Quasi-exact solutions to the quantum relativistic two-body problem are obtained for a one-dimensional Woods-Saxon-like potential. The quantised positive energy spectrum is obtained in the square well potential limit in terms of a set of simple transcendental equations. This potential is used to calculate excitonic states in narrow-gap single-walled carbon nanotubes and the binding energy is shown to scale with the band gap.

1.
O.
Klein
,
Zeitschrift für Physik
53
,
157
165
(
1929
).
2.
A. D.
Alhaidari
and
E. El
Aaoud
, in
Proceedings of the Fifth Saudi Physical Society Conference (SPS5)
, Vol.
1370
(
AIP Publishing
,
2011
) pp.
21
25
.
3.
V.
Alonso
and
S.
De Vincenzo
,
Journal of Physics A: Mathematical and General
30
,
8573
(
1997
).
4.
P.
Alberto
,
C.
Fiolhais
, and
V. M. S.
Gil
,
European Journal of Physics
17
,
19
(
1996
).
5.
U.
Percoco
and
V. M.
Villalba
,
Physics Letters A
140
,
105
107
(
1989
).
6.
H. Y.
Chen
,
V. M.
Apalkov
, and
T.
Chakraborty
,
Physical Review Letters
98
,
186803
(
2007
).
7.
A.
Matulis
and
F. M.
Peeters
,
Physical Review B
77
,
115423
(
2008
).
8.
B.
Thaller
,
The dirac equation
, Vol.
31
(
Springer-Verlag Berlin
,
1992
).
9.
V. A.
Volkov
and
I. V.
Zagorodnev
,
Low Temperature Physics
35
,
2
5
(
2009
).
10.
A. V.
Turbiner
,
Soviet Phys. JETP
67
,
230
236
(
1988
).
11.
A. G.
Ushveridze
,
Quasi-exactly solvable models in quantum mechanics
(
CRC Press
,
1994
).
12.
C. M.
Bender
and
S.
Boettcher
,
Journal of Physics A: Mathematical and General
31
,
L273
(
1998
).
13.
C. A.
Downing
,
Journal of Mathematical Physics
54
,
072101
(
2013
).
14.
R. R.
Hartmann
,
Journal of Mathematical Physics
55
,
012105
(
2014
).
15.
R. R.
Hartmann
and
M. E.
Portnoi
,
Physical Review A
89
,
012101
(
2014
).
16.
A. H. C.
Neto
,
F.
Guinea
,
N. M. R.
Peres
,
K. S.
Novoselov
, and
A. K.
Geim
,
Reviews of Modern Physics
81
,
109
(
2009
).
17.
P. R.
Wallace
,
Physical Review
71
,
622
(
1947
).
18.
V. N.
Kotov
,
B.
Uchoa
,
V. M.
Pereira
,
F.
Guinea
, and
A. H. C.
Neto
,
Reviews of Modern Physics
84
,
1067
(
2012
).
19.
D. V.
Khveshchenko
,
Physical Review Letters
87
,
246802
(
2001
).
20.
J. E.
Drut
and
T. A.
Lähde
,
Physical Review Letters
102
,
026802
(
2009
).
21.
T.
Stroucken
,
J. H.
Grönqvist
, and
S. W.
Koch
,
Physical Review B
84
,
205445
(
2011
).
22.
Y. E.
Lozovik
and
A. A.
Sokolik
,
JETP letters
87
,
55
59
(
2008
).
23.
O. L.
Berman
,
Y. E.
Lozovik
, and
G.
Gumbs
,
Physical Review B
77
,
155433
(
2008
).
24.
O. L.
Berman
,
R. Y.
Kezerashvili
, and
K.
Ziegler
,
Physical Review A
87
,
042513
(
2013
).
25.
O. L.
Berman
,
R. Y.
Kezerashvili
, and
K.
Ziegler
,
Physical Review B
85
,
035418
(
2012
).
26.
O. L.
Berman
,
R. Y.
Kezerashvili
, and
K.
Ziegler
,
Physical Review B
86
,
235404
(
2012
).
27.
L.
Yang
,
Physical Review B
83
,
085405
(
2011
).
28.
P. V.
Ratnikov
and
A. P.
Silin
,
Journal of Experimental and Theoretical Physics
114
,
512
528
(
2012
).
29.
C. A.
Downing
and
M. E.
Portnoi
, arXiv preprint arXiv:1506.04425 (
2015
).
30.
M.
Mahmoodian
and
M. V.
Entin
,
Nanoscale Research Letters
7
,
1
3
(
2012
).
31.
M. M.
Mahmoodian
and
M. V.
Entin
,
EPL (Europhysics Letters)
102
,
37012
(
2013
).
32.
L. L.
Marnham
and
A. V.
Shytov
,
Physical Review B
92
,
035437
(
2015
).
33.
A.
Iyengar
,
J.
Wang
,
H. A.
Fertig
, and
L.
Brey
,
Physical Review B
75
,
125430
(
2007
).
34.
O. L.
Berman
,
R. Y.
Kezerashvili
, and
Y. E.
Lozovik
,
Physical Review B
78
,
035135
(
2008
).
35.
J.
Sabio
,
F.
Sols
, and
F.
Guinea
,
Physical Review B
81
,
045428
(
2010
).
36.
M.
Zarenia
,
B.
Partoens
,
T.
Chakraborty
, and
F. M.
Peeters
,
Physical Review B
88
,
245432
(
2013
).
37.
J.
Wang
,
H. A.
Fertig
,
G.
Murthy
, and
L.
Brey
,
Physical Review B
83
,
035404
(
2011
).
38.
R. R.
Hartmann
,
I. A.
Shelykh
, and
M. E.
Portnoi
,
Physical Review B
84
,
035437
(
2011
).
39.
R. N.
Lee
,
A. I.
Milstein
, and
I. S.
Terekhov
,
Physical Review B
86
,
035425
(
2012
).
40.
M. E.
Portnoi
,
C. A.
Downing
,
R. R.
Hartmann
, and
I. A.
Shelykh
, in
2013 International Conference on Electromagnetics in Advanced Applications (ICEAA)
(
IEEE
,
2013
) pp.
231
234
.
41.
T.
Ando
,
Journal of the Physical Society of Japan
66
,
1066
1073
(
1997
).
42.
O. V.
Kibis
,
M.
Rosenau da Costa
, and
M. E.
Portnoi
,
Nano Letters
7
,
3414
3417
(
2007
).
43.
M. E.
Portnoi
,
O. V.
Kibis
, and
M.
Rosenau da Costa
,
Superlattices and Microstructures
43
,
399
407
(
2008
).
44.
R. R.
Hartmann
,
J.
Kono
, and
M. E.
Portnoi
,
Nanotechnology
25
,
322001
(
2014
).
45.
R. R.
Hartmann
and
M. E.
Portnoi
,
IOP Conference Series: Materials Science and Engineering
79
,
12014
12018
(
2015
).
46.
C. L.
Kane
and
E. J.
Mele
,
Physical Review Letters
78
,
1932
(
1997
).
47.
R. R.
Hartmann
and
M. E.
Portnoi
,
Optoelectronic Properties of Carbon-based Nanostructures: Steering electrons in graphene by electromagnetic fields
(
LAP LAMBERT Academic Publishing
,
Saarbrucken
,
2011
).
48.
L.
Brey
and
H. A.
Fertig
,
Physical Review B
73
,
235411
(
2006
).
49.
M. E.
Portnoi
,
M.
Rosenau da Costa
,
O. V.
Kibis
, and
I. A.
Shelykh
,
International Journal of Modern Physics B
23
,
2846
2850
(
2009
).
50.
D. S.
Miserev
and
M. V.
Entin
,
Journal of Experimental and Theoretical Physics
115
,
694
705
(
2012
).
51.
C. A.
Downing
and
M. E.
Portnoi
,
Physical Review A
90
,
052116
(
2014
).
52.
A.
Ronveaux
and
F. M.
Arscott
,
Heun’s differential equations
(
Oxford University Press
,
1995
).
53.
R. S.
Maier
,
Mathematics of Computation
76
,
811
843
(
2007
).
54.
T.
Ando
and
S.
Uryu
,
Physica Status Solidi (c)
6
,
173
180
(
2009
).
This content is only available via PDF.
You do not currently have access to this content.