We investigate the confined states and transport of three-dimensional Weyl electrons around a one-dimensional external rectangular electrostatic potential. The confined states with finite transverse wave vector exist at energies higher than the half well depth or lower than the half barrier height. The rectangular potential appears completely transparent to the normal incident electrons but not otherwise. The tunneling transmission coefficient is sensitive to their incident angle and shows resonant peaks when their energy coincides with the confined spectra. In addition, for the electrons in the conduction (valence) band through a potential barrier (well), the transmission spectrum has a gap of width increasing with the incident angle. Interestingly, the electron linear zero-temperature conductance over the potential can approach zero when the Fermi energy is aligned to the top and bottom energies of the potential, when only electron beams normal to the potential interfaces can pass through. The considered structure can be used to collimate the Weyl electron beams.

1.
M. Z.
Hasan
and
C. L.
Kane
,
Rev. Mod. Phys.
82
,
3045
(
2010
).
2.
X.-L.
Qi
and
S.-C.
Zhang
,
Rev. Mod. Phys.
83
,
1057
(
2011
).
3.
A. A.
Burkov
and
L.
Balents
,
Phys. Rev. Lett.
107
,
127205
(
2011
);
[PubMed]
A. A.
Burkov
,
M. D.
Hook
, and
L.
Balents
,
Phys. Rev. B
84
,
235126
(
2011
).
4.
P.
Hosur
and
X.
Qi
,
C. R. Phys.
14
,
857
(
2013
).
5.
W.
Witczak-Krempa
,
G.
Chen
,
Y. B.
Kim
, and
L.
Balents
,
Annu. Rev. Condens. Matter Phys.
5
,
57
(
2014
).
6.
O.
Vafek
and
A.
Vishwanath
,
Annu. Rev. Condens. Matter Phys.
5
,
83
(
2014
).
7.
Q. D.
Gibson
,
L. M.
Schoop
,
L.
Muechler
,
L. S.
Xie
,
M.
Hirschberger
,
N. P.
Ong
,
R.
Car
, and
R. J.
Cava
,
Phys. Rev. B
91
,
205128
(
2015
).
8.
A. M.
Turner
and
A.
Vishwanath
, e-print arXiv:1301.0330.
10.
X.
Wan
,
A. M.
Turner
,
A.
Vishwanath
, and
S. Y.
Savrasov
,
Phys. Rev. B
83
,
205101
(
2011
).
11.
G.
Xu
,
H.
Weng
,
Z.
Wang
,
X.
Dai
, and
Z.
Fang
,
Phys. Rev. Lett.
107
,
186806
(
2011
).
12.
Y.
Baum
,
E.
Berg
,
S. A.
Parameswaran
, and
A.
Stern
,
Phys. Rev. X
5
,
041046
(
2015
).
13.
14.
S. A.
Parameswaran
,
T.
Grover
,
D. A.
Abanin
,
D. A.
Pesin
, and
A.
Vishwanath
,
Phys. Rev. X
4
,
031035
(
2014
).
15.
P.
Hosur
and
X.-L.
Qi
,
Phys. Rev. B
91
,
081106
(
2015
).
16.
H.
Jiang
,
Z.
Qiao
,
H.
Liu
, and
Q.
Niu
,
Phys. Rev. B
85
,
045445
(
2012
).
17.
S. A.
Skirlo
,
L.
Lu
,
Y.
Igarashi
,
J.
Joannopoulos
, and
M.
Soljacic
,
Phys. Rev. Lett.
115
,
253901
(
2015
).
18.
Q.-D.
Jiang
,
H.
Jiang
,
H.
Liu
,
Q.-F.
Sun
, and
X. C.
Xie
,
Phys. Rev. Lett.
115
,
156602
(
2015
).
19.
S. A.
Yang
,
H.
Pan
, and
F.
Zhang
,
Phys. Rev. Lett.
115
,
156603
(
2015
).
20.
K.-Y.
Yang
,
Y.-M.
Lu
, and
Y.
Ran
,
Phys. Rev. B
84
,
075129
(
2011
).
21.
22.
J.
Liu
and
D.
Vanderbilt
,
Phys. Rev. B
90
,
155316
(
2014
).
23.
H.
Weng
,
C.
Fang
,
Z.
Fang
,
B. A.
Bernevig
, and
Xi.
Dai
,
Phys. Rev. X
5
,
011029
(
2015
).
24.
S.-Y.
Xu
,
I.
Belopolski
,
C.-C.
Lee
,
G.
Chang
,
B.
Wang
,
N.
Alidoust
,
G.
Bian
,
M.
Neupane
,
C.
Zhang
,
S.
Jia
,
A.
Bansil
,
H.
Lin
, and
M.
Zahid Hasan
,
Nat. Commun.
6
,
7373
(
2015
).
25.
C.
Shekhar
,
F.
Arnold
,
S.-C.
Wu
,
Y.
Sun
,
M.
Schmidt
,
N.
Kumar
,
A. G.
Grushin
,
J. H.
Bardarson
,
R.
Donizeth dos Reis
,
M.
Naumann
,
M.
Baenitz
,
H.
Borrmann
,
M.
Nicklas
,
E.
Hassinger
,
C.
Felser
, and
B.
Yan
, e-print arXiv:1506.06577.
26.
B. Q.
Lv
,
H. M.
Weng
,
B. B.
Fu
,
X. P.
Wang
,
H.
Miao
,
J.
Ma
,
P.
Richard
,
X. C.
Huang
,
L. X.
Zhao
,
G. F.
Chen
,
Z.
Fang
,
X.
Dai
,
T.
Qian
, and
H.
Ding
,
Phys. Rev. X
5
,
031013
(
2015
);
27.
C.
Zhang
,
S.-Y.
Xu
,
I.
Belopolski
,
Z.
Yuan
,
Z.
Lin
,
B.
Tong
,
N.
Alidoust
,
C.-C.
Lee
,
S.-M.
Huang
,
H.
Lin
,
M.
Neupane
,
D. S.
Sanchez
,
H.
Zheng
,
G.
Bian
,
J.
Wang
,
C.
Zhang
,
T.
Neupert
,
M. Z.
Hasan
, and
S.
Jia
, e-print arXiv:1503.02630.
28.
C.
Shekhar
,
A. K.
Nayak
,
Y.
Sun
,
M.
Schmidt
,
M.
Nicklas
,
I.
Leermakers
,
U.
Zeitler
,
Z.
Liu
,
Y.
Chen
,
W.
Schnelle
,
J.
Grin
,
C.
Felser
, and
B.
Yan
,
Nat. Phys.
11
,
645
(
2015
).
29.
Z. K.
Liu
,
B.
Zhou
,
Y.
Zhang
,
Z. J.
Wang
,
H. M.
Weng
,
D.
Prabhakaran
,
S.-K.
Mo
,
Z. X.
Shen
,
Z.
Fang
,
X.
Dai
,
Z.
Hussain
, and
Y. L.
Chen
,
Science
343
,
864
(
2014
).
30.
S.-Y.
Xu
,
C.
Liu
,
S. K.
Kushwaha
,
R.
Sankar
,
J. W.
Krizan
,
I.
Belopolski
,
M.
Neupane
,
G.
Bian
,
N.
Alidoust
,
T.-R.
Chang
,
H.-T.
Jeng
,
C.-Yi.
Huang
,
W.-F.
Tsai
,
H.
Lin
,
P. P.
Shibayev
,
F.-C.
Chou
,
R. J.
Cava
, and
M. Z.
Hasan
,
Science
347
,
294
(
2015
).
31.
Z. K.
Liu
,
J.
Jiang
,
B.
Zhou
,
Z. J.
Wang
,
Y.
Zhang
,
H. M.
Weng
,
D.
Prabhakaran
,
S-K.
Mo
,
H.
Peng
,
P.
Dudin
,
T.
Kim
,
M.
Hoesch
,
Z.
Fang
,
X.
Dai
,
Z. X.
Shen
,
D. L.
Feng
,
Z.
Hussain
, and
Y. L.
Chen
,
Nat. Mater.
13
,
677
(
2014
).
32.
Z.
Wang
,
H.
Weng
,
Q.
Wu
,
X.
Dai
, and
Z.
Fang
,
Phys. Rev. B
88
,
125427
(
2013
).
33.
T.
Liang
,
Q.
Gibson
,
M. N.
Ali
,
M.
Liu
,
R. J.
Cava
, and
N. P.
Ong
,
Nat. Mater.
14
,
280
(
2015
).
34.
Y.
Liu
,
C.
Zhang
,
X.
Yuan
,
T.
Lei
,
C.
Wang
,
L.
He
,
R.
Che
, and
F.
Xiu
,
NPG Asia Mater.
7
,
e221
(
2015
).
35.
Z.-G.
Chen
,
C.
Zhang
,
Y.
Zou
,
E.
Zhang
,
L.
Yang
,
F.
Xiu
, and
J.
Zou
,
Nano Lett.
15
,
5830
(
2015
).
36.
C.-Z.
Li
,
L.-X.
Wang
,
H.
Liu
,
J.
Wang
,
Z.-M.
Liao
, and
D.-P.
Yu
,
Nat. Commun.
6
,
10137
(
2015
).
37.
H. B.
Nielsen
and
M.
Ninomiya
,
Phys. Lett. B
130
,
389
(
1983
).
38.
X.
Huang
,
L.
Zhao
,
Y.
Long
,
P.
Wang
,
D.
Chen
,
Z.
Yang
,
H.
Liang
,
M.
Xue
,
H.
Weng
,
Z.
Fang
,
X.
Dai
, and
G.
Chen
,
Phys. Rev. X
5
,
031023
(
2015
).
39.
X.
Yang
,
Y.
Liu
,
Z.
Wang
,
Y.
Zheng
, and
Z.-A.
Xu
, e-print arXiv:1506.03190.
40.
J. M.
Pereira
,
V.
Mlinar
,
F. M.
Peeters
, and
P.
Vasilopoulos
,
Phys. Rev. B
74
,
045424
(
2006
).
41.
M.
Barbier
,
F. M.
Peeters
,
P.
Vasilopoulos
, and
J. M.
Pereira
,
Phys. Rev. B
77
,
115446
(
2008
);
M.
Barbier
,
F. M.
Peeters
,
P.
Vasilopoulos
, and
J. M.
Pereira
,
Phys. Rev. B
79
,
155402
(
2009
).
You do not currently have access to this content.