Textbooks often present the phenomenon of screening within the Thomas–Fermi model for three-dimensional free electron gases, but obtaining the dielectric response function and screening potential for dielectric systems of reduced dimensionality is also of pedagogical interest. In this work, we introduce a simple approach to investigate static screening in dielectric systems in the presence of an impurity charge for different dimensionalities. This approach is applicable to semiconductors and insulators alike. We demonstrate that, in 3D systems, the macroscopic dielectric function in reciprocal space is a constant, while in 2D and 1D systems, it strongly depends on the momentum transferred to the electrons in the dielectric. Through the proposed dielectric screening model, one can also determine binding energies in a hydrogenic model that can be used to describe excitations in real semiconductor systems.

1.
J. D.
Patterson
and
B. C.
Bailey
,
Solid-State Physics: Introduction to the Theory
(
Springer Berlin, Heidelberg
,
2007
) pp.
135
137
and 531–540.
2.
A.
Chernikov
,
T. C.
Berkelbach
,
H. M.
Hill
,
A.
Rigosi
,
Y.
Li
,
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
).
3.
A.
Tries
,
S.
Osella
,
P.
Zhang
,
F.
Xu
,
C.
Ramanan
,
M.
Kläui
,
Y.
Mai
,
D.
Beljonne
, and
H. I.
Wang
, “
Experimental observation of strong exciton effects in graphene nanoribbons
,”
Nano Lett.
20
(
5
),
2993
3002
(
2020
).
4.
G. D.
Mahan
,
Many-Particle Physics
, 3rd ed. (
Kluwer Academic/Plenum Publishers
,
New York
,
2000
) pp.
316
319
and 325–335.
5.
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
).
6.
G.
Onida
,
L.
Reining
, and
A.
Rubio
, “
Electronic excitations: Density-functional versus many-body Green's-function approaches
,”
Rev. Mod. Phys.
74
(
2
),
601
659
(
2002
).
7.
See the supplementary material online for (i) an example of the dimensionality effects on the polarizability. (ii) the details of the Fourier transform of Coulomb potentials, and (iii) a table with the parameters used to estimate the exciton binding energy.
8.
J. P.
Walter
and
M. L.
Cohen
, “
Wave-vector-dependent dielectric function for Si, Ge, GaAs, and ZnSe
,”
Phys. Rev. B
2
(
6
),
1821
1826
(
1970
).
9.
S.
Latini
,
T.
Olsen
, and
K. S.
Thygesen
, “
Excitons in van der Waals heterostructures: The important role of dielectric screening
,”
Phys. Rev. B
92
(
24
),
245123
(
2015
).
10.
W. C.
Dunlap
, Jr.
and
R. L.
Watters
, “
Direct measurement of the dielectric constants of silicon and germanium
,”
Phys. Rev.
92
(
6
),
1396
1397
(
1953
).
11.
T.
Olsen
,
S.
Latini
,
F.
Rasmussen
, and
K. S.
Thygesen
, “
Simple screened hydrogen model of excitons in two-dimensional materials
,”
Phys. Rev. Lett.
116
(
5
),
056401
(
2016
).
12.
H. T.
Nguyen-Truong
, “
Exciton binding energy and screening length in two-dimensional semiconductors
,”
Phys. Rev. B
105
,
L201407
(
2022
).
13.
A. C.
Neto
,
F.
Guinea
,
N.
Peres
,
K. S.
Novoselov
, and
A. K.
Geim
, “
The electronic properties of graphene
,”
Rev. Mod. Phys.
81
(
1
),
109
162
(
2009
).
14.
P.
Cudazzo
,
C.
Attaccalite
,
I. V.
Tokatly
, and
A.
Rubio
, “
Strong charge-transfer excitonic effects and the Bose-Einstein exciton condensate in graphane
,”
Phys. Rev. Lett.
104
(
22
),
226804
(
2010
).
15.
D.
Belitz
and
T. R.
Kirkpatrick
, “
The Anderson-Mott transition
,”
Rev. Mod. Phys.
66
(
2
),
261
380
(
1994
).
16.
S.
Brazovskii
and
N.
Kirova
, “
Physical theory of excitons in conducting polymers
,”
Chem. Soc. Rev.
39
(
7
),
2453
2465
(
2010
).
17.
H.
Haug
and
S. W.
Koch
,
Quantum Theory of the Optical and Electronic Properties of Semiconductors
, 4th ed. (
World Scientific Publishing Company
,
Singapore
,
2009
), pp.
169
183
.
18.
N.
Saigal
,
V.
Sugunakar
, and
S.
Ghosh
, “
Exciton binding energy in bulk MoS2: A reassessment
,”
Appl. Phys. Lett.
108
(
13
),
132105
(
2016
).
19.
B.
Yu
,
C.
Zhang
,
L.
Chen
,
Z.
Qin
,
X.
Huang
,
X.
Wang
, and
M.
Xiao
, “
Ultrafast dynamics of photoexcited carriers in perovskite semiconductor nanocrystals
,”
Nanophotonics
10
(
8
),
1943
1965
(
2021
).
20.
M. A.
Green
, “
Improved value for the silicon free exciton binding energy
,”
AIP Adv.
3
(
11
),
112104
(
2013
).
21.
D. D.
Sell
, “
Resolved free-exciton transitions in the optical-absorption spectrum of GaAs
,”
Phys. Rev. B
6
(
10
),
3750
3753
(
1972
).
22.
P. J.
Dean
,
E. C.
Lightowlers
, and
D. R.
Wight
, “
Intrinsic and extrinsic recombination radiation from natural and synthetic aluminum-doped diamond
,”
Phys. Rev.
140
(
1A
),
A352
A368
(
1965
).
23.
H.
Venghaus
, “
Valence-band parameters and g factors of cubic zinc selenide derived from free-exciton magnetoreflectance
,”
Phys. Rev. B
19
(
6
),
3071
3082
(
1979
).
24.
A. R.
Beal
,
J. C.
Knights
, and
W. Y.
Liang
, “
Transmission spectra of some transition metal dichalcogenides. II. Group VIA: Trigonal prismatic coordination
,”
J. Phys. C
5
(
24
),
3540
3551
(
1972
).
25.
D.
Novko
,
K.
Lyon
,
D. J.
Mowbray
, and
V.
Despoja
, “
Ab initio study of electromagnetic modes in two-dimensional semiconductors: Application to doped phosphorene
,”
Phys. Rev. B
104
(
11
),
115421
(
2021
).
26.
H.
Shi
,
H.
Pan
,
Y.-W.
Zhang
, and
B. I.
Yakobson
, “
Quasiparticle band structures and optical properties of strained monolayer MoS2 and WS2
,”
Phys. Rev. B
87
(
15
),
155304
(
2013
).
27.
F.
Iyikanat
,
E.
Torun
,
R. T.
Senger
, and
H.
Sahin
, “
Stacking-dependent excitonic properties of bilayer blue phosphorene
,”
Phys. Rev. B
100
(
12
),
125423
(
2019
).
28.
X. L.
Yang
,
S. H.
Guo
,
F. T.
Chan
,
K. W.
Wong
, and
W. Y.
Ching
, “
Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory
,”
Phys. Rev. A
43
(
3
),
1186
1196
(
1991
).
29.
X.
Ling
,
H.
Wang
,
S.
Huang
,
F.
Xia
, and
M. S.
Dresselhaus
, “
The renaissance of black phosphorus
,”
Proc. Natl. Acad. Sci. U. S. A.
112
(
15
),
4523
4530
(
2015
).
30.
F.
Grasselli
, “
Variational approach to the soft-coulomb potential in low-dimensional quantum systems
,”
Am. J. Phys.
85
(
11
),
834
839
(
2017
).

Supplementary Material

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