We present a comprehensive discussion of the so-called asymmetric Wigner bilayer system, where mobile point charges, all of the same sign, are immersed into the space left between two parallel, homogeneously charged plates (with possibly different charge densities). At vanishing temperatures, the particles are expelled from the slab interior; they necessarily stick to one of the two plates and form there ordered sublattices. Using complementary tools (analytic and numerical), we study systematically the self-assembly of the point charges into ordered ground state configurations as the inter-layer separation and the asymmetry in the charge densities are varied. The overwhelming plethora of emerging Wigner bilayer ground states can be understood in terms of the competition of two strategies of the system: net charge neutrality on each of the plates on the one hand and particles’ self-organization into commensurate sublattices on the other hand. The emerging structures range from simple, highly commensurate (and thus very stable) lattices (such as staggered structures, built up by simple motives) to structures with a complicated internal structure. The combined application of our two approaches (whose results agree within remarkable accuracy) allows us to study on a quantitative level phenomena such as over- and underpopulation of the plates by the mobile particles, the nature of phase transitions between the emerging phases (which pertain to two different universality classes), and the physical laws that govern the long-range behaviour of the forces acting between the plates. Extensive, complementary Monte Carlo simulations in the canonical ensemble, which have been carried out at small, but finite temperatures along selected, well-defined pathways in parameter space confirm the analytical and numerical predictions within high accuracy. The simple setup of the Wigner bilayer system offers an attractive possibility to study and to control complex scenarios and strategies of colloidal self-assembly, via the variation of two system parameters.

1.
E. P.
Wigner
,
Phys. Rev.
46
,
1002
(
1934
).
2.
C. C.
Grimes
and
G.
Adams
,
Phys. Rev. Lett.
42
,
795
(
1979
).
3.
D. C.
Tsui
,
H. L.
Stormer
, and
A. C.
Gossard
,
Phys. Rev. Lett.
48
,
1559
(
1982
).
4.
J. P.
Eisenstein
and
A. H.
MacDonald
,
Nature
432
,
691
(
2004
).
5.
Z.
Wang
,
Y. P.
Chen
,
L. W.
Engel
,
D. C.
Tsui
,
E.
Tutuc
, and
M.
Shayegan
,
Phys. Rev. Lett.
99
,
136804
(
2007
).
6.
Z.
Wang
,
Y. P.
Chen
,
Z.
Han
,
L. W.
Engel
,
D. C.
Tsui
,
E.
Tutuc
, and
M.
Shayegan
,
Phys. Rev. B
85
,
195408
(
2012
).
7.
D.
Zhang
,
X.
Huang
,
W.
Dietsche
,
K.
von Klitzing
, and
J. H.
Smet
,
Phys. Rev. Lett.
113
,
076804
(
2014
).
8.
D. S. L.
Abergel
and
T.
Chakraborty
,
Phys. Rev. Lett.
102
,
056807
(
2009
).
9.
G. E.
Morfill
and
A. V.
Ivlev
,
Rev. Mod. Phys.
81
,
1353
(
2009
).
10.
A.
Pertsinidis
and
X. S.
Ling
,
Phys. Rev. Lett.
87
,
098303
(
2001
).
11.
V. M.
Bedanov
and
F. M.
Peeters
,
Phys. Rev. B
49
,
2667
(
1994
).
12.
Yu. E.
Lozovik
,
Usp. Fiz. Nauk
153
,
356
(
1987
).
13.
Yu. E.
Lozovik
and
L. M.
Pomirchy
,
Phys. Status Solidi B
161
,
K11
(
1990
).
14.
Yu. E.
Lozovik
and
V. A.
Mandelshtam
,
Phys. Lett. A
145
,
269
(
1990
).
15.
Yu. E.
Lozovik
and
V. A.
Mandelshtam
,
Phys. Lett. A
165
,
469
(
1992
).
16.
F.
Bolton
and
U.
Rössler
,
Superlattices Microstruct.
13
,
139
(
1993
).
17.
M.
Baus
and
J.-P.
Hansen
,
Phys. Rep.
59
,
1
(
1980
).
18.
E. A.
Martinez
,
C. A.
Muschik
,
P.
Schindler
,
D.
Nigg
,
A.
Erhard
,
M.
Heyl
,
P.
Hauke
,
M.
Dalmonte
,
T.
Monz
,
P.
Zoller
, and
R.
Blatt
,
Nature
534
,
516
(
2016
).
19.
R.
Blatt
and
C. F.
Roos
,
Nat. Phys.
8
,
277
(
2012
).
20.
G.
Goldoni
and
F. M.
Peeters
,
Phys. Rev. B
53
,
4591
(
1996
).
21.
J.-J.
Weis
,
D.
Levesque
, and
S.
Jorge
,
Phys. Rev. B
63
,
045308
(
2001
).
22.
V.
Lobaskin
and
R. R.
Netz
,
Europhys. Lett.
77
,
36004
(
2007
).
23.
E. C.
Oǧuz
,
R.
Messina
, and
H.
Löwen
,
Europhys. Lett.
86
,
28002
(
2009
).
24.
L.
Šamaj
and
E.
Trizac
,
Europhys. Lett.
98
,
36004
(
2012
).
25.
L.
Šamaj
and
E.
Trizac
,
Phys. Rev. B
85
,
205131
(
2012
).
26.
T. B.
Mitchell
,
J. J.
Bollinger
,
D. H. E.
Dubin
,
X.-P.
Huang
,
W. M.
Itano
, and
R. H.
Baugham
,
Science
282
,
1290
(
1998
).
27.
D. H.
Winkle
and
C. A.
Murray
,
Phys. Rev. A
34
,
562
(
1986
).
28.
C. A.
Murray
and
D. H.
Van Winkle
,
Phys. Rev. Lett.
58
,
1200
(
1987
).
29.

The case σ1σ2 < 0 is somewhat simple and of little interest because we deal with ions that are all of the same charge. In this case, one plate repels them while the other is attractive, and structure I ensues. The interesting cases under study here are for σ1 and σ2 of the same sign, opposite to that of the point ions. Without loss of generality, we consider A = σ2/σ1 < 1, and both σ1, σ2, to be positive.

30.
S.
Earnshaw
,
Trans. Cambridge Philos. Soc.
7
,
97
(
1842
).
31.
D. E.
Goldberg
,
Genetic Algorithms in Search, Optimization, and Machine Learning
(
Addision-Wesley
,
Boston, MA
,
1989
).
32.
D.
Gottwald
,
G.
Kahl
, and
C.
Likos
,
J. Chem. Phys.
122
,
204503
(
2005
).
34.
J.
Fornleitner
,
F.
Lo Verso
,
G.
Kahl
, and
C. N.
Likos
,
Soft Matter
4
,
480
(
2008
).
35.
J.
Fornleitner
and
G.
Kahl
,
Europhys. Lett.
82
,
18001
(
2008
).
36.
G. J.
Pauschenwein
and
G.
Kahl
,
Soft Matter
4
,
1396
(
2008
).
37.
G.
Doppelbauer
,
E.
Bianchi
, and
G.
Kahl
,
J. Phys.: Condens. Matter
22
,
104105
(
2010
).
38.
G.
Doppelbauer
,
E. G.
Noya
,
E.
Bianchi
, and
G.
Kahl
,
Soft Matter
8
,
7768
(
2012
).
39.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation
, 2nd ed. (
Academic Press
,
Amsterdam
,
2001
).
40.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
, 2nd ed. (
Oxford
,
Oxford
,
2017
).
41.
P. J.
Steinhardt
,
D. R.
Nelson
, and
M.
Ronchetti
,
Phys. Rev. B
28
,
784
(
1983
).
42.
G.
Zhang
,
F. H.
Stillinger
, and
S.
Torquato
,
Phys. Rev. E
92
,
022119
(
2015
).
43.
J.
Gong
,
R. S.
Newman
,
M.
Engel
,
M.
Zhao
,
F.
Bian
,
S. C.
Glotzer
, and
Z.
Tang
,
Nat. Commun.
8
,
14038
(
2017
).
44.
S. C.
Glotzer
and
M. J.
Solomon
,
Nat. Mater.
6
,
557
(
2007
).
45.
E.
Bianchi
,
P. D. J.
van Oostrum
,
C. N.
Likos
, and
G.
Kahl
,
Curr. Opin. Colloid Interface Sci.
30
,
8
(
2017
).
46.
E.
Bianchi
,
B.
Capone
,
I.
Coluzza
,
L.
Rovigatti
, and
P. D. J.
van Oostrum
,
Phys. Chem. Chem. Phys.
19
,
19847
(
2017
).
47.
J.
Mikhael
,
J.
Roth
,
L.
Helden
, and
C.
Bechinger
,
Nature
454
,
501
(
2008
).
48.
Q.
Chen
,
S. C.
Bae
, and
S.
Granick
,
Nature
469
,
381
(
2011
).
49.
E.
Bianchi
,
C. N.
Likos
, and
G.
Kahl
,
ACS Nano
7
,
4657
(
2013
).
50.
E.
Bianchi
,
C. N.
Likos
, and
G.
Kahl
,
Nano Lett.
14
,
3412
(
2014
).
51.
M.
Antlanger
,
G.
Kahl
,
M.
Mazars
,
L.
Šamaj
, and
E.
Trizac
,
Phys. Rev. Lett.
117
,
118002
(
2016
).
52.
R.
Messina
and
H.
Löwen
,
Europhys. Lett.
91
,
146101
(
2003
).
53.
R. D.
Misra
,
Math. Proc. Cambridge Philos. Soc.
36
,
173
(
1940
);
M.
Born
and
R. D.
Misra
,
Math. Proc. Cambridge Philos. Soc.
36
,
466
(
1940
).
54.
R.
Byrd
,
P.
Lu
,
J.
Nocedal
, and
C.
Zhu
,
SIAM J. Sci. Stat. Comput.
16
,
1190
(
1995
).
55.
D.
Wales
and
J.
Doye
,
J. Phys. Chem. A
101
,
5111
(
1997
).
56.
M.
Mazars
,
Europhys. Lett.
84
,
55002
(
2008
).
57.
A.
Okabe
,
B.
Boots
,
K.
Sugihara
, and
S.
Nok Chiu
,
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams
, 2nd ed. (
John Wiley & Sons, Inc.
,
New York
,
2000
).
58.
W.
Mickel
,
S. C.
Kapfer
,
G. E.
Schröder-Turk
, and
K.
Mecke
,
J. Chem. Phys.
138
,
044501
(
2013
).
59.
H.
Leipold
,
E. A.
Lazar
,
K. A.
Brakke
, and
D. J.
Srolovitz
,
J. Stat. Mech.
2016
,
043103
;
H.
Leipold
,
E. A.
Lazar
,
K. A.
Brakke
, and
D. J.
Srolovitz
,
J. Stat. Mech.
2017
,
079901
.
60.
M.
Mazars
,
Europhys. Lett.
110
,
26003
(
2015
).
61.

With our code and for the choice of parameters for the Ewald method used in this study, central processing unit (CPU)-times that are required to perform 4.0 × 104 MC-cycles for a system with N ∼ 4000 particles typically amount to 48 CPU-hours on IBM IDataplex DX360 processors; this computer time also covers the numerical effort required for the Voronoi constructions57 which are performed after each MC-cycle.

62.
I. J.
Zucker
,
J. Math. Phys.
15
,
187
(
1974
).
63.
I. J.
Zucker
and
M. M.
Robertson
,
J. Phys. A: Math. Gen.
8
,
874
(
1975
).
64.
M.
Antlanger
,
M.
Mazars
,
L.
Šamaj
,
G.
Kahl
, and
E.
Trizac
,
Mol. Phys.
112
,
1336
(
2014
).
65.

Moiré patterns are created by superposing two simple lattices. Depending on the commensurability and the relative orientation of the two lattices, highly complex patterns can emerge.

66.
R.
Messina
,
C.
Holm
, and
K.
Kremer
,
Phys. Rev. E
64
,
021405
(
2001
).
67.
M.
Antlanger
,
G.
Doppelbauer
, and
G.
Kahl
,
J. Phys.: Condens. Matter
23
,
404206
(
2011
).
68.
B.
Grünbaum
and
G. C.
Shephard
,
Tilings and Patterns
, 2nd ed. (
Freeman
,
New York
,
1987
).
69.

In order to define these pathways, Ax=const.(η), suitable interpolating polynomials have been fixed, which are specified in  Appendix F.

70.
P. M.
Chaikin
and
T. C.
Lubensky
,
Principles of Condensed Matter Physics
(
Cambridge University Press
,
Cambridge
,
2013
).
71.
S.-K.
Ma
,
Modern Theory of Critical Phenomena
(
Westview Press
,
New York
,
1976
).
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