Transmission spectra and valley processing of graphene and carbon nanotube superlattices with inter-valley coupling

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by  Fuming Xu

Transmission spectra and valley processing of graphene and carbon nanotube
superlattices with inter-valley coupling
Fuming Xu,1 Zhizhou Yu,2 Yafei Ren,3,4 Bin Wang,1 Yadong Wei,1,∗ and Zhenhua Qiao3,4,†
1College of Physics and Energy, Shenzhen University, Shenzhen 518060, China.
2Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Hong Kong, China
3ICQD, Hefei National Laboratory for Physical Science at Microscale,
and Synergetic Innovation Center of Quantum Information and Quantum Physics,
7
University of Science and Technology of China, Hefei, Anhui 230026, China
1
4CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics,
0
University of Science and Technology of China, Hefei, Anhui 230026, China
2
n We numerically investigate the electronic transport properties of graphene nanoribbons and car-
a bon nanotubes with inter-valley coupling, e.g., in √3N √3N and 3N 3N superlattices. By
× ×
J takingthe√3 √3graphenesuperlatticeasanexample,weshowthattailoring thebulkgraphene
3 superlattice res×ults in rich structural configurations of nanoribbons and nanotubes. After studying
the electronic characteristics of the corresponding armchair and zigzag nanoribbon geometries, we

] find that the linear bands of carbon nanotubes can lead to the Klein tunnelling-like phenomenon,
l i.e., electrons propagatealong tubeswithout backscatteringevenin thepresenceofbarrier. Dueto
l
a the coupling between K and K′ valleys of pristine graphene by √3 √3 supercell, we propose a
h ×
valley-field-effect transistor based on the armchair carbon nanotube, where the valley polarization
-
ofthecurrentcanbetunedbyapplyingagatevoltageorvaryingthelengthofthearmchaircarbon
s
e nanotubes.
m
PACSnumbers: 72.10.-d,81.05.Ue,73.23.-b,73.63.-b
.
t
a
m
I. INTRODUCTION appear in graphene proximity-coupled with topological
- insulator substrates.28
d
n Valleytronics aim to design high-efficiency and low-
In this article, we explore the potential application of
o dissipation electronic devices by manipulating the Bloch
c these graphene superlattices in valleytronics and extend
electrons’ valley degree of freedom, which refers to the
[ our study to carbon nanotubes. Within top-adsorption
local minima of the electronic band structure in the
case,westudytherepresentative√3 √3superlatticeof
1 reciprocal space. In some traditional multi-valley sys- ×
graphene and carbon nanotubes without loss of general-
v tems, such as silicon,1–3 bismuth,4 and diamonds,5 the
2 itysincetheinter-valleycouplingmechanismsareuniver-
valley degree of freedom is shown to be controllable to
8 sal features in these superlattices. The √3 √3 super-
carry and transport information. In two-dimensional ×
5 latticewithtop-adsorptionintroducesmultiplestructural
0 materials, honeycomb-lattice systems are of special in- configurations of graphene nanoribbons and nanotubes.
0 terest in the study of valleytronics due to the presence
We focus on three kinds of zigzag ribbons, two kinds of
. of two inequivalent valleys K and K′.9–11 Particularly,
1 armchair ribbon, and typical armchair and zigzag sin-
graphene has attracted much attention due to its ex-
0 gle walled nanotubes, and theoretically investigate their
7 cellent electronic and mechanical properties6–8 Various electronic properties using the tight-binding model. Our
1 valleytronics devices have been proposed in graphene
numerical results show that there exist Klein tunneling-
: nanostructures12–16 utilizing, e.g., zigzag edges,17 zero-
v like phenomena in √3 √3 armchair and zigzag carbon
i line modes,18–20 topological line defects,21–23 strain and nanotubes evenin the×presence ofbarrier. By employing
X mechanical engineering,24,25 as well as temperature
the inter-valley coupling to induce valley processing, we
r gradient,26 to generate and control valley-polarized cur-
a propose a valley-field-effect transistor consisting of pris-
rents.
tineand√3 √3armchaircarbonnanotube,whichgen-
Recently, a new valley engineering mechanism is pro- erates nearly×fully valley-polarized current at large gate
posed in √3N √3N or 3N 3N superlattices of voltage and electron energy scale.
graphene.27 Due×to the band fold×ing in the superlattice,
the inequivalent K/K′ valleys in pristine graphene are The remaining ofour paper is organizedas follows. In
folded into the same Γ point and thus inter-valley cou- Sec.II,thetight-bindingHamiltonianofthebulk√3 √3
×
pling arises that act as valley-orbit coupling similar to graphene and the Green’s function method are intro-
spin-orbitcouplingprovidingpromisingvalley-processing duced. Thenumericalresultsonelectronicandtransport
mechanisms via electrical means. Reference [27] sug- properties of various confined graphene nanostructures,
gested that the √3N √3N or 3N 3N superlattices including zigzagandarmchairribbons,andsingle walled
× ×
could be realized in periodically doped graphene. More nanotubesareshowninSec.III.Abriefsummaryisgiven
recently,suchkindofspecialsupercellsarealsoshownto in Sec. IV.
2
II. TIGHT-BINDING HAMILTONIAN AND (a) (b)
3
THEORETICAL FORMALISM
2
1
V)1
In √3 √3 graphene superlattice, carbon atoms are E(ef0 2
still the m×ajority and π-orbital expansion is therefore a E--1
3 y
reasonable starting point. The nearest-neighbor tight- -2
binding Hamiltonian of the bulk √3 √3 graphene -3
reads:27 × -4M K M x
(c) (d)
H =−Xti,j c†icj +Xui c†ici, (1)
hi,ji i
where c†(c ) is the π-orbital creation (annihilation) op-
i i
erator on site i. To recover the single-valley metallic
phase of bulk √3 √3 graphene superlattice,27 the sys-
×
tem parameters are precisely determined. The nearest- FIG.1: Panel (a): bandstructureof bulk√3 √3graphene
×
neighbor hopping amplitudes t are t = 2.9 eV be- superlattice calculated from tight-binding Hamiltonian de-
i,j 1
tween top-absorption site and adjacent carbon atoms, fined in Eq.1 with periodic boundary conditions. Panel (b),
(c)and(d): schematicplotsofthreetypesof√3 √3zigzag
andt =2.6eVbetweencarbonatoms,respectively. The
2 ×
graphene ribbons. Green dots stand for top-absorption sites
on-site potentials u of different sites are chosen to be
i in the lattice. Blue rectangles indicate the unit cell in every
u = 4.79 eV for top-absorption site, u = 1.35 eV
1 − 2 − setup. The ribbons are finite in y direction and periodically
for its three nearest carbon sites, and u = 1.05 eV for
3 − infinitein x direction.
the rest carbon sites with no top-absorption neighbors.
These parameters perfectly recover the band structure
in Ref. [27], where periodic adatom or top-absorption
III. NUMERICAL RESULTS AND
introduces symmetry-breaking and valley-scattering in DISCUSSIONS
graphene superlattice. Detailed procedures on param-
eter selection can be found in Ref. [27].
In this Section, numerical results on the electronic
Based on this tight-binding Hamiltonian, the band
structures and transport properties of typical nanorib-
structureofbulk√3 √3grapheneisobtainedasshown bons and nanotubes of √3 √3 graphene supercell are
×
in Fig. 1(a) where the inequivalent K and K′ valleys in ×
presented, including both zigzag and armchair geome-
pristine graphene are folded at the same Γ point due to
tries.
the band folding. In our transport study, we focus on
an energy interval within several eV around the Fermi
energy. In Fig.1(a), it is obvious that the three bands
(labeled as ”1”, ”2” and ”3”) dominate in this energy A. Zigzag ribbons of √3 √3 graphene superlattice
×
range. Bands2and3formanidealquadraticcrossover.27
Moreover, one can also notice that bands ”1” and ”3” As displayed in Fig. 1, we exhibit three kinds of
are almost linear around Γ point. The influence of such √3 √3 zigzag graphene nanoribbons (ZGRs). Top-
×
kind of linear dispersion will be discussed when study- absorptionsites arehighlightedasgreendots. Blue rect-
ingthetransportpropertiesof√3graphenenanoribbons angles are used to indicate the unit cells in different
and carbon nanotubes. configurations. For simplicity, the setups in panels (b),
In below, we utilize the Green’s function method to (c), and (d) of Fig. 1 are respectively denoted as ZGR-
investigate the valley-related electronic transport prop- 1, ZGR-2, and ZGR-3. Their geometric differences can
erties. From the tight-binding Hamiltonian shown in be easily distinguished from the relative position of top-
Eq. (1), the transmission coefficient of electrons at en- absorptionsitesandtheirpresence/absenceattheribbon
ergy E can be expressed as: boundaries. The lower boundaries of ZGR-1 and ZGR-2
arepurelyconsistedofcarbonatomswhiletheabsorption
sites of these two samples are locate on different sublat-
T(E)=Tr[Γ GrΓ Ga], (2)
L R tices. Differently, adsorption sites appear at the lower
boundary of ZGR-3. As displayed in Fig. 1, these ZGRs
whereGr =[E H Σr]−1 istheretardedGreen’sfunc- havefinitewidthsalongydirectionandareperiodicalong
tion, and Ga =−(Gr−)† is the advanced Green’s function. xdirection. Thesezigzagribbonscanbe realizedbycut-
Σr = Σr +Σr is the self-energy from the left and right ting a large √3 √3 graphene sheet along proper direc-
L R
leads. Γ isthelinewidthfunctiondescribingthecou- tions, like the f×abrication of graphene nanoribbon29 or
L/R
pling betweenthe left/rightleadand the centralscatter- throughthe lithographymethod.30 Inourfirstprinciples
ingregion,andcanbedefinedasΓ =i[Σr Σa ]. calculation,31 we found that ZGR-1 exhibits the lowest
L/R L/R− L/R
3

1 8
V) 0 (a) wid= 24
e 1 ) 6 wid= 52
E-E (f--21 (a) (b ) (c ) 23 T(Ef4 wid=100
-3 2
1
V) 0
e 0 1 8
E-E ( f--21 (d) (e ) ( f) 23 (E)f6 (b) wwwiiiddd=== 1250420
T 4
-3
1
eV) 0 1 2
E ( f-1 (g) (h ) ( i) 2 08
E- (c) wid= 24
-2 3 E)f6 wwiidd== 15020
-3- 0 /- 0 /- 0 T( 4
kxa kxa kxa
2
FIG. 2: Band structures of three types of √3 zigzag ribbons
0
with various system sizes. Panels (a)-(c): band structures of
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
ZGR-1withribbonwidth24,52,and100latticesites;Panels
E (eV)
(d)-(f): dispersionrelationsofZGR-2withsystemsize24,52, f
and 100 sites; Panels (g)-(i): bands of ZGR-3with width 24,
FIG.3: TransmissioncoefficientasafunctionofFermienergy
52, and 100 sites. for three types of √3 √3 zigzag graphene ribbons (ZGRs)
×
at different widths. Panel (a), (b), and (c) corresponds to
ZGR-1,ZGR-2,and ZGR-3,respectively. In all panels, three
Gibbs free energy,while ZGR-3 exhibits the highestone. ribbon widths 24, 52, and 100 sites are evaluated.
Therefore,ZGR-1shouldtheoreticallybethemoststable
structureintheseZGRs. Inourfollowingdiscussions,all
these setups have been consideredandwe will show that
higher energy degeneracies. Compared with Fig. 1(a),
they possess distinct electronic properties.
one reasonable explanation is that these envelopes re-
Figure2displaysthebandstructuresofthreedifferent
flect the three bands in the corresponding bulk bands.
zigzaggrapheneribbonsfordifferentwidths(e.g.,24,52,
For ZGR-2, bands for system widths 24, 52, and 100
and 100 lattice sites). The first, second and third rows
sites are respectively plotted in Fig. 2(d), (e) and (f),
of Fig. 2 correspond respectively to ZGR-1, ZGR-2 and
which exhibit similar characters as those in ZGR-1 ex-
ZGR-3. One can find that Fig. 2(a) (for ZGR-1 with a
cept that these two edge modes are separable and di-
width of 24 lattice sites) exhibits a large energy gap ∆
vide the bandgapintothree narrowones. As the ribbon
locatedintheintervalof[ 1, 2]eV,deeplyunderneath
− − width increases,the upper and lowergaps close firstand
theFermienergy. Meanwhile,twonearlyflatbandsliein
the middle gap disappears at last. The cases for ZGR-
the gap. By projecting the local density of states of the
3 is similar to that of ZGR-2 as displayed in the last
twobandsontothe lattice sites,wefindthatthey arelo-
three panels of Fig. 2. However, different from ZGR-2,
calizedalongthezigzagribbonboundariesandthusthey
the upper surface bandofZGR-3 is more flat. Neverthe-
areedgemodes,whichoriginatefromthedanglingbonds
less, the band structures of these three types of zigzag
alongthezigzagedges,likethoseinzigzagribbonofpris-
ribbons share some common features, including surface
tine graphene. As clearly presented in Fig. 1, the upper
bands pinning at k = π and bands envelopes at large
and lower edges of these ZGRs are different, leading to x
±
systemsizes. Besides,these zigzagribbons areallmetal-
theformationoftwodifferentedgemodes. Astheribbon
lic since the Fermi level lies in the conduction band.
width increases, the energy gap above the edge modes
vanishes first as shown in Fig. 2(b) and the lower gap InFig.3, we plotthe transmissioncoefficientsofthese
closes at a larger ribbon width as displayed in Fig. 2(c). zigzagribbonsasafunctionofenergy,wheretheleftand
Therefore, these two band gaps arise from the finite-size right leads are exactly extended from the central region,
effect and disappear when the system size is sufficiently hence resulting in the quantized T(E ). Panels (a), (b)
F
large. and (c) correspond respectively to ZGR-1, ZGR-2 and
For ZGR-1 as the system width increases, another ob- ZGR-3. In each setup, three ribbon widths (i.e., 24, 52
servation is that the edge modes are pinned at k = π and 100 lattice sites) are considered. One can see an
x
±
but evolve with the bulk states around k 0. One can exact mapping between the transmission coefficients in
x
≈
also notice that, in Fig. 2(c), the band structure estab- Fig. 3 and the band structures in Fig. 2. At small sys-
lishes three envelopes, labeled as “1”-“3”. Upon larger temsize,alltheZGRshavezerotransmissioncoefficients
systemsize,theenvelopesbecomefurtherenhancedwith at certain energy regions below the Fermi level, where
4
(a) (b)
1 1
y
V) 0 0
e (a) (b) (c)
x E (-f1 wid=24 -1
-
E-2 wid=60 -2

-3 -3
1 1
)
V
e 0 (d) (e) (f) 0
(
E -f1 -1
-
E wid=24
FIG. 4: Sketches of two types of √3 armchair graphene rib- -2 wid=60 -2
bons: AGR-1andAGR-2,respectively. Bluerectangles show
-3 -3
the corresponding unit cells and green dots indicate top- - 0 /- 0 0 2 4 6 8 10 12
absorptionsites. Theribbonshavefinitewidthsinydirection kxa kxa T(Ef)
and extend to infinite x.
±
FIG. 5: Panels (a) and (b): band structures of AGR-1 with
systemwidths24and60sites. Panel(c)plotsthecorrespond-
the dispersion relations show energy gaps. For ZGR-1, ingT vsEF curvesforthesesystemsizes. Panels(d)and(e):
band structures of AGR-2 at ribbon widths 24 and 60 sites.
there are two zero-transmission-coefficient regions while
Panel (f) shows the transmission of AGR-2.
three gaps exist in ZGR-2 and ZGR-3 at ribbon width
(i.e., with 24 lattice sites in black lines). At system size
of 52, the zero conducting ranges shrink in all panels of
Fig.3. OnlyonegaparepresentintheZGR-2andZGR-
3, and two gaps still reside in the transmissionspectrum
of ZGR-3, shown in red lines. For a larger system with
100lattice sites, ZGR-1 has no zero transmissionareain lations, two ribbon widths (24 and 60 lattice sites) are
the whole energy interval and single narrowgaps appear consideredforbothAGR-1andAGR-2. Onenoticesthat
in panels (b) and (c) of Fig. 3. These finite-size gaps both armchair ribbons are good conductors, similar as
eventually disappear at even larger systems. As the rib- the ZGRs. Figures 5(a) and 5(b) plot the band struc-
bonwidthincreases,transmissioncoefficientatthe same turesofAGR-1. Apparently,thereisadirectbandgapat
energyincreases rapidlyfor allsystems. We alsoobserve kx =0aroundE 1.4eV.Asthesystemsizeincreases
∼−
some oscillationbehavior ofT(E ) aroundE 0.5 eV, from 24 to 60 lattice sites, this gap becomes narroweras
F F
≈
whichcanbeattributedtothebandoverlappingatthese shown in Fig. 5(b), indicating its finite-size nature. The
energies. One can deduce from Fig .2 that, the oscilla- transmissioncoefficientofAGR-1asafunctionofenergy
tions tend to be more wild at larger system sizes, which isdisplayedinFig. 5(c). Itis foundthatthe conducting-
is confirmed by our transport calculations. forbidden region matches exactly the band gaps in the
left panels. This gap is gradually reduced when system
size increases and finally vanishes as the ribbon width
is large enough. For AGR-2, the numerical results are
B. Armchair ribbons of √3 √3 graphene
× drawn in the lower panels of Fig. 5. The energy disper-
superlattice
sion of AGR-2 is rather similar to that of AGR-1 except
that AGR-2 has a larger band gap at the same system
Depending on the relative positions of top-absorption
size. The transmission coefficient versus energy curves
on the honeycomb lattice, there are two different config-
in Fig. 5(f) also confirm this observation. As a result, a
urationsof armchairgraphenenanoribbons(AGRs). We
wider ribbon width is required to close the band gap in
schematically plot these AGRs with widths of two full
AGR-2.
unitcellsinFig.4. Forsimplicity,wedenotethearmchair
ribbonshowninFig.4(a)asAGR-1andtheotheroneas
AGR-2. Unlikezigzagribbonsof√3 √3graphene,both ComparedwithZGRs,thereisnoedgemodeinAGRs.
×
armchairribbons reproduceperfect √3 √3 periodicity. This is a much natural expectation for armchair-edged
×
The first-principles calculation suggests that the Gibbs ribbons because of the valley mixture behaviour. An-
free energy of AGR-1 is higher than that of AGR-2.31 other striking difference from ZGRs is that there is less
Combining with the results of √3 √3 ZGRs, one can oscillationfeatureinthetransmissionspectrumofAGRs,
×
conclude that the ribbons for both zigzag and armchair which can be attributed to the absence of band-folding
formsarelessstablewhenthe top-absorptionsitesreside from 1 1 to √3 √3 supercells in the armchair con-
× ×
on the ribbon boundaries. figurations. From Figs. 5(b) and 5(e), one can observe
The electronic properties of these AGRs are numeri- that three band envelopesdevelopto reproduce the bulk
cally investigated and displayed in Fig. 5. In our calcu- bands of √3 √3 graphene superlattice.
×
5
(20,20)armchaircarbonnanotubes. Whentransforming
1
the zigzag ribbons into armchairs tubes, it is clear that
eV) 0 there are neither edge modes nor gaps in the bands of
E (f-1 √3 √3 armchair nanotubes. In Fig. 6(a), one can see
E--2 (a) (b) that×there is a band touching at E 1.4 eV. Near the
-3 touching point, the band density is≈lo−w for this (12,12)
1
V) armchair tube. For a larger (20,20) tube as shown in
e 0
E-E (f-1 (c) (d) Fbaign.d6(tbo)u,chthinegbapnodintd.enTsihtyebbeacnodmestsrduecntusererswoitfhthaefitxwedo
-2 √3 √3zigzagcarbonnanotubesareplottedinFig.6(c)
×
-3 and 6(d). The band touching also appear in these sys-
- 0 /- 0
kxa kx a tems. For the same energy, the bands of zigzag tubes
8 8
under the touching point reside around k = 0 point,
x
E)f6 (e) (f) 6 instead of spreading in the whole Brillouin zone as in
T( 4 4 the armchair tubes. This behavior indicates that cur-
2 (12,12) (12,0) 2 rent carrier in zigzag ribbons has a larger group velocity
(30,30) (45,0)
in this energy range. Except this minor difference, the
0 0
-3 -2 -1 0 1 -2 -1 0 1 bands between armchair and zigzag tubes share lots of
Ef (eV) Ef (eV) similarities.
FIG. 6: Panels (a) and (b): band structures of √3 √3 Figures 6(e) and 6(f) display the transmission coeffi-
armchair carbon nanotubes fabricated from ZGR-3 show×n in cients as a function of energy for the √3 √3 armchair
×
Fig. 1(d), at tube sizes (12,12) and (20,20). Panels (c) and and zigzag carbon nanotubes. The results for (12,12)
(d): dispersion relations of (12,0) and (45,0) zigzag carbon and (30,30) armchair nanotubes are shown in Fig. 6(e).
nanotubes, formed by folding armchair ribbons (Fig. 4) in y In a wide energy range of E [ 2.5, 1) eV, there is
F
direction. Panel (e) plots the T(EF) curves of (12,12) and only single conducting channe∈l in−the(−12,12) armchair
(30,30) armchair tubes at different Fermi energies. Panel
tube. This energy range corresponds to the two touched
(f) shows the transmission of zigzag tubes with system sizes
bands in Fig. 6(a) for the same system parameters. The
(12,0) and (45,0).
increase of system size narrowsthis T =1 region,as one
can see from the red line for the (30,30) armchair nan-
otube. Another interesting finding in Fig. 6(e) is that
C. Typical single wall carbon nanotubes of √3 √3 the transmission coefficients increase by 2 at lot of en-
×
graphene supercell
ergy points. This behaviour reveals that many energy
levels in the armchair tubes are doubly degenerate, re-
The carbon nanotubes of √3 √3 graphene super- gardless of the system size. The transmission spectra of
cellhavesimilarstructuresaspris×tinecarbonnanotubes. two√3 √3zigzagtubeswithsizesof(12,0)and(45,0)
×
Here we consider two representing configurations: sin- are displayed in Fig. 6(f). Despite their distinct geo-
gle wall armchair and zigzag carbon nanotubes. The- metric configurations, the √3 √3 zigzag and armchair
oretically, √3 √3 armchair carbon nanotubes can be carbon nanotubes have simila×r T(E ) profiles. We also
F
×
formedby connecting the upper and lowerboundariesof observe the large single mode region for small tube size
ZGR-3 as displayed in Fig. 1(d) with proper bonding. andlotsofdoublydegeneratebandsatzigzagnanotubes.
Meanwhile, √3 √3zigzagnanotubes canbe formedby In Figs. 6(a) and 6(c), the linear bands parallel
×
rolling up armchair ribbons as displayed in Fig. 4 and to the blue arrows are rather attractive. In single
linking their edges accordingly. Carbon nanotubes con- layer graphene, the massless Dirac fermion with lin-
structedinthesewayspossessthe full √3 √3periodic- ear energy dispersion leads to the counterintuitive Klein
×
ity. In our consideration,we adopt the classificationrule paradox,34,35 where incoming relativistic particles pen-
of pristine single wall carbon nanotubes32,33 to denote etrate high barrier with nearly perfect transmission at
thesetwotypesof√3 √3carbonnanotubes,whereno- certain angles. Klein tunnelling behaviour has been
×
tations (n,n) and (n,0) stand for armchair and zigzag revealed in various graphene-based systems, such as
tubes, respectively. Integer n in these notations refers graphene p n junctions,36,37 deformed single layer,38
to the number of unit vectors defined in the honeycomb and twisted−bilayer graphene,39 as well as spin-related
lattice of pristine bulk graphene. graphene system.40 Similar perfect transmission is also
Figure6displaysthebandstructuresofthese√3 √3 observed when electron propagates in pristine carbon
armchair and zigzag carbon nanotubes at different×tube nanotubes.41–43 Inspired by these findings, we carry out
sizeswhereonecanfindthat,similarto√3 √3graphene numericalcalculationtoshowtheexistenceofKleinpara-
nanoribbons, these two types of carbon ×nanotubes are dox in the √3 √3 carbon nanotubes.
×
also metallic and their Fermi energies lie deeply into the Oursysteminconsiderationisa√3 √3(12,12)arm-
×
conduction bands. Specifically, Figs. 6(a) and 6(b) dis- chairnanotube. ApotentialbarrierwithheightofV and
0
play the energy dispersion relations of the (12,12) and certainlengthisexertedonthe tube while otherpartsof
6

to quantized transmission coefficient T = 1. These nu-
4
mericalevidences stronglysuggestthe existence ofKlein
tunnelling-likebehaviorinthe √3 √3armchaircarbon
V0 = 0.0 eV ×
3 BL)1 V0 = 0.1 eV nanotubes, where electrons transport without backscat-
T( V0 = 0.2 eV tering in the system, independent of barrier length and
T(E)f2 0 VEf0 == -02..51 eeVV V0 = 0.5 eV hnneeailngliohnttgusbiusepsa.tlsHooohwoabelfvseearrnv,ewedlhecientnrottnhheevot√ult3b.e×Ssii√mze3ilainzricgprzeeaargfseeccst,arbtbuuonlnk-
0 20 40 60 80 100 states arise in the whole energy range and gradually
Barrier length(unit cell)
co-exist with the linear bands. As a result, the Klein
1 tunnelling-like phenomenon becomes obscure and even-
tually disappears.
0
-2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -0.9 -0.6 D. Valley processing in √3 √3 armchair
×
nanotubes
E (eV)
f
FIG. 7: Transmission vs Fermi energy of a √3 (12,12) arm- Due to the band folding of the √3 √3 superlattice,
chaircarbonnanotubewhereapotentialbarrierwithalength the inequivalent K and K′ valleys in×pristine graphene
of10unitcellsexists. Variouscoloredlinesstandfordifferent
and carbon nanotubes are folded into Γ point. There-
barrierheightsV0. Inset: transmission of thesamesystem at
fore,theintervalleycouplingandvalley-orbitcouplingef-
fixedenergy EF= 2.1eV and barrier height V0=0.5eV for
severalbarrierlengt−hs. Thelengthscale’unitcell’isindicated fectsemergeinthesenanotubesof√3 √3superlattices
in Fig.1(d). of graphene,27 which qualify them as×potential valley-
processing materials. Here, we propose a heterostruc-
ture composed of 1 1 and √3 √3 armchair carbon
× ×
nanotubes to act as a valley filter or valley polarizer as
thesystemremainsunchanged. Wecalculatetheelectron
illustrated in Fig. 8(a), which consists of two identical
transmission coefficient through the barrier. Figure 7
leads made of pristine armchair nanotubes and a cen-
plots the transmission coefficient as a function of energy
tral scattering regionmade of √3 √3 armchair carbon
for several different barrier heights. The barrier length ×
nanotubes. A gate voltage is applied in the central re-
is fixed at10 unit cells, and this length scale of√3 √3
× gion. We consider typical (8,8) armchair nanotubes at
armchairtube is depicted in Fig. 1(d), containing 4 sites
bothparts,whoseenergydispersionsarerespectivelydis-
along the x direction. It is found that, in a wide en-
played in Figs. 8(b) and 8(c). The intervalley coupling
ergy window of E [ 2.5, 1.7] eV, electrons in these
∈ − − and valley-orbit coupling mechanism in √3 √3 super-
armchair carbon nanotubes can almost perfectly pene-
×
lattices can manipulate valley polarization coherently in
trate the potential barrier. Quantizedtransmissioncoef-
analogy to real spin for spintronics. Electrons propagat-
ficients of T(E ) = 1 can be achieved for barrier height
F
ing in the left pristine armchair carbon nanotube con-
up to V =0.5 eV. Outside this region, the transmission
0
tain equivalent K and K′ valley components. When the
coefficientdramaticallydropsinthepresenceofpotential
valley-unpolarized current in pristine armchair CNT en-
barrier.
tersthecentral√3 √3armchairnanotuberegion,these
Wecarefullyexaminedthenumericalresultsandfound ×
mechanisms break the balance between the two compo-
that, the T(E )=1 energy window E [ 2.5, 1.7]eV
F ∈ − − nentsbyflippingelectronsofKvalleytoK’valleyorvice
corresponds to the linear dispersion region in Fig. 6(a),
versa. Hence in the outgoing current to the right lead,
suggesting a direct correlation between linear disper-
onevalleycomponentislargerthantheother. Inanother
sion and nearly perfect transmission. In energy range
word, the current is valley-polarized. Similar to the spin
E [ 2.5, 1.7] eV, the linear dispersion relation guar-
∈ − − polarization,avalleypolarizationfunctioncanbedefined
antees the electrons high group velocity, or so-called rel-
toevaluatethe efficiencyofthe device. Thevalleypolar-
ativistic electron. In our carbon nanotube system with
ization can be adjusted by external factors, such as bias
potential barrier, the incident electrons normally collide
voltage and gate voltage. The setup presented here can
with the barrier. The high-velocity relativistic electron
serve as a prototype valley field effect transistor, where
can penetrate the barrier without back-scattering, re-
valley-polarizedcurrentisturnedon/offviaapplyingthe
sulting almost perfect transmission or reflectionless tun-
gate voltage as illustrated below.
neling. We further checked the dependence of this al-
The efficiency of this valley field effect transistor is
most perfect tunnelling on barrier length, at fixed elec-
characterized by the valley polarization function, which
tron energy of E = 2.1 eV and barrier strength of
F − can be defined in terms of the valley-specified transmis-
V = 0.5 eV. The transmission function versus barrier
0 sion function as:
length is plotted in the inset of Fig. 7. Regardless of the
barrier length, the system with longer barrier remains TK TK′
P = − (3)
transparent to the incident electrons and also gives rise V TK+TK′
7
(a) Vgate
1.0 1.0
K K 0.8 0.8
.y.. K’ K’ ... /T /PKK’V 00..46 (a)(L8 eT,n8Kg)’ tthu boef 3 tube = 3 units (c ) (L1 eT2nK,g1’ t2h) o tfu b3e tube = 3 units 00..46
T 0.2 TK TK 0.2
x Pv Pv

0.0 0.0
(b) (c) 1.0 1.0
2 2
0.8 0.8
V
E (eV)f01 01 /T /PKK’00..46 (b) (L8e,n8g) tthu boef 3 tube= 6 units ( d) (L1e2n,g1t2h) otuf be3 tube= 6 units 00..46
E- T
-1 -1 0.2 0.2
K K’ 0.0 0.0
-2 -2
0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 0.6
E (eV) E (eV)
k a ( ) k a ( )
x x
FIG. 9: Panels (a) and (b): Valley-specified transmission
FIG. 8: Panel (a): Illustration of a valley-field-effect- functions and valley polarization versusenergy of (8,8) arm-
transistor based on pristine and √3 armchair carbon nan- chair carbon nanotube valley-field-effect-transistor. Lengths
otubes. Agatevoltage isappliedon the√3armchaircarbon of the √3 armchair carbon nanotube are respectively 3 and
nanotube region, which functions as valley-processing unit. 6 units. Panels (c) and (d): The same functions for (12,12)
The √3 armchair carbon nanotube is characterized by the armchair carbon nanotube valley-field-effect-transistor. Gate
blue-colored top-absorption sites. Panels (b) and (c): band voltage is set to be zero in thecalculations.
structures of (8,8) pristine and √3 √3 armchair carbon
×
nanotubes, respectively.
K and K′ valley modes45,46
ΓL,K/K′ =ΛK/K′~ΓL,K/K′Λ†K/K′,
e
whereTK andTK′ aretransmissionfunctionsofelectrons where ΛK,K′ is the eigenfunction of ΓL for K/K′ valley
belongingto K andK′ valleys,respectively. To separate
mode. ΓR,K/K′ of the right lead can be produced simi-
electrons from equivalent valleys, a simple and effective
larly. FollowingEq.(2),thevalley-specifiedtransmission
way is to consider their different group velocities. The
function can be straightforwardly expressed as
groupvelocity,v = 1∂E,isrelatedtothebandstructure,
~ ∂k
i.e., the slope of the band structures. When focusing on TK/K′ =Tr[ΓL,K/K′GrΓR,K/K′Ga]
the first subband of (8,8) pristine armchair carbon nan-
otubeshowninFig.8(b), itisclearthatabovetheFermi
Then one can calculate the valley polarization P via
V
energyelectroninK valleyhaslargergroupvelocitythan
Eq. (3).
that of K′ valley. The situation becomes reverse below
We first evaluate the valley-specified transmission co-
the Fermi level. Thus, we can calculate the transmission
efficient TK/K′ and valley polarization efficiency PV as
function contributed from any specific valley under the
function of the electron energy. The system under inves-
Green’s function frame. In the semi-infinite lead of pris-
tigation is armchair carbon nanotube-based valley-field-
tine armchair carbon nanotube, the velocity of incident
effect-transistor as schematically plotted in Fig. 8(a).
electrons is connected with the line width function Γ
L The numerically calculated transport properties are ex-
in the form of ~v = U†Γ U = Γ 44,45. Here v is a
L L L L hibited in Fig. 9. The energy interval of interest is the
diagonalvelocity matrix with nonzeero diagonalelements
first subband of pristine armchair carbon nanotube. For
contributed by electrons incoming from the left lead. U the(8,8)armchairsystemwith3unitslengthof√3 √3
is a unitary transformation matrix ranked by eigenfunc- ×
armchaircarbonnanotube,bothTK andTK′ arecontinu-
tions of ΓL, transforming it into a diagonal matrix ΓL. ousfunctionsoftheelectronenergyasshowninFig.9(a).
Obviously, there is an exact mapping between the ineci- One can see that T is rather smooth and larger than
K
dent electronvelocities and the eigenvalues of line width 0.8 in the focused energy regime, while the magnitude
function: vL =(1/~)ΓL. Considering only the propagat- of TK′ changes more abruptly and exhibits a fluctuating
ing modes of the firstesubband of (8,8) armchair carbon pattern. This observation reveals that incident electrons
nanotubes[SeeFig.8(b)],bothv andΓ are2 2diag- from K valley of pristine nanotube are less affected by
L L
onalmatrices. IncidentelectronfromKevalleyh×aslarger the central √3 √3 armchair carbon nanotube. And
×
velocity,correspondingtothelargeroneofthetwoeigen- this fact holds for all systems considered in Fig. 9. The
values of Γ . Based on this analysis, we can construct calculatedvalley polarizationP is very smallbelow the
L V
effective line width function using only the propagating Fermilevelandgrowswiththeincreasingelectronenergy.
8
PV also fluctuates like TK′ in the whole region, and its
maximum reaches about 0.3 for our considered system.
P
v
When the length of central scattering region increases 0.6 -0.10
from 3 units to 6 units, T is still smooth as plotted
K 0.05
in Fig. 9(b), but TK′ fluctuates more frequently in the 0.4
0.15
same energy window. Therefore, the resulting valley po-
larizationvibrateswithenergyforbothbelow andabove 0.25
0.2
the Fermi level. This result suggests that, by increas- ) 0.40
V
ing the length of central region, one can realize a more e
(0.0 0.45
effective manipulation of valley polarization in a rela- E
0.55
tively small energy range. Considering a larger system,
such as (12,12) armchair carbon nanotube valley-field- -0.2 0.65
effect-transistor, an interesting and important observa-
0.75
tion is: both TK and TK′, as well as Pv, are identically -0.4
0.85
the same as those in (8,8) system in the energy win-
dow of [ 0.6,0.6] eV, as shown in Fig. 9(c) and 9(d). 0.95
-0.6
−
We have examined setups from (6,6) to (12,12) carbon -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
nanotube systems and reached the following conclusion:
V (eV)
g
as long as the electron energy is in the first subband of
the pristine armchair carbon nanotube, both the valley- FIG. 11: Valley polarization function PV of a (8,8) armchair
specified transmission functions and valley polarization carbon nanotube valley-field-effect-transistor with 3 units
length, in terms of theelectron energy and gate voltage.
areindependentofthecircumferenceofthesystem. This
fantastic property guarantees a great freedom in fabri-
catingsuchkindofvalley-field-effect-transistor,sincethe
device’s performance is independent of its transverse di- pristine armchair carbon nanotube. The valley polariza-
mension. tion PV as a function of gate voltage Vg is calculated at
two energy points and different system sizes as shown in
Secondly, we investigatethe influence of the gate volt-
Fig.10. FromFig.10(a),onecanfindthatthevalleypo-
age on the device performance. The gate voltage is ap-
plied on the √3 √3 armchair carbon nanotube region, larizationfluctuatingly growsasthe appliedgatevoltage
× decreases from positive to negative. For a (8,8) valley-
which serves as the valley-processing unit. In our calcu-
lation, gate voltage simply shifts the on-site energies of field-effect-transistor with 3-unit length of the √3 √3
×
theaffectedatoms,i.e.,diagonalelementsoftheirHamil- armchair carbon nanotube, the valley polarization can
tonian. Electron energy is kept at the first subband of reach about 0.9 at the negative gate voltage, which in-
dicates that the K valley is almost fully polarized. The
wide P > 0.9 plateau shown in Fig. 10(a) qualifies the
V
deviceasastablevalley-polarizedcurrentgeneratorthat

Pv1.0 1.0 can operate in a broad gate voltage range.
n (8,8) (8,8)
atio 0.8 (a) (L8=,83) E=0.36 eV (b) (L8=,86) E=0.25 eV 0.8 Moreimportantly,PV jumpsabruptlyfrombelow0.02
z to above 0.9 at V 0.5 eV, showing a great on/off
ari 0.6 L=3 E=0.60 eV L=6 E=0.45 eV 0.6 g ≈ −
ol ratioofthisvalley-field-effect-transistor. Thefluctuation
P
ey 0.4 0.4 of PV exists at all electron energies. Comparing the re-
Vall 0.2 0.2 sults for E = 0.36 eV and E = 0.60 eV, obviously high
valley polarization can be easily achieved when the sys-
Pv0.0 (12,12) (12,12) 0.0 tem works at larger electron energy, which only requires
ation 0.8 (c) (L1=23,1 E2)= 0.36 eV (d) (L1=26,1 E2)= 0.25 eV 0.8 Vg ≈−0.25eV to reachPV >0.9 at E =0.60eV. When
Polariz 00..46 L=3 E=0.60 eV L=6 E=0.45 eV 00..46 iinncrtehaesidnegvitchee, mlenorgethfluocftuthaetiocnesntoraflPrVegaimreerteove6aleudniitns
y Fig. 10(b), suggesting a more effective gate modulation.
e
all 0.2 0.2 But the high valley polarization plateau shrinks at all
V
electron energies and a larger negative V is required for
0.0 0.0 g
-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 P > 0.9. We also performed calculations on a (12,12)
V
Vg (eV) Vg (eV) armchaircarbonnanotubesystemandthecorresponding
results are displayed in Fig. 10(c) and 10(d). It is found
FIG. 10: Panels (a) and (b): Valley polarization versus gate
voltageof(8,8)armchaircarbon nanotubevalley-field-effect- again that the device performance is independent of the
transistor at several electron energies. ’L’ in the legends diameter of the nanotubes.
stands for the lengths of the √3 armchair carbon nanotube, We summarize this part with the working map of a
whicharerespectively3and6unitsinthecalculation. Panels (8,8)armchaircarbonnanotubebasedvalley-field-effect-
(c) and (d): The same functions for (12,12) tubesystems.
transistor. ItsvalleypolarizationP asfunctionsofelec-
V
9
tron energy and gate voltage are displayed in Fig. 11. transmission through the tube is quantized even in the
P functionofthisdevicecanbeeffectivelytunedbyap- presence of a potential barrier, regardless of the barrier
V
plying an external gate voltage. The best working zone lengthandheightuptoafewhundredsofmeV.Avalley-
of this device is the large red triangular area highlighted field-effect-transistor consisting of pristine and √3 √3
×
in Fig. 11, where the K valley is nearly fully polarized. armchair carbon nanotubes is proposed, which can be
The valley-field-effect-transistor has a stable output in used to filter fully valley-polarized current and can be
this zone, and the on/off ratio is guaranteed by the nar- tuned by applying an external gate voltage or adjust-
row blue region adjacent to the red zone, whose valley ing the length of the central scattering regime. As long
polarization is below 0.05. The larger the electron en- as the electron energy is within the range of the first
ergy,the easierto getfully valley-polarizedcurrent. The subband of pristine armchair carbon nanotubes, perfor-
performance of this device is the same for systems with mance ofthis valley-field-effect-transistoris independent
different circumferences, as long as the energy is within of the tube circumference.
the first subband of pristine armchair carbon nanotube.
IV. CONCLUSION V. ACKNOWLEDGMENTS
In conclusion, we have numerically investigated the This work was financially supported by the NNSFC
electronic properties of typical √3 √3 graphene (GrantsNo. 11504240,No. 11574217,No. 11304205,and
×
nanoribbon and nanotube structures. We show that all No. 11474265), NSF of SZU (Grant No. 201550). Y.R.
the√3 √3nanostructuresaremetallicmaterials. Both and Z.Q. also acknowledge the financial supports from
×
the zigzag and armchair ribbons have finite-size energy theChinaGovernmentYouth1000-PlanTalentProgram,
gap below the Fermi energy more than 1 eV. Double- FundamentalResearchFundsfortheCentralUniversities
degeneracy in energy levels instead of energy gaps are (WK3510000001 and WK2030020027) and the National
found in the spectra of both armchair and zigzag car- Key R & D Program (Grant No. 2016YFA0301700) .
bon nanotubes. In small √3 √3 carbon nanotubes, The Supercomputing Center of USTC is gratefully ac-
×
there is a large energy range showing linear dispersion, knowledged for the high-performance computing assis-
whichleadstotheKleintunneling-likebehavior: electron tance.
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