Charged complexes
in 2D TMDC semiconductors
Vladimir Falko
DFT – parameterised k·p model for TMDCs
Kormányos, Zólyomi, Drummond, Rakyta, Burkard, VF - PRB 88, 045416 (2013)
Kormányos, Burkard, Gmitra, Fabian, Zólyomi, Drummond, VF - 2D Materials 2, 022001 (2015)
transitions


q  q  eA
z
z→-z even
z
z
q   q x  iq y
± for ± K valleys
SO splitting in DFT – parameterised k·p model
MoSe2
22meV
WS2
-32meV
bright excitons
and trions
dark excitons
and trions
MoS2
3meV
148 meV 186meV
WSe2
-37meV
430 meV 466meV
Kormányos, Burkard, Gmitra, Fabian, Zólyomi, Drummond, VF - 2D Materials 2, 022001 (2015)
3-body problems
in 2D systems
3-body problem
e
e
XA = A- + h + e
XD = D+ + e + h
X± trions
e

in atomic physics: 9 degrees of freedom = too many

in 2D systems
6
–
2
–
1
rotation
=
3
almost
like one
hydrogen
atom
Practical details: coordinates
  
rij  ri  rj



 1r13   2 r23
r'
/ r*
M1 2 / 3
and
 
~
r  r12 / r*
 (r , ,  )
0
isotropic solution
(ground state)
Schroedinger equation
3
 
2
3

 U (r , ,  )    
1
2
3
1
1
3
3
2
2
Logarithmic interaction at intermediate distances
similar to gapped bilayer graphene,
Cheianov, Aleiner, Falko - PRL 109, 106801 (2012)
r* 
bulk in-plane dielectric constant
of layered material
 ||  1
2
az
[~ (30  50) A]
distance between layers in
a layered material
Log-Coulomb interaction in 2D crystals: excitons
r* 
Excitons
 ||  1
2
az
e  h
m
e   h
Lapidus - Am.J.Phys. 49, 807 (1981)
Eveker, Grow, Jost, Monfort, Nelson - Am.J.Phys. 58, 1183 (1990)
Chernikov, Berkelbach, Hill, Rigosi, Li, Aslan, Reichman, Hybertsen, Heinz - PRL 113, 076802 (2014)
Binding energies of
three-particle complexes
We have developed
an unexpectedly
simple method to
study 3-particle
bound states in 2D
materials, and we
compared it to
diffusion quantum
Monte Carlo
calculations
r*  rX , X / 
Ganchev, Drummond, Aleiner, VF – PRL 114, 107401 (2015)
XA and XD are loosely
bound, as compared
to trions, despite that
their lines are
stronger red-shifted
with respect to the
free exciton.
calculated binding energies of three-particle
complexes across the parameter space r* ~ r
e
r* / re
e2
r*
E[ ]
trions
e
e  h
e / h
 e2 
E

r
r

 * e 
exciton
Szyniszewski, Drummond, Aleiner, VF (2015)
r*
r*  re
 Due to the relative signs of spin-orbit splitting for
electrons near the edge of conduction and valence
bands (around K-points), excitons and trions in MoX2
are bright and in WX2 are dark.
 We have calculated binding energies of trions and
acceptor/donor bound excitons, as a function of
electron/hole masses and polarizability of TMDC lattice.
with
B Ganchev, M Szyniszewski, N Drummond (Lancaster)
A Kormányos, G Burkard (Konstanz)
I Aleiner (Columbia U – NY)