Transmission in graphene-topological insulator heterostructures

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by  Christophe De Beule

Transmission in graphene-topological insulator heterostructures
C. De Beule,1, M. Zarenia,1 and B. Partoens1
∗
1Department of Physics, University of Antwerp, 2020 Antwerp, Belgium
We investigate scattering of the topological surface state of a three-dimensional time-reversal in-
varianttopologicalinsulatorwhengrapheneisdepositedonthetopological-insulatorsurface. Specif-
ically, we consider the (111) surface of a Bi Se -like topological insulator. We present a low-energy
2 3
modelforthebulkgraphene-topologicalinsulatorheterostructureandwecalculatethetransmission
probability at zigzag and armchair edges of the deposited graphene, and the conductance through
graphene nanoribbon barriers and show that its features can be understood from antiresonances in
7 the transmission probability.
1
0
2 I. INTRODUCTION mismatch (%) gap (eV) v /v
s g

n Bi Se 2.722 0.324 0.524, 0.320
2 3
a Topological insulators1–7 (TIs) are materials with Sb Te 0.123 0.323 0.423
J 2 3
metallic surface states that are topologically protected
Bi Te Se 0.922 0.325 0.525
0 by time-reversal symmetry and the insulating bulk. In 2 2
1 TlBiSe 0.221 0.3518, 0.319, 0.318, 0.419,
the simplest case, the topological surface state is given 2
0.220 0.720
by a single Dirac cone that is characterized by spin-
]
l momentum locking8,9. The topological surface states
al have potential applications in spintronics and quantum TABLE I. The lattice mismatch of the graphene-TI het-
h computation, and it is therefore desirable to tune their erostructure, band gap, and Fermi velocity vs for some TIs
- with a simple Dirac cone. We have taken a = 2.46 ˚A and
s properties to suit specific needs. Tailoring the surface v = 106 m/s for the lattice constant and Fermi velocity of
e states can also lead to new physics. For example, by g
graphene, respectively15.
m changingtheirdispersionrelation10,11 theybecomemore
. susceptibletowardsinteractionswhichcouldleadtonovel
t
a strongly correlated phases.
m
One possibility consists of depositing a thin layer of stronglyaffectthelow-energyphysicsifthechemicalpo-
- a non-topological metal on the topological-insulator sur- tential is tuned accordingly16. In this configuration, the
d
face (TIS), effectively changing the boundary conditions trigonal lattice of graphene and the TIS are rotated by
n
o at the surface12–14. The topological surface state mi- 30◦ with respect to each other and the surface unit cell
c grates to the new surface obtaining different properties contains six carbon atoms from graphene and one atom
[ depending on the type of deposited thin film. In par- from the TIS. The most promising currently known TIs

1 ticular, graphene, is a very interesting candidate, for forrealizingsuchaheterostructureareSb2Te3,whichhas
v a number of reasons. Graphene has been studied ex- recently been fabricated13, and TlBiSe218–20. Both have
2 tensively in the last decade and its properties are well only a lattice mismatch of the order of 0.1%20,21. While
3 known: It hosts four Dirac cones in its bulk whose Dirac the interlayer coupling of Sb2Te3 is van der Waals-like,
5 structure act on the sublattice pseudospin of the honey- that of TlBiSe2 is more covalent18, allowing for stronger
2 comb lattice15. The interplay between the Dirac cones couplingbetweengrapheneandtheTISinthelattercase.
0 of graphene and the topological Dirac cone can drasti- In Table I, we show a list of potential TIs together with
.
1 cally change the properties of the resulting topological thelatticemismatch,thebandgap,andtheFermiveloc-
0 surface state16. Moreover, the lattice mismatch between ity of the topological Dirac cone.
7
graphene and the natural surface of several TIs is very The paper is further organized as follows: In Sec. II,
1
: small, from a few percent to near perfect matching. we introduce the model for the graphene-topological in-
v In this work, we investigate transmission in het- sulator heterostructure. We consider different stacking
i
X erostructures made from depositing graphene on top of configurations,elucidatethephysicsbyblockdiagonaliz-
r the (111) surface of a Bi2Se3-like TI. This setup was re- ing the Hamiltonian, and derive a low-energy model. In
a cently experimentally realized17. The archetypal strong Sec. III, we solve the two-dimensional scattering prob-
topologicalinsulator, Bi2Se3, hasalayeredcrystalstruc- lem for different geometries. In particular, we consider
turewhereeachlayerhastrigonalsymmetryandthelay- the interface between the bare TIS and the heterostruc-
ers are generally only weakly coupled by van der Waals- ture for both zigzag and armchair graphene edges. We
likebonding. The(111)surfaceisparalleltotheselayers alsoconsiderbarriersconsistingofgraphenenanoribbons
and hosts a single Dirac cone at the center of the surface deposited on top of the TIS where the bare TIS acts as
Brillouin zone (BZ). If graphene is placed on top of this leads. We discuss our results for the transmission prob-
surface in the commensurate √3 √3 R30 stacking con- ability, the bound states, and the conductance through
×
figuration, the graphene Dirac cones are folded onto the thedifferentbarriersinSec.IVandpresentthesummary
topological Dirac cone so that even weak coupling can and conclusions of the paper in Sec. V.
2
where denotescomplexconjugationandτ isthePauli
x
K
matrix in valley space. Time-reversal symmetry gives
A Θh( k)Θ 1 =h(k) and constrains the coupling :
−
− V
(cid:32) (cid:33)
B t (k) t (k) λ (k) λ (k)
A B A B
(k)= ,
V λ ( k) λ ( k) t ( k) t ( k)
A ∗ B ∗ A ∗ A ∗
− − − − − −
y (7)
where t and t correspond to coupling between the
A B
x
same spins, and λ and λ to coupling between differ-
A B
T B H
ent spins. We do not consider the latter and hence we
put λ = λ = 0. The form of t and t depends on
A B A B
FIG.1. (Coloronline)Topviewofthedifferentcommensurate
√ √ the specific stacking: In Fig. 1, we show the three most
3× 3 R30 stacking configurations of the graphene (red)
symmetrical stacking configurations. Ab initio studies
andtopological-insulatorsurface(gray)heterostructure. The
ongraphenedepositedonthinfilmsofSb Te showthat
structures differ by the position of the TIS atom in the unit 2 3
the binding energy of these structures only differ by a
cell: (T) one sublattice on top, (B) bond on top, and (H) in
few meV with H the most stable configuration26.
the center of a graphene hexagon.
FortheTandBstructureshowninFig.1,thecoupling
is given, in lowest order, by
II. MODEL
(cid:32) (cid:33)
t t 0 0
A B
= , (8)
We consider the surface of a Bi2Se3-like time-reversal V 0 0 tA tB
invariant strong topological insulator on which a mono-
layer of graphene is deposited. The Hamiltonian reads
where t (t ) is the coupling matrix element between
A B
the TIS and the A (B) sublattice. Specifically, in lowest
H =H +H +V, (1)
G TIS
order, we have t = 0 for T stacking and t = t for
B A B
B stacking. However, for the H structure, also shown in
where H and H are, respectively, the Hamiltonians
G TIS
Fig. 1, the lowest-order coupling vanishes at k=0.
of graphene and the topological-insulator surface and V
represents the coupling between them. The energy spectrum of the T structure is shown in
For commensurate √3 √3 R30 stacking, illustrated Fig. 2 for µ = 0. A similar energy spectrum is obtained
in Fig. 1, the Dirac con×es at the K and K point of for the B structure. For the H structure, the spectrum
(cid:48)
graphene are folded onto the zone center Γ¯ of the TIS only shows a change in the Fermi velocity of the Dirac
BZwhichharborsthetopologicalDiraccone. Hence,the cones. The spectrum shown in Fig. 2 is thus generic for
low-energy Bloch Hamiltonian becomes any √3 √3 R30 stacking configuration at low energies
×
withtheexceptionofHstacking. Sinceweareinterested
  in strong coupling between the Dirac cones, we restrict
h 0
K †
V ourselves to the T structure with µ = 0. Thus, we put
h(k)= 0 hK(cid:48) † , (2) t =t and t =0 in the remainder of the article.
V A B
h
TIS
V V
where are the coupling matrix elements of V between
V Valley exchange
thep orbitalsofgrapheneandtheTIS.Inthecoordinate
z
system shown in Fig. 1, we have
From the energy spectrum for the T structure, shown
hK(k)=(cid:126)vgs0 (σ k) µ (3) in Fig. 2, we observe that two of the four Dirac cones
⊗ · −
hK(cid:48)(k)=(cid:126)vgs0 ( σ∗ k) µ (4) of graphene do not couple at all with the TIS. This sug-
h (k)=(cid:126)v (zˆ⊗ s−) k,· − (5) geststhatthegrapheneDiracconespartlydecouple. The
TIS s × · symmetry that enables this block diagonalization is val-
leyexchange: K K . Statesthatareevenundervalley
where vg and vs are respectively the Fermi velocity of ↔ (cid:48)
exchangecoupletotheTIS,whilestatesthatareoddun-
graphene and the bare TIS, µ is the chemical potential
der valley exchange do not. Formally, we can write
difference between graphene and the TIS, and σ and s
are the Pauli matrices corresponding to pseudospin and
spin, respectively. In the remainder of this article, we UhU† =h+ h , (9)
put (cid:126)=1 unless otherwise stated. ⊕ −
In our basis, the time-reversal operator becomes where U = U(k) is a suitable unitary transformation,
whoseexplicitformisgivenintheAppendixforTstack-
Θ=(τ is σ ) is , (6) ing. ThisisillustratedinFig. 3. ForTstacking,thetwo
x y 0 y
⊗ ⊗ ⊕ K
3
1.0
(a) (b)
0.5
UhU =
† ⊕
h :TIS (+, ) h :( , )
+
-0.5 × ↑↓ − − ↑↓
FIG. 3. (Color online) Graphical representation of the block
diagonalizationofh(k)intosubspacesthatareeven(+,blue
-1.0
-1.5 1.5 cone) and odd (−, red cone) under valley exchange. The
spectra are shown for t=0. Only the even subspace couples
FIG. 2. (Color online) (a) Momentum space of the commen- to the topological-insulator surface (green cone).
√ √
surate 3× 3R30stackingconfigurationshowninFig.1in
the extended zone scheme. The small (gray) hexagons corre-
spondtotheTIS,wherethedotsarereciprocallatticepoints, are symmetric-like superposition of K and K which are
(cid:48)
andthelarge(red)hexagonisthefirstBZofgraphene;theK
given explicitly in the Appendix. Note that these states
andK(cid:48) pointofgraphenearefoldedtotheΓ¯ pointofthesur-
correspondtothesublatticethatdoesnotcoupledirectly
face BZ. (b) Energy spectrum of the T structure with µ=0,
with the TIS in lowest order of vk/t. Accordingly, the
t = 0.3 eV, t = 0, and v = v /2. The dashed curve is
A B s g low-energy physics is understood in terms of an interme-
theoriginaltopologicalDiracconeandtheindexn=1,...,5
diate virtual process: In lowest order, the spin states of
refers to the n-th scattering channel.
the B+ sublattice couple to each other via the A+ sub-
lattice and the original topological surface state, lead-
blocks h and h can be written as ing to the cubic dispersion. Apart from the cubic dis-
+
− persion, two uncoupled valley odd cones remain. The
 
0 vgk presence of boundaries, however, can induce coupling to
 vgk+ − 0 − √2t  theseconesandtheyarenotrobustagainsttime-reversal
−  invariant perturbations in general. Similarly, an AB-
 √2t 0 v ik 
h+ = vsik+ s0− √2t  sthtaecTkeIdSlgeraadpshteoneaqbuilianyteicrdtihsapterissiosnuaittalbolwy edneeprogsiietse,dnoown
 − 
 √2t 0 v k  localized on a single sublattice of the top layer of the bi-
 g 
− layer, together with two quadratic cones corresponding
v k 0
g + to the odd subspace of the bilayer16.
(10)
InFig. 4,weshowthetwo-dimensionalbandsobtained
from h together with the corresponding spin expecta-
h =v (σ k σ k), (11) +
g ∗
− · ⊕− · tion value. While the decoupled Dirac cones from h
with k =kx iky. We find that h+ is equivalent to the remain sz eigenstates, the other bands inherit their spi−n
low-en±ergy Ha±miltonian of spinless ABC-stacked trilayer structure from the original topological surface state. Be-
grapheneforwhichthemiddlelayeristriaxiallystrained, sides the cubic Dirac bands, there are two bands from
while h is like a spinless version of graphene27. We the valley even subspace that have a Rashba-like disper-
can und−erstand the decoupling as follows: The matrix sionwithoppositespin-momentumlocking. Thesestates
elements between the odd subspace and the topological arise from proximity-induced Rashba coupling since re-
surfacestatepickupaminussignundertimereversal,so flection symmetry about the graphene plane is broken
that they have to be zero because the coupling is time- whendepositedontheTIS.Byexpandingthedispersion
reversal invariant. relation to second order in k, we find that the Rashba
InanalogywithABCtrilayergraphene,theenergydis- momentumandenergysplittingareapproximatelygiven
persion is cubic at low energies (vk/t 1)27. Moreover, by (2√2tv )/(4v2+v2) and (tv2)/[√2(cid:0)4v2+v2(cid:1)].
(cid:28) s g s s g s
wefindthatthetopologicalsurfacestatemigratestothe
graphene. For our choice of unitary transformation, the
effectivelow-energyHamiltonianclosetotheΓ¯ pointbe-
III. TRANSMISSION
comes
(cid:32) (cid:33)
vg2vs 0 k3 In this section, we consider elastic scattering of the
2t2 k3 0− ⊕h−, (12) topological surface state at a graphene-topological insu-
+
lator heterostructure for the T structure. First, we con-
where the basis of the first 2 2 block is sider scattering at a graphene step terminated by zigzag
i(cid:12)(cid:12)ψ+ (cid:11),(cid:12)(cid:12)ψ+ (cid:11) . The + indicates th×at these states or armchair edges, where an incident wave on the bare
{ B ↑ B ↓ }
4
A. Graphene step
1. Scattering states
Inthesemi-infiniteTregion, thewavefunctioncanbe
written as
5
(cid:88)
Φ (x)= t ψ eiqnxx, (16)
II n n
n=1
where t , ψ , q = q xˆ + k yˆ are, respectively, the
n n n nx y
transmission coefficient, the spinor, and the momentum
ofthenthscatteringchanneloftheheterostructure. The
sign of q is chosen such that for scattering modes the
nx
groupvelocityispositiveandthewavepropagatestothe
right, while for evanescent modes it is chosen such that
the imaginary part is positive since otherwise the solu-
tionfromEq.(16)wouldblowupforx . Thebands
FIG. 4. (Color online) Low-energy spectrum for T (or B) corresponding to the different transmis→sion∞channels are
stackingwherethecorrespondingspinexpectationvaluesare
shown in Fig. 2: ψ corresponds to the cubic dispersion,
1
shown as arrows. All bands except the two valley odd Dirac
ψ and ψ to the Rashba-like bands, while ψ and ψ
cones that decouple in the bulk are shown. 2 3 4 5
correspondtothetwouncoupledDiraccones. Scattering
to a particular channel only takes place if q is real, oth-
x
erwisethecorrespondingwavefunctionisevanescentand
TIS coming in from the left (x < 0) is transmitted to
doesnotcontributetotransmission. Wealsoexpectthat
the right (x > 0) into a semi-infinite region of the het-
thereisnotransmissionintothechannelsψ andψ that
erostructure. Next, we consider transmission through a 4 5
are decoupled from the TIS in the bulk. The presence of
graphene nanoribbon barrier of finite width.
certain boundaries, however, allows for transmission to
WeworkintheoriginalbasisinwhichtheHamiltonian
ψ and ψ , as we show below.
4 5
takes the form given in Eq. (2). In the basis where the
The spinors ψ and ψ can be explicitly written as
4 5
Hamiltonian is block diagonal, the boundary conditions
at a graphene edge can couple the two blocks and we  E/v   0 
g
prefer to work in the original basis where the boundary
   
conditions are straightforward. q4x+iky  0 
   
If we take the coordinate system shown in Fig. 5, the  0   E/vg 
scattering state for the bare TIS is given by an incident  0  q5x+iky
and reflected wave    
 E/v   0 
ψ = − g , ψ = , (17)
4   5  
ΦI(x)=φieikxx+rφre−ikxx, (13) q4x−iky  0 
 0   E/v 
where r is the reflection coefficient and    − g 
 0  q5x iky
(cid:32) (cid:33) (cid:32) (cid:33)    − 
E/v E/v  0   0 
s s    
φ = , φ = , (14)
i r
k ik k +ik 0 0
y x y x
−
with
are the corresponding spinors with E the energy relative
(cid:113)
to the Dirac point. We have left out normalization con-
q =q =sign(E) (E/v )2 k2. (18)
stants since they are irrelevant for our calculation. The 4x 5x g − y
longitudinal and transverse momentum are given by k
x It is clear that the spinors ψ and ψ are s eigenstates
4 5 z
and k , respectively. The latter is conserved because of
y and have odd valley parity since they are antisymmetric
translation symmetry in the y direction. The longitudi-
superpositions of states at K and K . The other spinors
(cid:48)
nal momentum is given by
ψ , ψ , and ψ and the corresponding wave vectors are
1 2 3
found numerically. The secular equation H(q ,k )
(cid:113) x y
| −
k =sign(E) (E/v )2 k2, (15) E =0 yields a bicubic equation:
x s − y |
v4v2
kwhemreakEes=suvrsektfhoartthteheDiinracicdceonntewoafvteheprToIpSa.gTatheesstiognthoef Eg2sqm6 −vg2(cid:0)vg2+2vs2(cid:1)qm4 (19)
x
right and the reflected wave propagates to the left. +(cid:2)(cid:0)2v2+v2(cid:1)E2 4v2t2(cid:3)q2 (cid:0)E2 2t2(cid:1)2 =0,
g s − g m− −
5
(cid:113)
where q = q2 k2 with m = 1,2,3. The sign
mx ± m− y
is determined so that scattering modes propagate to the
right and evanescent modes decay inside the T region. y y
x x
2. Boundary conditions
B A A
Theboundaryconditionsatx=0aregivenbythecon- B
tinuity of the TIS spinor components together with the
ZZ1 ZZ2
appropriate open boundary conditions for the graphene
(a) zigzag (b) armchair
componentsdependingonthetypeofedge28,29. Wecon-
sider three different edge geometries, shown in Fig. 5.
FIG. 5. (Color online) Basic edge geometries of graphene
FortheTstructuretherearetwodistincttypesofzigzag (red, small dots) on top of a TIS (gray, large dots) for the T
edges: one terminated by sublattice A (ZZ1) and one stackingconfiguration. (a)Zigzagedges: twotypesdepending
terminatedbysublatticeB(ZZ2). Forthearmchairedge onwhethertheedgeisterminatedbytheA(ZZ1)orB(ZZ2)
(AC) there are three different edge configurations, but sublattice. (b) One of the three physically distinct armchair
thecontinuummodelcannotdistinguishanyofthembe- edges which the continuum model cannot distinguish.
cause the armchair edge contains both sublattices. In
case of B stacking, shown in Fig. 1, there is also no dis-
in Fig. 5(a). Thus, we find that the armchair boundary
tinction between the ZZ1 and ZZ2 edges within the con-
condition at x=0 is given by
tinuum model.
The continuity of the TIS spinor components gives Φ (0) + Φ (0) =0, α=A,B. (24)
II |α ( ) II |α(cid:48) ( )
↑ ↓ ↑ ↓
Φ (0)= Φ (0) . (20) Ingeneral,thecombinedboundaryconditionsfromEq.
I II |TIS
(20)andEqs.(21)or(24)resultinsixequationsthatare
Next, we consider the boundary conditions for the solved numerically and yield the reflection coefficient r
graphene components. For the zigzag edge, shown in and the five transmission coefficients t .
n
Fig. 5 (a), the boundary condition is satisfied by putting
the spinor component of the relevant sublattice equal to
zero at the edge for the two valleys separately30. For a 3. Transmission channels
zigzag edge at x=0, this gives
There are five scattering channels in the heterostruc-
ΦII(0)|α ( ) = ΦII(0)|α(cid:48) ( ) =0, (21) tureregion for thegraphenestep, whilethere isonlyone
↑ ↓ ↑ ↓
reflection channel for the bare topological-insulator sur-
where α = A,B for the ZZ2 and ZZ1 boundary condi- face. In order to obtain the transmission probability of
tions, respectively. For the armchair edge, shown in Fig. the different scattering channels, we consider the proba-
5 (b), the boundary condition only yields a nontrivial bility current in the x direction. The probability-current
solution if the K and K(cid:48) valleys of graphene are cou- operator in the x direction is given by
pledbytheedgebecauseanarmchairedgecontainsboth
sublattices30. The boundary condition for the armchair j =(vgs0 σx) ( vgs0 σx) ( vssy). (25)
⊗ ⊕ − ⊗ ⊕ −
edge is thus given by
By definition, the transmission probability of the nth
(cid:12) scattering channel is given by
ΨKeiK·r+ΨK(cid:48)eiK(cid:48)·r(cid:12)(cid:12) =0, (22)
edge ψ jψ ψ jψ
T = n† n t 2 = n† n t 2, (26)
n n n
where ΨK and ΨK(cid:48) are the graphene spinors. For the φ†ijφi | | 2Ekx | |
c4oπoxˆrdwinhaetreesaysistetmheshgroawpnheinneFliagt.t5ice(bc)o,nasntdanKt,(cid:48)w=e h−aKve= andthetotaltransmissionprobabilityT =(cid:80)5n=1Tn. For
3a scattering modes of the valley odd graphene Dirac cones
(cid:32) (cid:33) (cid:32) (cid:33) (E2 > v2k2), that are decoupled in the bulk, we further
g y
ψA ψA(cid:48) obtain from Eq. (17),
ΨK = , ΨK(cid:48) = , (23)
ψB ψB(cid:48) 2q 2q
T = 4x t 2, T = 5x t 2, (27)
4 4 5 5
k | | k | |
for both spin components. Note that we have chosen the x x
Hamiltonian in such a way that no phase factors arise in while T = T = 0 for evanescent modes (E2 < v2k2).
4 5 g y
thecomponents. Inthezigzagcase,thisisofnoconcern, The reflection probability R is given by
since relative phase factors between valleys drop out of
the boundary condition. Hence, it does not matter that R= φ†rjφr r 2 = r 2, (28)
weusedrotatedcoordinatesforthezigzagcase,asshown −φ†ijφi | | | |
6
where conservation of the probability current requires for α=A,B, where ∆K =8π/3a.
that R+T = 1. Before we discuss our results for the The boundary conditions for the barrier give twelve
step geometry, we consider the boundary conditions for equations that are solved numerically and yield the re-
the nanoribbon barrier. flection coefficient r, the ten barrier coefficients a and
n
b , and the transmission coefficient t.
n
B. Graphene nanoribbon barrier
3. Bound states
Here, we consider a barrier composed of a graphene
nanoribbon deposited on the TIS in the T stacking con- States of the TIS for which E2 < v2k2 are evanescent
s y
figuration. The ribbon is infinite along the y direction and as such we can have bound states, localized in the
and finite in the x direction with width W. This is illus- graphene nanoribbon. In this case, the wave functions
trated for the zigzag barrier in Fig. 5 (a). outside the ribbon become
(cid:32) (cid:33) (cid:32) (cid:33)
E/v E/v
Φ (x)=c s eκx, Φ (x)=d s e κx,
1. Scattering states I III −
k κ k +κ
y y
−
(37)
ThescatteringstateoftheTISforx<0isagaingiven (cid:113)
where κ = k2 (E/v )2 and the wave function inside
byEq.(13). Inthebarrierregion(0<x<W),thewave y− s
function can be written as theribbonisgivenbyEq.(29). Theboundaryconditions
andthenormalizationgivetwelveindependentequations
5
ΦII(x)= (cid:88)anψn+eiqnxx+bnψn e−iqnxx, (29) for the coefficients an, bn, c, and d.
−
n=1
wherethewavevectorsqnx arefoundfromEqs.(18)and IV. RESULTS
(19) and the spinor ψ corresponds to q . Note that
n nx
we do not need to wo±rry about the cor±rect sign of the
In this section, we discuss our numerical results for
wavevectorbecausebothareadmissibleinthefinitebar-
transmission through a graphene step and nanoribbon
rier. Behind the barrier (x>W), the solution becomes
barrier deposited on the TIS in the T stacking configu-
Φ (x)=tφ eikxx, (30) ration. Wealwaysputvs =vg/2,whichisrepresentative
III t
for the TIs listed in Table I, and we present our results
where t is the reflection coefficient, the spinor φ =φ is for t=0.3 eV as an example, unless stated explicitly.
t i
given in Eq. (14), and k is given in Eq. (15).
x
A. Graphene step
2. Boundary conditions
Out of the three edges we have considered for the
The boundary conditions of the barrier consist of the graphene step, only one of the zigzag edges, ZZ1,
continuity of the TIS spinor components and the appro- shows interesting features in the transmission probabil-
priate open boundary conditions for the graphene spinor ityT(E,k ). Interestingly,theresultfortheZZ2andAC
y
components at x=0 and x=W. The former become edges is exactly the same and shows near perfect trans-
mission, even at oblique angles. As seen in Fig. 5, only
Φ (0)= Φ (0) (31)
I II |TIS fortheZZ1boundarydoestheterminatedgrapheneedge
ΦIII(W)= ΦII(W)|TIS. (32) couple directly to the TIS lattice. We find that only the
ZZ1 edge induces coupling to the valley odd cones that
First, we consider the zigzag ribbon. We take the ZZ1
are decoupled for the bulk heterostructure. The trans-
edgeatx=0sothattheedgeatx=W isautomatically
missionprobabilityofthedifferentscatteringchannelsat
ZZ2. In this case, the boundary conditions become
the ZZ1 edge is shown in Fig. 6, together with the total
Φ (0) = Φ (0) =0 (33) transmission probability. For E < √2t, the main trans-
II |B↑(↓) II |B(cid:48)↑(↓) mission channel is T , and the ZZ∼1 edge allows for some
1
Φ (W) = Φ (W) =0. (34)
II |A ( ) II |A(cid:48) ( ) transmission to channels 4 and 5, corresponding to the
↑ ↓ ↑ ↓
valley odd cones. At higher energies, the Rashba chan-
Analogoustothediscussiononthearmchairedgeabove,
nels T and T become available and the transmission
we find that the boundary conditions for the armchair 2 3
via T reduces inside the region E2 < v2k2 defined by
ribbon are given by 1 g y
the graphene Dirac cone. Interestingly, the channels T
4
ΦII(0)|α ( )+ ΦII(0)|α(cid:48) ( ) =0 (35) andT5,whicharesz eigenstatesandcompletelylocalized
↑ ↓ ↑ ↓ in graphene for the bulk heterostructure, are mirrored
Φ (W) +ei∆KW Φ (W) =0, (36)
II |α↑(↓) II |α(cid:48)↑(↓) with respect to each other about ky =0. Moreover, they
7
FIG. 6. (Color online) (a)-(e) Transmission probabilities T for scattering at the ZZ1 step with t = 0.3 eV for the scattering
n
channels n=1,...,5, respectively, and (f) the total transmission probability T =(cid:80)5 T .
n=1 n
showapreferenceforeitherleftorrightmovingstatesfor In Figs. 7 and 8, we show the transmission proba-
bothelectronsandholes,creatingabulkspin-momentum bility for the zigzag and armchair barrier, respectively.
locked state in the graphene originating from valley odd The transmission probability is always equal to unity at
statesthataredecoupledinthebulk. NotethatonlyT , normal incidence for both zigzag and armchair ribbons,
1
and therefore also the total transmission probability, is which is what we expect for a nonmagnetic scatterer on
not symmetric with respect to zero energy. This asym- theTIS. Moreover,weobservetworesonancesatlowen-
metry originates from the fact that a step graphene-TIS ergies for the zigzag ribbon and antiresonances for both
system has only one interface which breaks the symme- the zigzag and armchair ribbons. The low-energy reso-
try of the lattice structure, resulting in an asymmetric nancesforthezigzagribbons,showninFig.7,arecaused
transmission for electrons and holes, in contrast to the byedgestates, thatareabsentforanarmchairribbon.
graphene-TIS barrier structure. To understand the nature of these edge states and the
antiresonances,weconsidertheevolutionofthetransmis-
sion probability as a function of the coupling t between
B. Graphene nanoribbon barrier
graphene and the TIS. In Fig. 9, we plot the transmis-
sion probability for (a)-(b) zigzag and (c)-(d) armchair
Now we discuss our results for the transmission across ribbons with t = 0.1 eV and t = 0.2 eV. We see that
the graphene nanoribbon. The results for the barrier are the two positive-energy edge states for the zigzag ribbon
symmetric with respect to zero energy and we only show split with increasing t. The upper branch is localized on
theresultsforpositiveenergy. Thewidthofthegraphene theZZ1edgewhichcouplesdirectlytotheTIS,whilethe
ribbons, including dangling bonds, is given by lower branch is localized on the ZZ2 edge which has no
direct coupling to the TIS. In Fig. 10, we show the elec-
a
WZZ = 2√3(3N +2), (38) tron density for a fixed value of ky for both edge states
corresponding to Fig. 7 (b). Note that the upper branch
a
W = (N +1), (39) is actually a hybridized state of graphene and the TIS,
AC
2
localized near the ZZ1 edge. The energy splitting of the
where a is the graphene lattice constant and N is the edge states is shown in Fig. 11 as a function of t for
numberoftwo-atomunitcellsalongthefinitexdirection. N =10 and N =20. For N =10, there is a confinement
8
FIG. 7. (Color online) Transmission probability T(E,k ) for
y
a zigzag ribbon with t = 0.3 eV, and (a) N = 10 and (b)
N = 20. The red lines outside the cone are bound states
and the density corresponding to the states marked with an
asterix is shown in Fig. 10.
FIG. 9. (Color online) Transmission probability T(E,k ) for
y
the nanoribbon barrier. (a)-(b) Zigzag ribbon with N = 30
for (a) t = 0.1 eV and (b) t = 0.2 eV. (c)-(d) Armchair
ribbon with N = 30 for (c) t = 0.1 eV and (d) t = 0.2 eV.
Theredlinesareboundstates,localizedinthebarrier,while
theorangedashedlinesin(a)and(c)aretheboundstatesof
a bare graphene nanoribbon.
FIG. 8. (Color online) Transmission probability T(E,k ) for
y
an armchair ribbon with t = 0.3 eV, and (a) N = 30 (insu-
lating) and (b) N =41 (metallic). The red lines outside the
cone are bound states.
effect near t = 0 which is absent for N = 20. How-
ever, this confinement splitting is lifted when t increases
becausetheenergydifferenceofstateslocalizedatdiffer-
FIG. 10. (Color online) Projected electron density of the (a)
ent edges increases, and the lower branch returns to zero
lowerand(b)upperbranchofedgestatesforthezigzagribbon
energy. The energy of the upper branch grows linearly withN =20andt=0.3fork =0.7nm−1. Thesestatesare
y
with t, since the coupling with the TIS splits the two marked in Fig. 7 with an asterisk.
formerlys eigenstateslocalizedontheZZ1edge. More-
z
over, if the barrier is wide enough or the coupling strong
enough, there are also bound states that are delocalized Rashba-split bands while those that remain very sharp
over the entire ribbon, both in the zigzag and armchair and almost at the same energy correspond to the valley
case, as is shown in Figs. 7, 8, and 9. oddcones. Indeed,thelatteraremissingforthearmchair
Furthermore, in Fig. 9 (a) and (c), we have superim- barrier because the AC edge does not induce coupling
posed the bound states of a bare graphene ribbon on to these states. Note that the coupling due to the ZZ1
the transmission probability for t = 0.1 eV for both an edgealsoinducessomespinsplittingintothequasibound
armchair and zigzag barrier. In this case, the antireso- states originating from the valley odd states. At these
nancesareverysharpandcoincidealmostperfectlywith energies, the wave function is either strongly hybridized,
the bound states of the bare ribbon. These antireso- whichisthecasefortheRashba-likestates,orcompletely
nancesarequasiboundstatesoriginatingfrombothvalley localized in the graphene, which is the case for the val-
even and valley odd states. With increasing t, the quasi- ley odd states. In the latter case, which only occurs for
bound states split into two classes: Those that broaden zigzag ribbons, tunneling is impossible since the ribbon
and move in energy with increasing t correspond to the contains at least one edge that does not allow tunnel-
9
armchair graphene nanoribbons deposited on the TIS in
theTstackingconfigurationareshowninFig.12forsev-
eral values of the width W and the coupling t.
The plateaus in the conductance are caused by the
antiresonances in the transmission probability discussed
above. They are more pronounced for the zigzag bar-
rier than the armchair barrier. With increasing N, the
number of plateaus increase and they move towards zero
energybecauseofthereducedconfinement. Ontheother
hand, if we increase t, more plateaus appear in the con-
ductance and it is suppressed overall due to backscatter-
FIG. 11. (Color online) Energy of the zigzag edge states at
k =2 nm−1 as a function of t for N =10 and N =20. We ing at oblique angles.
y
onlyshowonestateforN =20,sincetheotherstateremains
at zero energy for all t.
V. SUMMARY AND CONCLUSIONS
In summary, we have considered the electronic trans-
mission, using a continuum model, of the topological
surface state of a three dimensional time-reversal invari-
ant topological insulator through heterostructures made
by depositing a monolayer graphene on the topological-
insulator surface. We obtained the transmission of the
topologicalsurfacestatethroughasemi-infinitegraphene
stepandagraphenenanoribbonforbothzigzagandarm-
chair boundaries. We found that the transmission de-
pends strongly on the type of edge: In the case of a
graphene step, we found that the transmission exhibits
electron-hole asymmetry for the ZZ1 edge configuration
while the transmission is perfect at all energies for arm-
chairandZZ2junctions. Moreover,ourresultsshowthat
FIG.12. (Coloronline)(a)-(b)Conductanceforthe(a)zigzag
theconductancethroughagraphenenanoribbonexhibits
and (b) armchair barrier for several widths with t = 0.3 eV.
The widths of the armchair ribbon are chosen so that it is plateaus caused by antiresonances in the transmission
insulatingandmatchesthecorrespondingwidthsinthezigzag probabilityatenergiesofthequasiboundstatesofthede-
case. (c)-(d)Conductanceforthe(c)zigzagbarrierwithN = posited nanoribbon for both zigzag and armchair edges.
20and(d)armchairbarrierwithN =34forseveralt, whose The heterostructures we considered are commensu-
values are shown in eV. rate by less than one percent with at least two well-
known topological insulators, Sb Te and TlBiSe . Hy-
2 3 2
bridgraphene-TIdevicescouldbefabricatedusingame-
ing to these states. On the other hand, the Rashba-like
chanical transfer method where the chemical potential
boundstatesofthegrapheneribbon, inducedbytherib-
differenceandelectrondensitycanbetunedbygatevolt-
bon confinement, can only lead to more possibilities for
ages. Furtherstudiesarerequiredtoaddresstheeffectof
backscattering, and thus antiresonances. In Figs. 7 and
an external magnetic field and the number of graphene
8 the antiresonances are broadened compared to Fig. 9
layers on the transport properties.
because the coupling to the TIS is stronger.
Conductance ACKNOWLEDGMENTS
The zero-temperature conductance through a barrier This research was supported by the Flemish Research
of width W and length L is given by Foundation (FWO). We thank B. Van Duppen for inter-
esting discussions.
(cid:90) |E|
L (cid:126)vs
G(E)=G dk T(E,k ), (40)
0 y y
2π |E|
−(cid:126)vs Appendix A: Unitary transformation
where G = 2e2/h is the conductance quantum and
0
where we have used dimensionful units. This is a Here, we give the explicit expression for the unitary
weighted sum over the available incident transverse transformation U that block diagonalizes the Hamilto-
k
modes LE /(π(cid:126)v ). The conductance for zigzag and nian from Eq. (2) for the case t =0 (T structure), into
s B
| |
10
the form shown in Eqs. (9), (10), and (11). We find basis states are
(cid:12) (cid:11) 1 (cid:0)(cid:12) (cid:11) (cid:12) (cid:1)(cid:11)
(cid:12)ψA±↑(↓) = √2 (cid:12)ψA↑(↓) ±(cid:12)ψA(cid:48)↑(↓) (A4)
 
Uk =AAkk BBkk 00, (A1) (cid:12)(cid:12)ψB±↑(cid:11)= √12(cid:0)∓e−2iθk|ψB↑(cid:105)+|ψB(cid:48)↑(cid:105)(cid:1) (A5)
−
0 0 1 (cid:12)(cid:12)ψB± (cid:11)= 1 (cid:0)ψB e2iθk ψB(cid:48) (cid:1), (A6)
↓ √2 | ↓(cid:105)∓ | ↓(cid:105)
where correspondstotheevenandoddvalleysubspace.
±
with Under time reversal, the new basis transforms as
(cid:12) (cid:11) (cid:12) (cid:11)
Θ(cid:12)(cid:12)ψA±↑(↓)(cid:11)=(cid:12)±(cid:12)ψA±↓((cid:11)↑) (A7)
Ak = 1 diag(cid:0)1, e−2iθk,1,1(cid:1) (A2) Θ(cid:12)ψB±↑(↓) =(cid:12)ψB∓↓(↑) ,
√2 −
so that
Bk = √12diag(cid:0)1,1,1,−e2iθk(cid:1), (A3) (cid:10)ψA− ( )(cid:12)(cid:12)V (cid:12)(cid:12)φ ( )(cid:11)=(cid:10)ψA− ( )(cid:12)(cid:12)Θ−1VΘ(cid:12)(cid:12)φ ( )(cid:11)
↑ ↓ ↑ ↓ (cid:10) ↑ ↓ (cid:12) (cid:12) (cid:11)↑ ↓
=−(cid:10)ψA−↓(↑)(cid:12)(cid:12)V (cid:12)(cid:12)φ↓(↑)(cid:11) (A8)
where θ =arctan(k /k ). =− ψA−↑(↓)(cid:12)V (cid:12)φ↑(↓) =0,
k y x
and the matrix element between the odd subspace and
After performing this unitary transformation, the new the topological surface state vanish.
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