha o >0 he pump
exci a ion occu s be o e pho oemission: eco e y o equilib-
ium equi es ha ˜
I(k,E, ) →+∞
−−−−→I(k,E).
075417-2
EXCITED-STATE BAND STRUCTURE MAPPING PHYSICAL REVIEW B 105, 075417 (2022)
Op ical
Pump
Conduc ion band
k-space map bounda y, 20 eV
Occupied S a es Unoccupied s a es
EF
Ene gy
Momen um
BAND STRUCTURE
MAPPING
EXCITED-STATE
BAND STRUCTURE MAPPING
Ul a as sca e ing
Time
XUV p obe e-
XUV p obe
e-
(b) (d)
(a)
XUV p obe
21.7 eV
e-
Pump
3.1 eV
Θ
e-
Conduc ion band
Angula scan
2H:WSe
HEA
k||
EK
(e)
E=-0.6 eV
-1.5
-0.5
0.5
1.5
I
CB
(a b.u)
I
VB
(a b.u.)
kx (Å-1)
-1.0
1.0
ky (Å-1)
0.0
5
4
3
2
1
4
3
2
1
0
E=1.6 eV
E=2 eV
Sli k||
-0.8 -0.4 0.0
-50 s
0.10.0
-0.8 -0.4 0.0
100 s
0.20.0
4.0
3.2
2.4
1.6
0.8
E (eV)
-0.8 -0.4 0.0
1 ps
1.00.50.0
k|| k||
Γ
M
K
∑
Γ
∑
Γ
∑Γ
∑
(c)
Γ
∑
K
M
FIG. 1. (a) Band s uc u e mapping in ecip ocal space by angle- esol ed pho oelec on spec oscopy (ARPES). The ecip ocal space
egion measu ed by he hemisphe ical ene gy analyze (HEA) o wo sample il angles is indica ed by a g een line, he maximum pa allel
momen um which can be accessed by 20 eV pho ons is indica ed by a iole dashed line. (b) ARPES expe imen s on 2H −WSe2: an op ical
pump pulse a an ene gy o 3.1 eV exci es he sys em. A a delay , an XUV p obe pulse a an ene gy o 21.7 eV gene a es pho oelec ons,
which a e measu ed as a unc ion o he emission angle θwi h a HEA. The sample angle is scanned ac oss he analyze sli o collec ARPES
maps. (c) Exci ed-s a e band s uc u e mapping. (d) ARPES da a collec ed in he conduc ion band o WSe2 o pump-p obe delays o −50 s,
100 s, and 1 ps. Inse : The su ace B illouin zone o WSe2. (e) Pho oelec on in ensi y dis ibu ion as a unc ion o pa allel momen um o
h ee ene gies a a pump-p obe delay o 100 s; VB and CB ene gy dis ibu ion cu es ha e been independen ly in ensi y no malized o be e
isualiza ion. The expe imen al da a is collec ed in a egion delimi ed by he dashed line. Ou side his egion, he esul s o G0W0calcula ions
a e displayed, and he heo e ical band dispe sion along he kzdi ec ion was in eg a ed; he conduc ion bands we e igidly o se by a scisso
ope a o o −0.16 eV o ma ch he expe imen al ene gy.
As illus a ed in Fig. 1(b), ARPES p o ides access o
s a es unoccupied a equilib ium. This can be unde s ood as
a wo-s ep p ocess, whe e in a i s s ep he em osecond
pump pulse c ea es an op ical pola iza ion in allowed mo-
men um and ene gy egions co esponding o e ical op ical
ansi ions in he ma e ial (k=0) [39]. In a second s ep,
mic oscopic sca e ing e en s wi hin a ew hund ed em osec-
onds edis ibu e he elec onic popula ion o mul iple s a es
ac oss he conduc ion band (CB) [Fig. 1(a)]. Elec ons elax
hei excess ene gy ia mul iple elec on-phonon sca e ing
e en s owa ds he band edges and accumula e a he CB min-
ima on imescales ypically sho e han a ew picoseconds. By
measu ing he pho oelec on ene gy and angula dis ibu ion
be o e signi ican ene gy elaxa ion o he la ice has occu ed,
he in o ma ion encoded in ˜
A−can be e ealed in a ange
E<μ+hνp, whe e hνpis he pump pho on ene gy.
Exci ed-s a e band mapping o unoccupied s a es is pa ic-
ula ly demanding and s ongly bene i s om high- epe i ion-
a e (>100 kHz) XUV sou ces. Fi s , a su icien ly sho XUV
pulse is undamen al o accessing he ou -o -equilib ium
s a e be o e i s decay h oughou he BZ. In addi ion, space
cha ge e ec s, which a e inhe en in ARPES wi h sho XUV
pulses, a e mi iga ed in high- epe i ion- a e expe imen s [40].
Fu he mo e, he highe he pump exci a ion ene gy den-
si y, he s onge many-body in e ac ions modi y he unc ion
˜
I(k,E, ) ela i e o he equilib ium case. ARPES expe i-
men s a high epe i ion a es bene i om highe coun ing
s a is ics and hence da a can be acqui ed a weake pe u ba-
ion s eng hs.
III. EXPERIMENTAL METHODS
To mee he simul aneous equi emen s o an ul asho
XUV sou ce wi h a high epe i ion a e, in his wo k we gene -
a e p obe pulses by high-ha monic gene a ion wi h an op ical
pa ame ic chi ped pulse ampli ie ope a ing a 500 kHz [41].
This esul s in XUV pulses a an ene gy o 21.7 eV and
wi h cha ac e is ic ime-bandwid h p oduc s o app oxima ely
20 s ×110 meV [23], which a e empo ally sho enough
o access he exci ed s a es be o e signi ican ca ie ene gy
elaxa ion has occu ed and, a he same ime, ha e an en-
e gy bandwid h su icien ly na ow o esol e he exci ed-s a e
ene gy ea u es. ARPES expe imen s we e pe o med on
single-c ys alline samples o bulk WSe2clea ed in ul ahigh
acuum condi ions. Comme cial WSe2single c ys als whe e
p epa ed by ex olia ion in si u unde UHV condi ions. The
base p essu e du ing he expe imen s was below 1 ×10−10
mba . The ma e ial was exci ed by a pump pulse wi h a pho-
on ene gy o 3.1 eV and a an exci a ion ene gy densi y o
40 μJ/cm2. All he expe imen s we e pe o med a oom em-
pe a u e, whe e no su ace pho o ol age o cha ging e ec s
we e obse ed.
To illus a e he abili y o ARPES o isualize s a es
which a e unoccupied a equilib ium, we show in Fig. 1(d)
075417-3
M. PUPPIN e al. PHYSICAL REVIEW B 105, 075417 (2022)
ene gy e sus momen um da a collec ed in an ene gy window
in he CB along he high-symme y di ec ion -K. Th ee se-
lec ed ime delays (–50 s, 100 s, and 1 ps) a e plo ed side by
side. The su ace BZ o WSe2, wi h he high-symme y poin s
ma ked, is shown as an inse o Fig. 1(d). Du ing he ising
edge o he pump pulse (–50 s), he CB signal is localized a
−0.35 Å−1 om he BZ cen e (poin ). This sugges s ha in
his egion popula ion is ans e ed ia an op ical ansi ion a
he pho on ene gy o 3.1 eV a he han indi ec ly by sca e -
ing. The in ensi y o his ea u e as a unc ion o ime was used
as a measu e o he pump-p obe empo al c oss-co ela ion,
and he empo al maximum was used o de ine he ime ze o.
The FWHM o he c oss-co ela ion is 95 s, domina ed by
he pump pulse du a ion. Fu he de ails conce ning he c oss-
co ela ion i s a e shown in he Appendix. Th oughou his
wo k, he ze o ene gy was se o con enience o he alence-
band ene gy a he Kpoin , he co ne o he hexagonal BZ.
A a ime delay o 100 s, popula ion can be obse ed
h oughou he conduc ion s a es, up o a an ene gy ≈2.5 eV
[Fig. 1(d), cen al panel]. This delay was selec ed o pe o m
he exci ed-s a e band s uc u e mapping. Relaxa ion owa ds
he conduc ion-band alley minimum is indeed al eady
appa en a a delay o 1 ps [Fig. 1(d), igh panel].
An ene gy window om −1.5 o 3.5 eV was selec ed o
simul aneously obse e alence and conduc ion bands a ound
he band gap, which is a unique ea u e o ARPES. Th ee
exempla y cons an ene gy cu s o he da a a =100 s a e
shown in Fig. 1(e), which display in alse colo s he pho oelec-
on in ensi y dis ibu ion as a unc ion o pa allel momen um
o ene gies o −0.6 eV in he alence band (VB), 1.6 eV and
2 eV in he conduc ion band (CB). The measu emen egion
is indica ed by a dashed line and comp ises he whole i s BZ
o WSe2.InFig.1(e), wo di e en alse colo scales a e used
o conduc ion and alence s a es. Ene gy dis ibu ion cu es
(EDCs) in he VB we e no malized o he same a ea as a
unc ion o pa allel momen um. This was chosen o educing
he impac o he ma ix elemen in he display o he cons an
ene gy map and o a clea e compa ison wi h heo e ical
calcula ions. The same p ocedu e was applied independen ly
o EDCs in he CB (i.e., on he da a o E >1eV),bu
p io o he a ea no maliza ion, an exponen ial backg ound
ail om he unde lying occupied s a es was sub ac ed. No
no maliza ion p ocedu e was pe o med on he da a displayed
in he o he igu es o he ex .
IV. RESULTS AND DISCUSSION
To a ionalize he expe imen al da a we pe o m ab
ini io densi y unc ional heo y (DFT) calcula ions o he
elec onic band s uc u e. The sys em was modeled using a
hexagonal supe cell wi h he expe imen al la ice cons an s
a=b=3.28 Å and c=12.98 Å [42]. DFT calcula ions
we e pe o med using he gene alized g adien app oxima ion
(GGA) wi h he Pe dew-Bu ke-E nze ho (PBE) unc ional
[43], as implemen ed in he QUANTUM ESPRESSO package
[44]. To imp o e he ag eemen wi h expe imen al da a, we
use many-body pe u ba ion heo y a he one-sho G0W0
le el [45,46] on op o he DFT esul s. The B illouin zone
was sampled wi h a 9×9×9k-poin g id, and he spin-o bi
coupling was included di ec ly in he DFT calcula ions and
pe u ba i ely a he G0W0le el using he Be keleyGW
package [47]. Finally, we pe o med DFT calcula ions using a
24×24×9 BZ sampling and in e pola ed linea ly he 9×9×9
GW band s uc u e in o his ine k-poin g id. We used a
o al o 1000 conduc ion bands and a 18 Ry ene gy cu o
o he compu a ion o he in e se dielec ic ma ix. Fo he
e alua ion o he sc eened and ba e Coulomb pa s o he
sel -ene gy ope a o , we used ene gy cu o s o 18 Ry and 160
Ry, espec i ely. All employed cu o alues, BZ sampling,
and numbe o bands we e sys ema ically and independen ly
inc eased un il esul s we e con e ged wi hin a ew ens o
meV o he conduc ion- and alence-band ene gy di e ence.
The G0W0me hod compu es quasipa icle ene gies, co ec -
ing o lowes o de he unsc eened elec onic G een’s unc ion
G0by he Coulomb in e ac ion W0. The quasipa icle ene gy
dispe sion is calcula ed as a unc ion o he h ee-dimensional
wa e ec o (kx,ky,kz). Fo a di ec compa ison wi h da a
in Fig. 1(d), he heo e ical bands a e in eg a ed along he
ecip ocal space di ec ion o hogonal o he sample su ace
(kz). This choice is jus i ied by he s ong su ace sensi i i y o
XUV-based pho oemission due o he sho mean ee pa h o
pho oelec ons. Elec on momen um conse a ion is elaxed
o he kzcomponen , adding an addi ional sou ce o ene gy
b oadening o bands wi h dispe sion ou o he su ace plane.
The e is s ong e idence ha in WSe2 he pho oemission p ob-
ing dep h a 21.7 eV is mos ly limi ed o he uppe mos laye
(≈0.5 nm), in ac , in e sion-symme ic WSe2su p isingly
exhibi s s ong spin-pola ized bands [48] and alley pola -
iza ion in ci cula ly pumped -ARPES [15]. The impo ance
o inal-s a e e ec s in he ma e ial is e idenced by one-s ep
pho oemission calcula ions [34] and will be discussed u he
below.
The expe imen al da a con ains he exci ed-s a e CB and
VB ene gy-momen um dispe sion o a bi a y ecip ocal
space di ec ions, which can be compa ed wi h ou ab ini io
calcula ions and wi h o he expe imen s. Fo his pu pose,
ene gy e sus momen um pho oelec on dis ibu ions a e plo -
ed along h ee high-symme y di ec ions --K,K-M,M-
in Fig. 2and compa ed wi h he esul s o he calcula-
ions. The heo e ical kzdispe sion is indica ed by a shading,
highligh ing wo-dimensional (low-kzdispe sion) and h ee-
dimensional s a es. The expe imen al pho oelec on in ensi y
is plo ed wi hou addi ional no maliza ion, and in ensi y
modula ions a e a ibu able o he momen um-dependen ma-
ix elemen . The a e age in ensi y o he conduc ion-band
signal is a ac o 10−3 ha o he alence s a es, and we
use wo dis inc alse colo scales o conduc ion and alence
s a es, espec i ely.
The ze o ene gy e e ence is se o he highes ene gy VB
a he Kpoin also o he heo e ical da a o minimize any
alignmen unce ain y due o kzdispe sion. The heo e ical
conduc ion s a es we e shi ed by −160 meV o ma ch he
measu ed CB ene gy a he Kpoin , bo h in Fig. 1(d) and in
Fig. 2; he same ene gy shi was applied o e e y elec on
momen um ( igid scisso ope a o ).
Theo y p edic s wo alence and wo conduc ion bands
in he obse ed ene gy window, as all calcula ed bands a e
spin degene a e, consis en wi h he in e sion-symme ic bulk
c ys al s uc u e o 2H −WSe2. The spin-o bi spli ing o he
VB band a he Kpoin is ≈500 meV, in good ag eemen
075417-4
EXCITED-STATE BAND STRUCTURE MAPPING PHYSICAL REVIEW B 105, 075417 (2022)
E-EVBK (eV)
k
||
(Å-1)
0.8
Γ∑KMΓ
ICB (a b.u.) IVB (a b.u.)
-1.5
-1.0
-0.5
0.0
0.5
1.0
0.80.0
3.5
3.0
2.5
2.0
1.5
1.0
3
2
1
0
10
5
x104
0.0 0.0
-0.16 eV
Γ
M
K
∑
FIG. 2. Measu ed ARPES in ensi y as a unc ion o ene gy and pa allel momen um showing he VB and CB along he --K,K-M,
M-di ec ions, indica ed in he uppe panel. Conduc ion-band s a es a e displayed by a di e en colo scale. Blue and ed cu es indica e he
quasipa icle ene gies calcula ed wi h he G0W0me hod o he CB and VB, espec i ely. The heo e ical band s uc u e ene gy ze o was se
o he VB posi ion a he Kpoin , and he CBs (blue) we e igidly shi ed by a −0.16-eV scisso ope a o o ma ch he  alley cen e ene gy.
The momen um dispe sion along he kzdi ec ion is indica ed by he shaded a ea.
wi h pas li e a u e [49–51]. Despi e being a laye ed quasi-2D
ma e ial, WSe2displays some inhe en ly h ee-dimensional
ea u es. In pa icula , he  alley, as well as he alence band
a he poin , ha e conside able kzdispe sion. In con as ,
he ou -o -plane band dispe sion is low in he icini y o he
Kpoin , as con i med by ene ge ically na owe ea u es in
ARPES. Ou G0W0calcula ions p edic an o hogonal mo-
men um dispe sion on he o de o 40 meV o he VB and
30 meV o he CB a he Kpoin . Calcula ions place he indi-
ec band gap be ween he maximum o he VB a he poin
and he  alley. In ou da a he conduc ion-band minimum
(CBM) is unambiguously loca ed a he poin ; howe e , he
appa en alence-band maximum (VBM) is obse ed a he K
poin , and a b oad con inuum o s a es is obse ed a he 
poin . I is widely accep ed ha he absolu e VB maximum is
loca ed a he poin and ha ma ix elemen e ec s cancel
he con ibu ion o he uppe VB a [48,49]. A e he igid
o se o −160 meV men ioned abo e, he G0W0calcula-
ions a e in quali a i e ag eemen wi h he exci ed-s a e band
s uc u e and ep oduce he main ea u es o he expe imen al
conduc ion band.
Fo a quan i a i e compa ison, he quasipa icle ene gy
mus be de e mined om he ARPES in ensi y. Final-s a e e -
ec s usually complica e he e ie al o quasipa icle ene gies
and o many-body e ec s in he spec al unc ion. Howe e ,
he p oblem is absen in a s ic ly wo-dimensional s a e
[dispe sion only along k=(kx,ky)] [52]. Bo h alence and
conduc ion s a es a he di ec op ical band gap a he Kpoin
a e quasi- wo-dimensional, enabling a obus compa ison o
he expe imen al exci ed-s a e band gap wi h heo y and o he
expe imen al echniques.
The CB and VB ene gies a e ex ac ed om he expe -
imen al da a by a i o he ene gy dis ibu ion cu e a
he Kpoin o =100 s. The p ocedu e is illus a ed in
Fig. 3(a); he pho oelec on spec um o he VB is well i ed
by wo Gaussian peaks and by a Shi ley backg ound. The
wo, nea ly degene a e conduc ion bands p edic ed by heo y
a e no esol ed wi hin he expe imen al linewid h, and a
single Gaussian peak desc ibes well he CB signal. Due o i s
highe in ensi y, he highe ene gy ail o he VB spec um
appea s as a backg ound on he CB and is modeled by an
exponen ial decay. We de ine he expe imen al band gap as
he dis ance be ween he uppe mos VB peak posi ion (E=0
by de ini ion) o he cen e o he CB peak, as highligh ed
by he ed line in Fig. 3(a), and we measu e a band gap o
1.78 ±0.03 eV o da a collec ed a a luence o 40 μJ/cm2.
We no e ha his p ocedu e, alid o quasi-2D bands, di e s
om he me hod adop ed o h ee-dimensional semiconduc-
o s, whe e he band edge is ound by linea ex apola ion o
he pho oelec on spec al edge [53]. The exci ed-s a e quasi-
pa icle ene gy, an ou -o -equilib ium quan i y, can change
as a unc ion o he exci a ion ene gy densi y [18–20]. To
in es iga e he impac on he band gap, we ollow i s e o-
lu ion o inc easing inciden op ical ene gy densi y up o
320 μJ/cm2and obse e a dec ease o he band gap
[Fig. 3(b)]. The maximum e ec is ≈50 meV, wi h a linea
slope o −2±1×10−1meV/(μJ/cm2); he ex apola ed limi
a ze o exci a ion densi y is 1.77 ±0.01 eV.
075417-5

M. PUPPIN e al. PHYSICAL REVIEW B 105, 075417 (2022)
(a)
1.0
0.8
0.6
0.4
0.2
0.0
1.00.50.0-0.5
2.0
1.5
1.0
0.5
0.0
x103
2.52.01.5
In ensi y (a b.u.)
Ene gy (eV)
(b)
Fluence (μJ/cm2)
Eg,exc(eV)
Eg,exc
VL
CB
VB
(c)
ARPES ARIPES
CB
VB
OPTICAL GAPFUNDAMENTAL GAP
(e)
VL
Time
Eg, Eg,o
Eg,exc( )
ARPES
EXCITED-STATE GAP
1.85
1.80
1.75
1.70
1.65
400.0300.0200.0100.00.0
CB
VB
(d)
CB*
VB*
FIG. 3. (a) Ene gy dis ibu ion cu e a he Kpoin , oge he wi h he i used o de e mine he exci ed-s a e band gap Eg,exc. The conduc ion-
band signal in ensi y, displayed on he igh -hand axis, was scaled by a ac o 103 o cla i y. (b) Fluence dependence o he exci ed-s a e di ec
band gap a he Kpoin o a ime delay o 100 s. Schema ic desc ip ion o he (c) undamen al, (d) op ical, and (e) exci ed-s a e di ec band
gaps a he K alley, whe e VL indica es he acuum le el. Only he di ec gap is conside ed, i.e., he ene gy o emi ed (abso bed) elec ons
is measu ed a he same pa allel momen um alue. The pho oexci a ion occu s a ime ze o a a di e en ecip ocal space loca ion, abo e he
di ec band gap. The pho oexci ed elec on and hole dis ibu ions eno malize he exci ed-s a e bands, indica ed by CB* and VB*.
I is in e es ing o compa e his expe imen al band gap,
which we call he exci ed-s a e band gap Eg,exc, wi h ab
ini io calcula ions and o he expe imen al echniques. Se -
e al expe imen s ha e been designed o esol e he elec onic
s uc u e abo e he chemical po en ial [54]. In e se pho-
oemission [55], scanning unneling spec oscopy [56], and
e y low-ene gy elec on di ac ion [57] access unoccupied
conduc ion s a es by adding an elec on o he sys em and
p obing he complemen a y one-elec on-addi ion spec al
unc ion A+(k,E)[58]. Angle- esol ed in e se pho oemis-
sion (ARIPES), in pa icula , has momen um esolu ion [54].
Un o una ely, due o he small c oss-sec ion o he p ocess
and, unlike ARPES, due o he lack o pa allel de ec o s
wi h mul iple angula and ene gy channels, ARIPES has no
e ol ed o a simila ly widesp ead echnique [55]. Ano he
app oach can used in pho oemission o obse e o he wise un-
occupied s a es, namely, sample doping by alkali me al a oms
[50,59]. A limi a ion o alkali doping is he possibili y o
chemical modi ica ion o he band s uc u e [50]. Addi ionally,
esonan inelas ic x- ay sca e ing echniques ha e also been
used o map he dispe sion o unoccupied s a es [60,61]. The
di ec gap a he Kpoin o WSe2 om a ious me hods is
displayed in Table I.
The undamen al o quasipa icle band gap Eg, is usu-
ally de ined as he di e ence be ween he elec on a ini y,
i.e., he ene gy gained by adding a single elec on o an
N elec on sys em, and he ioniza ion ene gy, needed o e-
mo e an elec on lea ing N-1 elec ons behind [74]. The
quasipa icle gap should no be con used wi h he so-called
op ical band gap, which will be discussed la e on. The so-
called anspo band gap, de e mined by elec ical anspo
measu emen s, coincides wi h he undamen al band gap;
howe e , in he case o semiconduc o s such as bulk WSe2,
possessing an indi ec band gap and mul iple conduc ion-
band alleys, momen um- esol ed echniques p o ide a mo e
comple e pic u e. In iew o compa ison wi h op ical spec-
oscopy, we es ic discussion he e o he case o he di ec
band gap a he K alley and mo e loosely conside he
band gap as a momen um-dependen quan i y which a ains
i s minimum a he di ec undamen al band gap. Expe imen-
ally, he momen um-dependen quasipa icle band gap can be
measu ed by compa ing he VB measu ed by pho oemission
(N−1 elec on inal s a e) wi h he CB measu ed by in e se
pho oemission (N+1 elec on inal s a e). This p ocedu e is
schema ized in Fig. 3(c) and necessi a es a common ene gy
e e ence be ween he wo expe imen al se ups. In pa icula ,
he di ec undamen al gap o WSe2a he Kpoin was expe -
imen ally measu ed o be Eexp
g, =1.7±0.1 eV by combining
ARPES and ARIPES [49].
When compa ing he expe imen al gap wi h heo e ical
esul s, an impo an ques ion is o wha ex en one is allowed
o compa e ab ini io calcula ions such as DFT wi h ene gies
de e mined by ( ime- esol ed) pho oelec on spec oscopy.
DFT compu es he g ound-s a e elec onic densi y and e u ns
a se o sel -consis en Kohn-Sham (KS) bands [75]. E en
in an idealized case whe e he exac densi y unc ional is
known, a di ec compa ison be ween he KS bands and he
TABLE I. Compa ison be ween expe imen al (uppe pa ) and
heo e ical di ec band gap o WSe2(lowe pa ) a he Kpoin .
∗Measu ed a 77 K; a oom empe a u e he gap is educed by
≈60 meV [65]. ‡bilaye WSe2.
Me hod Band gap (eV) Re e ences
ARPES+ARIPES 1.7, 1.4 [49],[62]
ARPES+Doping 1.62 [59]
ARPES 1.77 This wo k
Op ics, A exci on 1.697∗, 1.60, 1.626 [63], [64], [65]
Op ics, In e band 1.752∗, 1.686 [63], [65]
EELS, A exci on 1.75 [66]
DFT 1.25, 1.17-1.55 This wo k, [67–71]
G0W01.90, 1.75, 2.08‡This wo k, [72], [73]
BSE, A exci on 1.86‡[73]
BSE, In e band 2.02‡[73]
075417-6
EXCITED-STATE BAND STRUCTURE MAPPING PHYSICAL REVIEW B 105, 075417 (2022)
ARPES measu emen s is no jus i ied [76]. None heless, in
many cases, wi hin a cons an ene gy o se , he KS bands
a e in good ag eemen wi h ARPES da a o he alence band.
Fo WSe2, in pa icula , DFT bands ep oduce easonably well
he ARPES VB ene gy dispe sion [48,49,51,77]. Howe e , i
Eg, is di ec ly calcula ed om he KS bands, heo y g ossly
unde es ima es he band gap. Be o e applying he G0W0co -
ec ion, ou calcula ions p edic a gap alue o 1.25 eV, in
line wi h o he DFT esul s, epo ed in Table I. This well-
known band-gap p oblem is in insic o DFT [78] and is a
eminde ha KS ene gies a e indeed no quasipa icle ene -
gies. Con e sely, Hedin’s GW me hod [46,79] can be used o
calcula e quasipa icle exci a ions in a solid, such as measu ed
in ARPES (elec on emo al) o ARIPES (elec on addi ion).
GW calcula ions co ec he DFT ene gies by an app oxima e
elec onic sel -ene gy, ypically pe o med o he lowes o de
(G0W0). We ind a conside able imp o emen in he calcu-
la ed undamen al gap and ob ain a alue EGW
g, =1.90 eV, in
line wi h p e ious calcula ions [72].
A second commonly de ined band gap is he so-called
op ical band gap Eg,o, which co esponds o he lowes ene gy
equi ed o a e ical (k=0) elec onic ansi ion in he
sys em. This is a neu al exci a ion whe e bo h he ini ial
and inal s a es ha e Nelec ons, in con as wi h he case
o he undamen al gap, which is calcula ed as he ene gy
di e ence be ween an N+1 and an N−1 elec on s a e. The
op ical band gap can be expe imen ally measu ed by op ical
abso p ion spec oscopy. A ema kable ea u e in op ical ab-
so p ion spec a is he appea ance o exci onic esonances a
ene gies below he onse o elec onic in e band ansi ions,
as shown in Fig. 3(c). The obse a ion o an exci onic peak
is he hallma k o he elec on-hole in e ac ion. To p edic
he op ical abso p ion spec um ab ini io, one mus sol e
he Be he-Salpe e equa ion [80]. In he op ical abso p ion
spec a o bulk WSe2 he so-called A exci on is he lowes
esonance a an ene gy o 1.68 eV, he exci on binding ene gy
Exwas de e mined o be 50 meV, and he in e band ansi ion
has an ene gy o 1.73 eV [63]. This se s he scale o he
elec on-hole in e ac ion in bulk TMD semiconduc o s, and
one expec s Eg,o≈Eg, −Ex.
In he exci ed-s a e band-gap measu emen [Fig. 3(d)], a
neu al op ical exci a ion is ollowed by an ioniza ion s ep a
ime , leading o a N−1 elec on exci ed inal s a e wi h an
addi ional hole in he VB, which is gene a ed o =0 and is
ollowed by a elaxa ion dynamics o >0. The band gap is
measu ed by compa ing he kine ic ene gy o pho oelec ons
o igina ing om he CB and he VB. Gene ally speaking,
Eg,exc( ) is a ime-dependen quan i y in luenced by many-
body e ec s and can be eno malized by elec on-elec on
in e ac ions, leading o sc eening and exci onic e ec s, and by
he elec on-phonon coupling wi h he (non he mal) phonon
dis ibu ion.
Ou da a shows ha in he low-exci a ion limi , Eg,exc(100
s) is in good nume ical ag eemen wi h he undamen al
band gap de e mined by o he expe imen s. Fu he mo e, we
obse e no signa u es o he A exci onic peak a he Kpoin ,
which appea s in op ical measu emen s a a lowe ene gy o
≈1.62 eV [63–65]. A de ia ion om he single-quasipa icle
pic u e is expec ed when elec on and hole a e bound o o m
exci ons [81–83] and pho oelec on spec a bea he signa u e
o such in e ac ions as a eno malized ene gy and momen um
dispe sion [2,7]. The ag eemen wi h he heo e ical G0W0
bands in he p esen case can be a ionalized by he ac ha
he pump pho on ene gy is well abo e he gap and su i-
cien ly o - esonance o app oxima e he ini ial ( ≈0) ca ie
dis ibu ion as an elec on-hole plasma, whe e exci on quasi-
pa icles a e no o med [39]. In bulk WSe2 he o ma ion o
s able A exci ons a he Kpoin is hinde ed by he possibili y
o elec on (hole) sca e ing o he poin (poin ), which
a e he global band ene gy edges. Howe e , i ins ead he
exci a ion ene gy is esonan wi h he exci onic peak obse ed
by op ics, exci onic e ec s can be obse ed [2].
We no e ha G0W0calcula ions o e es ima e he band gap
obse ed in ou ou -o -equilib ium expe imen by ≈160 meV.
Howe e , he ag eemen wi h he obse ed band dispe sion is
s ill sa is ac o y upon a igid shi o he conduc ion bands o
lowe pho on ene gies, sugges ing ha a single-quasipa icle
pic u e holds well o he exci ed-s a e band s uc u e in i s
app oxima ion. Band-gap eno maliza ion is expec ed o oc-
cu due o ca ie sc eening and ia elec on-phonon coupling
[18,19,84]. Time- esol ed di ac ion s udies e eal ha a
nonequilib ium phonon dis ibu ion ises on he imescale o a
ew picoseconds [85]. A a pump-p obe delay o 100 s, whe e
ou da a was collec ed, a signi ican ho phonon popula ion
has no ye de eloped. We conclude ha elec onic sc eening
mus domina e in band s uc u e mapping expe imen s, and
we a ibu e o his e ec he obse ed band-gap educ ion a
highe exci a ion densi ies [Fig. 3(b)].
Ha ing es ablished ha he exci ed-s a e band gap well
app oxima es he undamen al band gap in ou expe imen al
condi ions, we now ex ac he momen um- esol ed ene gy
dispe sion con ained in he expe imen al maps o he whole
2D K alley. The K alley ene gy is shown in Fig. 4(a),
and o compa ison we plo he heo e ical dispe sion o he
lowes CB in Fig. 4(b). The h ee old symme y o he alley
is e iden om he da a, and he aniso opy o he K alley
can be quan i ied by ex ac ing he dispe sion along he high-
symme y di ec ions K-and K-M, indica ed in Fig. 4(b).Fo
his pu pose we employ he p e iously desc ibed i ing p oce-
du e o EDCs su ounding he K alley. The band dispe sion
o bo h conduc ion and alence bands was es ima ed by i ing
a pa abola in a ange o 0.15 Å−1, as illus a ed in Fig. 4(c)
o he case o he CB. We ob ain a alue o mK
e=0.38m0
(mK
h=−0.52 m0) and mKM
e=0.55m0(mKM
h=−0.56m0)
o he CB (VB) in he di ec ions K-and K-M, espec i ely,
whe e m0is he elec on mass. The expe imen al dispe sion
is somewha smalle han e ec i e masses epo ed o DFT,
mh=−0.625m0and me=0.821m0[86]. Calcula ed e ec i e
masses om DFT depend s ongly on compu a ional de ails
and also on he compu a ional band gap [87]; la ge heo e ical
masses migh be he e o e linked o he unde es ima ion o he
gap in he a o emen ioned wo k.
By obse ing he hole and elec on quasipa icle inde-
penden ly, one can calcula e e ec i e [M=me+mh] and
educed [μ =memh/(me+mh)] exci on masses. The exci-
on e ec i e masses a e MK=0.9m0and MKM =1.1m0,
which can be compa ed wi h expe imen al esul s om elec-
on ene gy loss spec oscopy, M=0.91 m0[66], and wi h
op ical measu emen s unde a magne ic ield, which e-
po M=0.7m0[88]. The exci on educed mass de e mined
075417-7
M. PUPPIN e al. PHYSICAL REVIEW B 105, 075417 (2022)
-0.4
-0.2
0.0
0.2
0.4
1.71.51.31.10.9
-0.4
-0.2
0.0
0.2
0.4
1.61.41.21.0
kx (Å-1)
ky (Å-1)
ky (Å-1)
Ene gy (eV) Ene gy (eV)
2.3
2.2
2.1
2.0
1.9
1.8
kll (Å-1)
KΣ KM
kx (Å-1)
Ene gy (eV)
(a) (b) (c)
0.0 0.2
-0.2
2.42.11.8 2.82.42.0
KΣ
KM
K
FIG. 4. (a) Conduc ion-band cen e ene gy a he K alley, (b) G0W0ene gy o he K alley, and (c) dispe sion along he di ec ions K-
(nega i e xaxis) and K-M (posi i e xaxis). The ull line indica es he esul o pa abolic i s o he da a.
om ou da a is μK
=0.22 m0and μKM
=0.28m0.This
can be compa ed wi h op ical abso p ion spec oscopy da a
om which μ =0.21m0was de e mined [63]. We s ess,
howe e , ha despi e he easonable nume ical ag eemen ,
o he echniques do no iden i y he hole and elec on masses
independen ly. Fu he mo e, band aniso opy along di e en
symme y di ec ions can be eadily iden i ied and accoun ed
o wi hin he exci ed-s a e band s uc u e. This is pa ic-
ula ly ele an , o example, in alley onic applica ions in
he e olaye s, whe e ene gy-degene a e alleys appea a di -
e en momen um loca ions [89]. The de ailed e ec s o laye
s acking on he momen um dispe sion and on he op ical and
anspo p ope ies is as ye poo ly unde s ood and can be di-
ec ly cha ac e ized by exci ed-s a e band s uc u e mapping.
V. CONCLUSIONS
The exci ed-s a e band s uc u e is isualized o he TMD
WSe2by -ARPES. The expe imen p o ides simul ane-
ous access o alence and conduc ion s a es h oughou he
BZ, he eby comple ely mapping he ma e ial’s band gap.
The exci ed-s a e di ec gap a he Kpoin ag ees in he
low-exci a ion limi wi h undamen al quasipa icle gap, as
ob ained by s a ic expe imen s. Ou expe imen shows ha
he exci ed-s a e band s uc u e ag ees in he low-exci a ion
limi wi h he single-quasipa icle bands, and we ob ain ex-
pe imen ally conduc ion- and alence-band dispe sion o he
Kpoin o a ious high-symme y di ec ions. Thanks o
XUV ligh sou ces a high epe i ion a e, we an icipa e ha
he measu emen o he exci ed-s a e band s uc u e in he
whole BZ can be pe o med o a b oad class o samples.
G0W0calcula ions p o ide a good quali a i e desc ip ion o
he da a bu p edic he expe imen al ou -o -equilib ium band
gap only wi hin 160 meV. Exci e-s a e band s uc u e map-
ping can p o ide an expe imen al benchma k o quan i a i ely
ine une compu a ions, e.g., o accu a ely p edic he band
gap in high- h oughpu compu a ional ma e ial disco e y o
op oelec onic applica ions [71,90]. Au oma ed me hods o
compa ison wi h heo y, demons a ed o mul idimensional
ARPES da a [91], a e applicable also o exci ed-s a e band
s uc u e da a. Impo an ly, he me hod p o ide access o un-
occupied s a es o quan um ma e ials o esol e opological
ea u es abo e he Fe mi le el [10], and o cha ge densi y
wa e ma e ials, whe e he exci ed-s a e band s uc u e can be
ollowed ac oss a pho on-induced phase ansi ion [92–94].
A u u e open ques ion is he applicabili y o he me hod
o s ongly co ela ed ma e ials, e.g., o access he spec al
unc ion o unoccupied s a es in co ela ed oxides, o e eal
he symme y o he momen um dis ibu ion o he uppe
Hubba d bands in cup a e supe conduc o s [95]. Calcula ions
ou o equilib ium in such sys ems is a challenge o cu en
heo e ical me hods [96], and a de ailed knowledge o he
unoccupied s a es is lacking. In his case, sho -li ed exci ed-
s a e ea u es migh be accessible ac oss he B illouin zone
by sui ably uning he ime-bandwid h p oduc o imp o e
he empo al esolu ion, which is necessa y o ully exploi
exci e-s a e band s uc u e mapping.
ACKNOWLEDGMENTS
This wo k was unded by he Max-Planck-Gesellscha ,
by he Ge man Resea ch Founda ion (DFG), wi hin he
Emmy Noe he P og am (G an No. RE 3977/1), and G an s
No. FOR1700 (P ojec E5), No. SPP2244 (P ojec No.
443366970), and om he Eu opean Resea ch Council, G an
4
3
2
1
E (eV)
-0.8 -0.4 0.0
k║ (Å-1)
= -50 s 300
200
100
-400 -200 0 200 400
I (a b. u.)
Delay ( s)
(a) (b)
FIG. 5. (a) ARPES in ensi y as a unc ion o ene gy and pa -
allel momen um showing he conduc ion s a es along he −
di ec ion a a ime delay o −50 s. The pump-p obe empo al c oss-
co ela ion is de e mined by in eg a ing he signal in he ec angula
box. (b) Tempo al ace showing he in eg a ed in ensi y in he box o
panel (a) as a unc ion o ime. Red cu e, Gaussian i o he ising
edge, whe e he FWHM is 95 s.
075417-8
EXCITED-STATE BAND STRUCTURE MAPPING PHYSICAL REVIEW B 105, 075417 (2022)
No. ERC-2015-CoG-682843. M.P. acknowledges inancial
suppo om he Swiss Na ional Science Founda ion (SNSF)
h ough G an No. CRSK-2_196756. C.W.N. and C.M. ac-
knowledge inancial suppo by Swiss Na ional Science
Founda ion (SNSF) G an No. P00P2_170597. A.R. and H.H.
acknowledge inancial suppo om he Eu opean Resea ch
Council (G an No. ERC-2015-AdG-694097) and he Clus e
o Excellence “CUI: Ad anced Imaging o Ma e ” o he
Deu sche Fo schungsgemeinscha (G an EXC 2056, P ojec
No. 390715994).
APPENDIX: DETERMINATION OF TEMPORAL
PUMP-PROBE CROSS-CORRELATION
The empo al ime ze o and pump-p obe c oss-co ela ion
o 95 s we e measu ed by i ing he ising edge o he i s
obse able signal in he exci ed-s a e band s uc u e, as illus-
a ed in Fig. 5. The second maximum obse ed a e 100 s
is a esul o elec on popula ion sca e ed om o he s a es
du ing he ene gy elaxa ion p ocess.
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