PHYSICAL REVIEW B 101, 127401 (2020)
Comment on “Excitons, trions, and biexcitons in transition-metal dichalcogenides:
Magnetic-field dependence”
Ngoc-Tram D. Hoang
*
and Duy-Nhat Ly
†
Department of Physics, Ho Chi Minh City University of Education,
280 An Duong Vuong Street, Ward 4, District 5, Ho Chi Minh City, Vietnam
Van-Hoang Le
‡
Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science,
Ton Duc Thang University, Ho Chi Minh City, Vietnam
and Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam
(Received 23 August 2019; accepted 11 February 2020; published 18 March 2020)
To separate the motion of the center of mass for the exciton in a constant magnetic field, Donk et al. [Phys.
Rev. B 97, 195408 (2018)] considered the case of zero angular momentum and assumed the equality of the
electron and hole masses. Within these approximations, they removed the angular momentum operator from the
Schrödinger equation and were able to obtain a Hamiltonian describing the electron-hole relative motion. In this
Comment paper, we show that the assumptions mentioned above are not justified, and, as a consequence, the
given Hamiltonian for the relative motion is incorrect. In addition, we present an analytical procedure (without
any assumptions) to separate the center of mass and obtain the exact Hamiltonian for the relative motion, which
agrees with the previous theoretical studies.
DOI: 10.1103/PhysRevB.101.127401
Recently, energy spectra of excitons in monolayers such as
transition-metal dichalcogenides (TMDs) with the presence
of constant magnetic field are of great interest, both experi-
mentally and theoretically [1–3]. For solving the Schrödinger
equation, one needs to separate the motion of the electron-
hole mass center. Because of the magnetic field, the variable-
separation procedure for this system is not trivial but exists
and leads to the exact equation for the relative motion [4–6].
However, in Ref. [1], the authors presented an approximate
separation based on the assumption that the effective masses
of the hole and the electron are equal to each other. In this
Comment paper, first, we will show that the approximations
made in Ref. [1] are inappropriate, which lead to the incor-
rect equation for the hole-electron relative motion. Then we
will also present an analytical procedure to obtain an exact
equation for the relative motion, which is quite different in
comparison with the approximate equation of Ref. [1].
The Hamiltonian for the system of one electron and one
hole interacting with each other by the Keldysh potential in
the presence of a constant magnetic field has the following
form:
ˆ
H
ex
=
1
2m
h
p
2
h
+
1
2m
e
p
2
e
+
eB
2m
e
ˆ
l
e
z
−
eB
2m
h
ˆ
l
h
z
+
e
2
B
2
8m
h
r
2
h
+
e
2
B
2
8m
e
r
2
e
+ V
he
(| r
h − r
e
| ). (1)
*
tramhdn@hcmue.edu.vn
†
Also at Vietnam Atomic Energy Institute, 59, Ly Thuong Kiet,
Hoan Kiem, Hanoi, Vietnam.
‡
Corresponding author: levanhoang@tdtu.edu.vn
For reference, see the review [7]. However, this equation was
used in Ref. [1] with the elimination of the term
ˆ
A =
eB
2m
e
ˆ
l
e
z
−
eB
2m
h
ˆ
l
h
z
considering only excitons with zero angular momentum
[Eq. (6) in Ref. [1]]. This elimination of
ˆ
A in the Hamiltonian
(1) is not plausible since this operator does not commute
with the Hamiltonian, i.e., [
ˆ
H
ex
,
ˆ
A] = 0. Physically, for the
considered system, only the total angular momentum for the
hole-electron system, but neither the angular momentum of
each electron nor hole, conserves. Therefore, ones cannot
exclude operators
ˆ
l
e
z
and
ˆ
l
h
z
from the Hamiltonian.
Equation (6) of Ref. [1] then was rewritten in the center
of mass and relative coordinates. For the variable separation,
the authors assume the masses of electron and hole are equal
to each other: m
e
= m
h
to eliminate the term
e
2
B
2
4
m
e
−m
h
Mμ
R ·
r in the Hamiltonian. Here M = m
h
+ m
e
and μ =
m
h
m
e
m
h
+m
e
.
However, in monolayer TMDs, the masses are effective; thus,
this equality does not hold in general. In fact, Table I of
Ref. [1] shows the data for MoS
2
, MoSe
2
, WS
2
, and WSe
2
with the relative difference between the effective masses of
electron and hole from 6% to 15%. Therefore, the assumption
that m
e
= m
h
is very crude and not justified for the case of
an exciton in a constant magnetic field where the energy
calculations are very accurate.
By using the two assumptions mentioned above, the au-
thors of Ref. [1] were able to separate the Schrödinger equa-
tion, and then expressed the motion of the exciton center of
mass as an oscillation, and described the relative motion by
the Hamiltonian given by Eq. (11) in Ref. [1] as
ˆ
H
rel
=
1
2μ
ˆ p
2
+
e
2
B
2
32μ
r
2
+ V
he
(r ). (2)
2469-9950/2020/101(12)/127401(2) 127401-1 ©2020 American Physical Society