Monolayers of transition metal dichalcogenides present an intriguing platform to investigate the interplay of excitonic complexes in two-dimensional semiconductors. Here, we use optical spectroscopy to study the light–matter coupling and non-equilibrium relaxation dynamics of three-particle exciton states, commonly known as trions. We identify the consequences of the exchange interaction for the trion fine structure in tungsten-based monolayer materials from variational calculations and experimentally determine the resulting characteristic differences in their oscillator strength. It allows us to quantitatively extract trion populations from time-resolved photoluminescence measurements and monitor their dynamics after off-resonant optical injection. At liquid helium temperature, we observe a pronounced non-equilibrium distribution of the trions during their lifetime with comparatively slow equilibration that occurs on time-scales up to several hundreds of ps. In addition, we find an intriguing regime of population inversion at lowest excitation densities, which builds up and is maintained for tens of picoseconds. At a higher lattice temperature, the equilibrium is established more rapidly and the inversion disappears, highlighting the role of thermal activation for efficient scattering between exchange-split trions.

1.
H.
Haug
and
S. W.
Koch
, “
Semiconductor laser theory with many-body effects
,”
Phys. Rev. A
39
(
4
),
1887
1898
(
1989
).
2.
C.
Klingshirn
,
Semiconductor Optics
, 3rd ed. (
Springer
,
Berlin, Heidelberg; New York
,
2007
).
3.
T. C.
Berkelbach
,
M. S.
Hybertsen
, and
D. R.
Reichman
, “
Theory of neutral and charged excitons in monolayer transition metal dichalcogenides
,”
Phys. Rev. B
88
(
4
),
045318
(
2013
).
4.
P.
Cudazzo
,
I. V.
Tokatly
, and
A.
Rubio
, “
Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane
,”
Phys. Rev. B
84
(
8
),
085406
(
2011
).
5.
L. V.
Keldysh
, “
Coulomb interaction in thin semiconductor and semimetal films
,”
J. Exp. Theor. Phys. Lett.
29
,
658
(
1979
), available at http://www.jetpletters.ac.ru/ps/1458/article_22207.shtml.
6.
N. S.
Rytova
, “
Screened potential of a point charge in a thin film
,”
Moscow Univ. Phys. Bull.
3
,
30
(
1967
); arXiv:1806.00976.
7.
G.-B.
Liu
,
D.
Xiao
,
Y.
Yao
,
X.
Xu
, and
W.
Yao
, “
Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides
,”
Chem. Soc. Rev.
44
(
9
),
2643
2663
(
2015
).
8.
G.
Wang
,
A.
Chernikov
,
M. M.
Glazov
,
T. F.
Heinz
,
X.
Marie
,
T.
Amand
, and
B.
Urbaszek
, “
Colloquium: Excitons in atomically thin transition metal dichalcogenides
,”
Rev. Mod. Phys.
90
(
2
),
021001
(
2018
).
9.
J.
Zipfel
,
M.
Kulig
,
R.
Perea-Causín
,
S.
Brem
,
J. D.
Ziegler
,
R.
Rosati
,
T.
Taniguchi
,
K.
Watanabe
,
M. M.
Glazov
,
E.
Malic
, and
A.
Chernikov
, “
Exciton diffusion in monolayer semiconductors with suppressed disorder
,”
Phys. Rev. B
101
(
11
),
115430
(
2020
).
10.
A.
Chernikov
,
T. C.
Berkelbach
,
H. M.
Hill
,
R.
Albert
,
Y.
Li
,
O. B.
Aslan
,
D. R.
Reichman
,
M. S.
Hybertsen
, and
T. F.
Heinz
, “
Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2
,”
Phys. Rev. Lett.
113
(
7
),
076802
(
2014
).
11.
J.
Singh
, “
Theory of excitons
,” in
Excitation Energy Transfer Processes in Condensed Matter
(
Springer US
,
1994
), pp.
1
45
.
12.
A. V.
Stier
,
N. P.
Wilson
,
K. A.
Velizhanin
,
J.
Kono
,
X.
Xu
, and
S. A.
Crooker
, “
Magnetooptics of exciton Rydberg states in a monolayer semiconductor
,”
Phys. Rev. Lett.
120
(
5
),
057405
(
2018
).
13.
G. H.
Wannier
, “
The structure of electronic excitation levels in insulating crystals
,”
Phys. Rev.
52
(
3
),
191
(
1937
).
14.
M.
Barbone
,
A. R.-P.
Montblanch
,
D. M.
Kara
,
C.
Palacios-Berraquero
,
A. R.
Cadore
,
D.
De Fazio
,
B.
Pingault
,
E.
Mostaani
,
H.
Li
,
B.
Chen
,
K.
Watanabe
,
T.
Taniguchi
,
S.
Tongay
,
G.
Wang
,
A. C.
Ferrari
, and
M.
Atatüre
, “
Charge-tuneable biexciton complexes in monolayer WSe2
,”
Nat. Commun.
9
(
1
),
3721
(
2018
).
15.
P.
Nagler
,
M. V.
Ballottin
,
A. A.
Mitioglu
,
M. V.
Durnev
,
T.
Taniguchi
,
K.
Watanabe
,
A.
Chernikov
,
C.
Schüller
,
M. M.
Glazov
,
P. C. M.
Christianen
, and
T.
Korn
, “
Zeeman splitting and inverted polarization of biexciton emission in monolayer WS2
,”
Phys. Rev. Lett.
121
(
5
),
057402
(
2018
).
16.
A.
Steinhoff
,
M.
Florian
,
A.
Singh
,
K.
Tran
,
M.
Kolarczik
,
S.
Helmrich
,
A. W.
Achtstein
,
U.
Woggon
,
N.
Owschimikow
,
F.
Jahnke
, and
X.
Li
, “
Biexciton fine structure in monolayer transition metal dichalcogenides
,”
Nat. Phys.
14
(
12
),
1199
1204
(
2018
).
17.
Z.
Ye
,
L.
Waldecker
,
E. Y.
Ma
,
D.
Rhodes
,
A.
Antony
,
B.
Kim
,
X.-X.
Zhang
,
M.
Deng
,
Y.
Jiang
,
Z.
Lu
,
D.
Smirnov
,
K.
Watanabe
,
T.
Taniguchi
,
J.
Hone
, and
T. F.
Heinz
, “
Efficient generation of neutral and charged biexcitons in encapsulated WSe2 monolayers
,”
Nat. Commun.
9
(
1
),
3718
(
2018
).
18.
Y.
You
,
X.-X.
Zhang
,
T. C.
Berkelbach
,
M. S.
Hybertsen
,
D. R.
Reichman
, and
T. F.
Heinz
, “
Observation of biexcitons in monolayer WSe2
,”
Nat. Phys.
11
(
6
),
477
481
(
2015
).
19.
K.
Kheng
,
R. T.
Cox
,
M. Y.
d’ Aubigné
,
F.
Bassani
,
K.
Saminadayar
, and
S.
Tatarenko
, “
Observation of negatively charged excitons X− in semiconductor quantum wells
,”
Phys. Rev. Lett.
71
(
11
),
1752
1755
(
1993
).
20.
K. F.
Mak
,
K.
He
,
C.
Lee
,
G. H.
Lee
,
J.
Hone
,
T. F.
Heinz
, and
J.
Shan
, “
Tightly bound trions in monolayer MoS2
,”
Nat. Mater.
12
(
3
),
207
211
(
2012
).
21.
J. S.
Ross
,
S.
Wu
,
H.
Yu
,
N. J.
Ghimire
,
A. M.
Jones
,
G.
Aivazian
,
J.
Yan
,
D. G.
Mandrus
,
D.
Xiao
,
W.
Yao
, and
X.
Xu
, “
Electrical control of neutral and charged excitons in a monolayer semiconductor
,”
Nat. Commun.
4
(
1
),
1474
(
2013
).
22.
C.
Zhang
,
H.
Wang
,
W.
Chan
,
C.
Manolatou
, and
F.
Rana
, “
Absorption of light by excitons and trions in monolayers of metal dichalcogenide MoS2: Experiments and theory
,”
Phys. Rev. B
89
,
205436
(
2014
).
23.
B.
Ganchev
,
N.
Drummond
,
I.
Aleiner
, and
V.
Fal’ko
, “
Three-particle complexes in two-dimensional semiconductors
,”
Phys. Rev. Lett.
114
(
10
),
107401
(
2015
).
24.
G.
Plechinger
,
P.
Nagler
,
J.
Kraus
,
N.
Paradiso
,
C.
Strunk
,
C.
Schüller
, and
T.
Korn
, “
Identification of excitons, trions and biexcitons in single-layer WS2
,”
Phys. Status Solidi RRL
9
(
8
),
457
461
(
2015
).
25.
S.
Mouri
,
Y.
Miyauchi
, and
K.
Matsuda
, “
Tunable photoluminescence of monolayer MoS2 via chemical doping
,”
Nano Lett.
13
(
12
),
5944
5948
(
2013
).
26.
A.
Singh
,
G.
Moody
,
K.
Tran
,
M. E.
Scott
,
V.
Overbeck
,
G.
Berghäuser
,
J.
Schaibley
,
E. J.
Seifert
,
D.
Pleskot
,
N. M.
Gabor
,
J.
Yan
,
D. G.
Mandrus
,
M.
Richter
,
E.
Malic
,
X.
Xu
, and
X.
Li
, “
Trion formation dynamics in monolayer transition metal dichalcogenides
,”
Phys. Rev. B
93
(
4
),
041401(R)
(
2016
).
27.
K. F.
Mak
,
K.
He
,
J.
Shan
, and
T. F.
Heinz
, “
Control of valley polarization in monolayer MoS2 by optical helicity
,”
Nat. Nanotechnol.
7
(
8
),
494
(
2012
).
28.
T.
Cao
,
G.
Wang
,
W.
Han
,
H.
Ye
,
C.
Zhu
,
J.
Shi
,
Q.
Niu
,
P.
Tan
,
E.
Wang
,
B.
Liu
, and
J.
Feng
, “
Valley-selective circular dichroism of monolayer molybdenum disulphide
,”
Nat. Commun.
3
(
1
),
887
(
2012
).
29.
D.
Xiao
,
G.-B.
Liu
,
W.
Feng
,
X.
Xu
, and
Y.
Wang
, “
Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides
,”
Phys. Rev. Lett.
108
(
19
),
196802
(
2012
).
30.
G.
Plechinger
,
P.
Nagler
,
A.
Arora
,
A.
Granados del Águila
,
M. V.
Ballottin
,
T.
Frank
,
P.
Steinleitner
,
M.
Gmitra
,
J.
Fabian
,
P. C. M.
Christianen
etet al, “
Excitonic valley effects in monolayer WS2 under high magnetic fields
,”
Nano Lett.
16
(
12
),
7899
7904
(
2016
).
31.
A.
Torche
and
G.
Bester
, “
First-principles many-body theory for charged and neutral excitations: Trion fine structure splitting in transition metal dichalcogenides
,”
Phys. Rev. B
100
(
20
),
201403(R)
(
2019
).
32.
E.
Courtade
,
M.
Semina
,
M.
Manca
,
M. M.
Glazov
,
C.
Robert
,
F.
Cadiz
,
G.
Wang
,
T.
Taniguchi
,
K.
Watanabe
,
M.
Pierre
,
W.
Escoffier
,
E. L.
Ivchenko
,
P.
Renucci
,
X.
Marie
,
T.
Amand
, and
B.
Urbaszek
, “
Charged excitons in monolayer WSe2: Experiment and theory
,”
Phys. Rev. B
96
(
8
),
085302
(
2017
).
33.
D.
Vaclavkova
,
J.
Wyzula
,
K.
Nogajewski
,
M.
Bartos
,
A. O.
Slobodeniuk
,
C.
Faugeras
,
M.
Potemski
, and
M. R.
Molas
, “
Singlet and triplet trions in WS2 monolayer encapsulated in hexagonal boron nitride
,”
Nanotechnology
29
(
32
),
325705
(
2018
).
34.
F.
Cadiz
,
E.
Courtade
,
C.
Robert
,
G.
Wang
,
Y.
Shen
,
H.
Cai
,
T.
Taniguchi
,
K.
Watanabe
,
H.
Carrere
,
D.
Lagarde
 et al, “
Excitonic linewidth approaching the homogeneous limit in MoS2-based van der Waals heterostructures
,”
Phys. Rev. X
7
(
2
),
021026
(
2017
).
35.
A.
Raja
,
L.
Waldecker
,
J.
Zipfel
,
Y.
Cho
,
S.
Brem
,
J. D.
Ziegler
,
M.
Kulig
,
T.
Taniguchi
,
K.
Watanabe
,
E.
Malic
,
T. F.
Heinz
,
T. C.
Berkelbach
, and
A.
Chernikov
, “
Dielectric disorder in two-dimensional materials
,”
Nat. Nanotechnol.
14
(
9
),
832
837
(
2019
).
36.
D.
Rhodes
,
S. H.
Chae
,
R.
Ribeiro-Palau
, and
J.
Hone
, “
Disorder in van der Waals heterostructures of 2D materials
,”
Nat. Mater.
18
(
6
),
541
549
(
2019
).
37.
X.-X.
Zhang
,
Y.
You
,
S. Y. F.
Zhao
, and
T. F.
Heinz
, “
Experimental evidence for dark excitons in monolayer WSe2
,”
Phys. Rev. Lett.
115
(
25
),
257403
(
2015
).
38.
T. C.
Berkelbach
,
M. S.
Hybertsen
, and
D. R.
Reichman
, “
Bright and dark singlet excitons via linear and two-photon spectroscopy in monolayer transition-metal dichalcogenides
,”
Phys. Rev. B
92
(
8
),
085413
(
2015
).
39.
J. P.
Echeverry
,
B.
Urbaszek
,
T.
Amand
,
X.
Marie
, and
I. C.
Gerber
, “
Splitting between bright and dark excitons in transition metal dichalcogenide monolayers
,”
Phys. Rev. B
93
(
12
),
121107(R)
(
2016
).
40.
X.-X.
Zhang
,
T.
Cao
,
Z.
Lu
,
Y.-C.
Lin
,
F.
Zhang
,
Y.
Wang
,
Z.
Li
,
J. C.
Hone
,
J. A.
Robinson
,
D.
Smirnov
,
S. G.
Louie
, and
T. F.
Heinz
, “
Magnetic brightening and control of dark excitons in monolayer WSe2
,”
Nat. Nanotechnol.
12
(
9
),
883
888
(
2017
).
41.
E.
Malic
,
M.
Selig
,
M.
Feierabend
,
S.
Brem
,
D.
Christiansen
,
F.
Wendler
,
A.
Knorr
, and
G.
Berghäuser
, “
Dark excitons in transition metal dichalcogenides
,”
Phys. Rev. Mater.
2
(
1
),
014002
(
2018
).
42.
G.
Plechinger
,
P.
Nagler
,
A.
Arora
,
R.
Schmidt
,
A.
Chernikov
,
A. G.
del Águila
,
P. C. M.
Christianen
,
R.
Bratschitsch
,
C.
Schüller
, and
T.
Korn
, “
Trion fine structure and coupled spin–valley dynamics in monolayer tungsten disulfide
,”
Nat. Commun.
7
(
1
),
12715
(
2016
).
43.
M.
Koschorreck
,
D.
Pertot
,
E.
Vogt
,
B.
Fröhlich
,
M.
Feld
, and
M.
Köhl
, “
Attractive and repulsive Fermi polarons in two dimensions
,”
Nature
485
(
7400
),
619
622
(
2012
).
44.
M.
Sidler
,
P.
Back
,
O.
Cotlet
,
A.
Srivastava
,
T.
Fink
,
M.
Kroner
,
E.
Demler
, and
A.
Imamoglu
, “
Fermi polaron-polaritons in charge-tunable atomically thin semiconductors
,”
Nat. Phys.
13
(
3
),
255
261
(
2016
).
45.
D. K.
Efimkin
and
A. H.
MacDonald
, “
Many-body theory of trion absorption features in two-dimensional semiconductors
,”
Phys. Rev. B
95
(
3
),
035417
(
2017
).
46.
M. M.
Glazov
, “
Optical properties of charged excitons in two-dimensional crystals
,”
J. Chem. Phys.
034703
(in press).
47.
A.
Castellanos-Gomez
,
M.
Buscema
,
R.
Molenaar
,
V.
Singh
,
L.
Janssen
,
H. S. J.
van der Zant
, and
G. A.
Steele
, “
Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping
,”
2D Mater.
1
(
1
),
011002
(
2014
).
48.
A.
Raja
,
A.
Chaves
,
J.
Yu
,
G.
Arefe
,
H. M.
Hill
,
A. F.
Rigosi
,
T. C.
Berkelbach
,
P.
Nagler
,
C.
Schüller
,
T.
Korn
,
C.
Nuckolls
,
J.
Hone
,
L. E.
Brus
,
T. F.
Heinz
,
D. R.
Reichman
, and
A.
Chernikov
, “
Coulomb engineering of the bandgap and excitons in two-dimensional materials
,”
Nat. Commun.
8
(
1
),
15251
(
2017
).
49.
S. J.
Byrnes
, “
Multilayer optical calculations
,” arXiv:1603.02720 (
2016
).
50.
A.
Arora
,
T.
Deilmann
,
T.
Reichenauer
,
J.
Kern
,
S.
Michaelis de Vasconcellos
,
M.
Rohlfing
, and
R.
Bratschitsch
, “
Excited-state trions in monolayer WS2
,”
Phys. Rev. Lett.
123
(
16
),
167401
(
2019
).
51.
Y.
Cho
and
T. C.
Berkelbach
, “
Environmentally sensitive theory of electronic and optical transitions in atomically thin semiconductors
,”
Phys. Rev. B
97
(
4
),
041409(R)
(
2018
).
52.
A.
Chernikov
,
A. M.
van der Zande
,
H. M.
Hill
,
A. F.
Rigosi
,
A.
Velauthapillai
,
J.
Hone
, and
T. F.
Heinz
, “
Electrical tuning of exciton binding energies in monolayer WS2
,”
Phys. Rev. Lett.
115
(
12
),
126802
(
2015
).
53.
Z.
Wang
,
J.
Shan
, and
K. F.
Mak
, “
Valley- and spin-polarized Landau levels in monolayer WSe2
,”
Nat. Nanotechnol.
12
(
2
),
144
149
(
2016
).
54.
G.
Sallen
,
L.
Bouet
,
X.
Marie
,
G.
Wang
,
C. R.
Zhu
,
W. P.
Han
,
Y.
Lu
,
P. H.
Tan
,
T.
Amand
,
B. L.
Liu
 et al, “
Robust optical emission polarization in MoS2 monolayers through selective valley excitation
,”
Phys. Rev. B
86
(
8
),
081301
(
2012
).
55.
A. M.
Jones
,
H.
Yu
,
J. R.
Schaibley
,
J.
Yan
,
D. G.
Mandrus
,
T.
Taniguchi
,
K.
Watanabe
,
H.
Dery
,
W.
Yao
, and
X.
Xu
, “
Excitonic luminescence upconversion in a two-dimensional semiconductor
,”
Nat. Phys.
12
(
4
),
323
327
(
2015
).
56.
H.
Yu
,
X.
Cui
,
X.
Xu
, and
W.
Yao
, “
Valley excitons in two-dimensional semiconductors
,”
Natl. Sci. Rev.
2
(
1
),
57
70
(
2015
).
57.
M. V.
Durnev
and
M. M.
Glazov
, “
Excitons and trions in two-dimensional semiconductors based on transition metal dichalcogenides
,”
Phys.-Usp.
61
(
9
),
825
845
(
2018
).
58.
M.
Danovich
,
V.
Zólyomi
, and
V. I.
Fal’ko
, “
Dark trions and biexcitons in WS2 and WSe2 made bright by e–e scattering
,”
Sci. Rep.
7
(
1
),
45998
(
2017
).
59.
M.
Goryca
,
J.
Li
,
A. V.
Stier
,
T.
Taniguchi
,
K.
Watanabe
,
E.
Courtade
,
S.
Shree
,
C.
Robert
,
B.
Urbaszek
,
X.
Marie
, and
S. A.
Crooker
, “
Revealing exciton masses and dielectric properties of monolayer semiconductors with high magnetic fields
,”
Nat. Commun.
10
(
1
),
4172
(
2019
).
60.
J.
Zipfel
,
J.
Holler
,
A. A.
Mitioglu
,
M. V.
Ballottin
,
P.
Nagler
,
A. V.
Stier
,
T.
Taniguchi
,
K.
Watanabe
,
S. A.
Crooker
, and
P. C. M.
Christianen
 et al, “
Spatial extent of the excited exciton states in WS2 monolayers from diamagnetic shifts
,”
Phys. Rev. B
98
(
7
),
075438
(
2018
).
61.
E.
Liu
,
J.
van Baren
,
T.
Taniguchi
,
K.
Watanabe
,
Y.-C.
Chang
, and
C. H.
Lui
, “
Magnetophotoluminescence of exciton Rydberg states in monolayer WSe2
,”
Phys. Rev. B
99
(
20
),
205420
(
2019
).
62.
A.
Esser
,
E.
Runge
,
R.
Zimmermann
, and
W.
Langbein
, “
Photoluminescence and radiative lifetime of trions in GaAs quantum wells
,”
Phys. Rev. B
62
(
12
),
8232
8239
(
2000
).
63.
H. H.
Fang
,
B.
Han
,
C.
Robert
,
M. A.
Semina
,
D.
Lagarde
,
E.
Courtade
,
T.
Taniguchi
,
K.
Watanabe
,
T.
Amand
,
B.
Urbaszek
,
M. M.
Glazov
, and
X.
Marie
, “
Control of the exciton radiative lifetime in van der Waals heterostructures
,”
Phys. Rev. Lett.
123
(
6
),
067401
(
2019
).
64.
B.
Stébé
,
E.
Feddi
,
A.
Ainane
, and
F.
Dujardin
, “
Optical and magneto-optical absorption of negatively charged excitons in three- and two-dimensional semiconductors
,”
Phys. Rev. B
58
(
15
),
9926
9932
(
1998
).
65.
A.
Molina-Sánchez
and
L.
Wirtz
, “
Phonons single-layer and few-layer MoS2 and WS2
,”
Phys. Rev. B
84
(
15
),
155413
(
2011
).
66.
A.
Berkdemir
,
H. R.
Gutiérrez
,
A. R.
Botello-Méndez
,
N.
Perea-López
,
A. L.
Elías
,
C.-I.
Chia
,
B.
Wang
,
V. H.
Crespi
,
F.
López-Urías
,
J.-C.
Charlier
,
H.
Terrones
, and
M.
Terrones
, “
Identification of individual and few layers of WS2 using Raman spectroscopy
,”
Sci. Rep.
3
(
1
),
1755
(
2013
).
67.
M.
Selig
,
G.
Berghäuser
,
M.
Richter
,
R.
Bratschitsch
,
A.
Knorr
, and
E.
Malic
, “
Dark and bright exciton formation, thermalization, and photoluminescence in monolayer transition metal dichalcogenides
,”
2D Mater.
5
(
3
),
035017
(
2018
).
68.
M. M.
Glazov
,
T.
Amand
,
X.
Marie
,
D.
Lagarde
,
L.
Bouet
, and
B.
Urbaszek
, “
Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides
,”
Phys. Rev. B
89
(
20
),
201302(R)
(
2014
).
69.
M. R.
Carbone
,
M. Z.
Mayers
, and
D. R.
Reichman
J. Chem. Phys.
152
,
194705
(
2020
).

Supplementary Material

You do not currently have access to this content.