Resonant and non-dissipative tunneling in independently contacted graphene structures

0.26 MB·English

by  F. T. Vasko

Resonant and non-dissipative tunneling
in independently contacted graphene structures
F. T. Vasko∗
QK Applications, 3707 El Camino Real, San Francisco, CA 94033, USA
(Dated: January 24, 2013)
The tunneling current between independently contacted graphene sheets separated by boron ni-
tride insulator is calculated. Both dissipative tunneling transitions, with momentum transfer due
3 to disorder scattering, and non-dissipative regime of tunneling, which appears due to intersection
1 ofelectronandholebranchesofenergyspectrum,aredescribed. Dependenciesoftunnelingcurrent
0 on concentrations in top and bottom graphene layers, which are governed by the voltages applied
2 through independent contacts and gates, are considered for the back- and double-gated structures.
The current-voltage characteristics of the back-gated structure are in agreement with the recent
n
experiment[Science335,947(2012)]. Forthedouble-gatedstructures,theresonantdissipativetun-
a
neling causes a ten times enhancement of response which is important for transistor applications.
J

2
PACSnumbers: 72.80.Vp,73.40.Gk,85.30.Mn
2

]
l I. INTRODUCTION
l
a
h
In contrast to the tunneling processes between bulk
-
s materials,[1]thetunnelingbetweenlow-dimensionalsys-
e
temsmustbeassistedbyscatteringinordertosatisfythe
m
momentumandenergyconservationlaws,seeresultsand
t. discussions for double quantum wells or wires in Refs. 2
a
or3,respectively. Whenthesplittingenergybetween2D
m
states(thisenergy∆isdeterminedbytransversevoltages
- appliedacrossstructure)exceedsthecollisionbroadening
d
energyΓ(h¯/Γisthedeparturetime),thetunnelingprob-
n
o abilityappearstobeproportionalΓ. Inconditionsoftun-
c nelingresonanse,when∆≪Γ,thisprobabilityispropor-
[ tional to ¯h/Γ, [4] i.e. the tunnel current depends on the

scatteringtimeinthesamewayasthecurrentinmetallic
2
v conductor. The breakdown of the dissipative tunneling
7 regime is possible if the energy spectrum branches are
4 intersectedandthe energy-momentumconservationlaws
8 are satisfied without scattering. For example, the in- FIG. 1: (Color online) (a) Gt/BN/Gb structure under volt-
1 ages Vtb and Vbg applied to top (t-) and bottom (b-) layers
tersection of the parabolic electron branches in double
. through independent contacts; back gate (blue) is separated
2 quantumwellstakesplaceifthemagneticfieldisapplied
by substrate of thickness ds and top gate under voltage Vtg
1 perpendicular to the tunneling direction, see [5] and [6] is shown by blue dashed lines. (b) Dispersion laws of t- and
2
1 for the experimental data and theory. Similar intersec- b-layers(blackandgraycrossedlines)withFermienergiesεFt
tionbetweenthelinearbranchesofgaplessenergyspectra and ε for the electron-electron tunneling regime. (c) The
: Fb
v should take place in graphene/boron nitride/graphene same as in (b) for theelectron-hole tunneling regime.
i (G/BN/G) heterostructure. Such a structure was re-
X
ported recently [7] and the tunneling transistor, which
r
a is based on the independently-contacted G-sheets con- acteristics were not analyzed) and a problem is timely
nected through a few monolayer BN, was demonstrated. now.
[8]Incontrasttothesemiconductorheterostructurecase, In the paper, we calculate the tunneling current I be-
the independently-contacted G/BN/G structures can be tweenthe independently-contactedtop (Gt) and bottom
easily realized with the use of the single-layer-transfer (Gb) graphene layers separated by BN. We analyze the
technology. [9] But a complete theoretical investigation dependenciesofI onthe sheetconcentrations(Fermien-
oftunneling currentinsucha structure is notperformed ergies)andon∆, whicharedeterminedbythegatevolt-
yet(somenumericalresultsonthetunnelingconductance ages applied to the contacts, see Fig. 1a. Depending on
were reportedrecently [10] but the current-voltagechar- voltagesapplied,onecanrealizedeitherelectron-electron
(hole-hole) or electron-hole regimes of tunneling, as it is
shown in Figs. 1b and 1c, respectively. In the latter
case the cross-point E =0 is located between the Fermi
∗Electronicaddress: [email protected] energies (εFt > 0 > εFb or vise versa) and the non-
2
dissipative regime of tunneling takes place in addition Similarly to Ref. 6, one express {ρ} through the carrier
to the resonant dissipative tunneling transitions. The distributionsdeterminedby[ρ]andthetunnelingcurrent
current-voltagecharacteristicsappearstobedifferentfor density takes form
these regimes. For the back-gated structure, the results
are in agreement with the experimental data. [8] For 0
double-gatedstructures, the resonantdissipative tunnel- I ≈ 4i dtSp [ρˆ] Tˆ,e−iHˆGt/h¯IˆeiHˆGt/h¯ . (4)
L2¯h
ingregimecanberealized,withatentimesenhancement −Z∞ (cid:16) h i(cid:17)
of response.
The paper is organized as follows. In Sec. II, Here Sp(...) means both summations over states of car-
we present the basic equations which describe the two riers in G - and G -layers and averaging over lateral
t b
regimes of interlayer tunneling. In Sec. III we analyze disorder which should be included in the Hamiltonians
the current-voltage characteristics and compare the re- hˆ . Calculations of Sp(...) are performed below with
Gt,b
sultsforthe back-gatedstructurewiththe ofexperimen- the use of the basis formed by 2-row wave functions in
tal data. [8] The last section includes the discussion ap- kth layer Ψ(kα) determined by the eigenvalue problem
x
proximationsused,andtheconclusions. InAppendix,we hˆ Ψ(kα) = ε Ψ(kα). We introduce the spectral density
evaluatetheeffectivetunnelingHamiltonianforG/BN/G Gk x α x
matrix, labeled by l,l′ =1 and 2, as
structure.
A(k) (x−x′)= δ(E−ε )Ψ(kα)Ψ(kα)∗ , (5)
II. BASIC EQUATIONS E,ll′ * kα lx l′x′ +
Xα k
Under consideration of G /BN/G structure, we use where the averaging over random disorder in kth layer
t b
the tunneling Hamiltonian which connects Gt and Gb h...ik is performed. Using the Fermi distribution for
layersdescribedby 2×2matriceshˆ , i.e. we introduce heavily-doped layers, when [ρ] is replaced by the θ-
Gt,b
4×4 matrix [11] function θ(εFk − εkα) with the Fermi energies εFk in
Eq.(4), we transform I into
hˆ τˆ
HˆGBNG =(cid:12) τˆGt hˆGb (cid:12)≡HˆG+Tˆ. (1) I = 8π|e| εFtdE dx dx′ (6)
Hlaeyreersw, eHˆha,vaendsepthareatt(cid:12)(cid:12)(cid:12)uednntehliengHacmo(cid:12)(cid:12)(cid:12)nitltroibnuiatniono,fTuˆ,ncworuipttleedn ¯hL2 εZFb Z Z
through tGhe 2×2 matrix τˆ= τˆ+ which is determined by ×tr τˆ+AˆE(b)(x′−x)τˆAˆ(Et)(x−x′) .
a stackinggeometryofthe structure,seeAppendix. The h i
chargedensityinG andG layers,Q(k) (hereandbelow wheretr(...)meanssummationoverthematrixvariable.
t b t
k = t,b), and the tunnel current density, I , are deter- Below, we express Aˆ(k) in the momentum representa-
t E
mine throughthe 4×4density matrix ρˆt by the formulas tion as Aˆ(k) = i GˆR(k)−GˆR(k) + /2π, where GˆR(k)
[12] E,p E,p E,p E,p
is the retarded Gr(cid:16)een’s function of(cid:17)kth layer with the
QI(tk) = L42Sp ePIˆˆk ρˆt . (2) csproliststinpgoinenteregnye.rgiWesit±hi∆n/t2hewmritotdeenl othfrsohuogrht-rtahnegeledveisl--
(cid:12) t (cid:12) (cid:18)(cid:12) (cid:12) (cid:19) order with the same statistically independent character-
(cid:12) (cid:12) (cid:12) (cid:12)
Here L2 is th(cid:12)(cid:12)e norm(cid:12)(cid:12)alization ar(cid:12)(cid:12)ea, Pˆ(cid:12)(cid:12)k is the projection istics for k = t and b, the Green’s function in the Born
approximationtakes form [13]
operator on the k-states, and the interlayer current op-
erator Iˆis determined from the charge conservation re-
quirement I =dQ(t)/dt=−dQ(b)/dt (see similar calcu- GˆR(k) =Pˆ(+)G(k) +Pˆ(−)G(k) ≡G(k)+ σˆ ·pG(k),
t t t E,p p E,p p E,−p Ep p Ep
lations in Refs. 6), so that
G(k) ≈ (E∓∆/2)(1+Λ +ig)−υp −1, (7)
E,p E∓∆/2
e 0 −iτˆ
Iˆ= . (3)
¯h iτˆ 0 where −∆/(cid:2)2 and +∆/2 are correspondent to(cid:3)t- and b-
(cid:12) (cid:12)
(cid:12) (cid:12) layers and υ ≃ 108 cm/s is the carrier velocity. The
As a result, tunneling pr(cid:12)ocesses a(cid:12)re described by the
(cid:12) (cid:12) projection operators on the conduction (+) and valence
above-introducedmatricesTˆandIˆ,aswellasthedensity
(−) band states, Pˆ(±) = [1±(σˆ ·p)/p]/2, are writ-
matrixgovernedbythestandardequation: i¯h(∂ρˆ/∂t)= p
t ten through the 2×2 isospin Pauli matrix σˆ. The self-
[Hˆ ,ρˆ]. [12]
GBNG t energy contributions (E ∓∆/2)(Λ +ig) are writ-
Further, we separate the diagonal and non-diagonal E∓∆/2
tenthroughthe logarithmically-divergentrealcorrection
partsofthedensitymatrixρˆ ≡[ρˆ]+{ρˆ}whichdescribe
t t t which is proportional to Λ = (gπ)ln(E /|E|) and the
the distribution of carriers in G - and G -layers and the E c
t b coupling constant g. This approach corresponds to the
tunneling current, It = (4/L2)Sp Iˆ{ρˆt} , respectively. short-range scattering with the cut-off energy Ec. [14]
(cid:16) (cid:17)
3
The interlayer tunneling is described by the parameters
1
T2 =tr(τˆ+τˆ), T2 = tr(τˆ+σˆ τˆσˆ ), (8)
s 2 µ µ
µ
X
whichappearundercalculationofthematrixtraceinEq.
(6). Using Eqs. (7) and (8) we transform the tunneling
current density (6) into the form
εFt
2e dE
I = T2ImG(t)ImG(b)
π¯h L2 Ep Ep
εZFb Xp (cid:16)
+T2ImG(t)ImG(b) , (9)
s Ep Ep
(cid:17)
where the explicit expressions for G(k) and G(k) are de-
Ep Ep
termined by Eq. (7). Due to the in-plane symmetry of
FIG.2: (Coloronline)(a)TunnelingcurrentdensityI versus
the problem, I is written as the double-integral over the
p-plane and the energy interval (ε ,ε ). levelsplitting∆fortheresonanttunnelingregimeatεF =200
Ft Fb meV and δε=50 meV (1) or 100 meV (2) for g =0.3. Solid
Integrations in (9) are performed analytically for the
and dashed curves are plotted for χ =1 and χ =0, respec-
collisionless case, g →0, and the tunneling current den-
tively. Curve (3) is plotted for parameters (1) at g = 0.15.
sityisgivenbythesumofthedissipativecurrent,whichis Dottedcurvefitsdependencies(2)withtheuseEq. (10). (b)
∝δΓ(∆), and the non-dissipative contribution (∝χ|∆|): The same as in panel (a) for the non-dissipative regime, if
εF =0 and δε=200 meV (1) or 100 meV (3). Curve (2) cor-
ε2Ft−ε2Fb δΓ(∆) , εFtεFb >0 responds εF =200 meV, δε =100 meV and dotted line gives
I ≃J sign(ε ) ε2 +ε2 , (10) ∝∆ contribution in Eq. (10).
T (cid:0) Ft (cid:1)Ft Fb
×δ (∆)+χ|∆|/2 , ε ε <0
 Γ Ft Fb
(cid:0) (cid:1)
J = |e| T2+Ts2 , χ= T2−Ts2. Fermi energies determined by εF = (εFt +εFb)/2 and
T 2h¯3υ2 T2+T2 δε = ε −ε at χ =1 or 0. The resonant dissipa-
(cid:0) (cid:1) s F Ft Fb
tive regime of tunneling is realizedat |ε |>|δε /2|and
F F
Here the resonant dissipative contribution (at ε ε >
Ft Fb peak value I(∆ = 0) increases both with doping levels
0) is written through the δ-like function δ (∆) =
Γ and with g, as it is shown in Fig. 2a. If |ε | < |δε /2|,
F F
Γ/[π(∆2 + Γ2)] with the phenomenological broadening
the electron-hole tunneling regime takes place and the
Γ ≃ g(ε +ε ). At ε ε < 0, the non-dissipative
Ft Fb Ft Fb non-dissipative contribution becomes dominant with in-
contribution is written through the factor χ (0 < χ < 1
creasing of ∆, where I ∝ χ|∆|. At small ∆ the de-
because T2 ≥ T2) determined by the tunneling energies
s pendency I(∆) is transformed into narrow peak due to
T and T . These parameters depend on the stacking
s the dissipative contribution with weak broadening, see
order of the G /BN/G structure and rough estimate
t b Eq. (10) and Fig. 2b. The numerical results given by
for the case of N-layer BN barrier [15] is determined
Eq. (9)areinagoodagreementwiththeapproximations
by Eq. (A.4), see Appendix. As a result, we obtain
(10), shown by the dotted asymptotics, because of weak
T,T ∼ γ(γ/ε )N, where γ ∼0.4 eV is the interlayer
s BN (≤10%)contributionsfromtherenormalizationofenergy
overlap integral and ε ≤2.5 eV is of the order of the
BN spectra.
c- and v-band energies in BN. We use below T,T ∼13
e s Neglectingthequantumcapacitancecontributionsand
µeV for N = 4 and ∼6 µeV for 6-layer BN barrier,
e the near-contactdrops of potentials, we use εFt−εFb ≃
which are in agreement with the experimental data of
eV andtheGausstheoremconnectedthecarrierconcen-
tb
Ref. 8. The scattering parameters used in Eqs. (9, 10)
trations in graphene with interlayer electric fields, V /d
tb
are taken from the conductivity measurements, see sim-
and V /d (here d and d are thicknesses of BN layer
bg s s
ilar considerations in Refs. 16. We choose the cut-off
andSiO substrate). As a result, the Fermienergiesand
2
energy E ∼0.2 eV and the coupling constants g ≃0.3 or
c ∆ are connected with drops of voltages as follows:
0.15correspondedtothe maximalsheetresistance∼4or
2 kΩ per square. 2εFt,b ≃±eVtb−F(eVtbεd)−F(eVbgεds −eVtbεd),
ThecurrentdensityI isdependentonεFt, εFb and∆, ∆≃eVtb+F(eVtbεd)−F(eVbgεds −eVtbεd),
whicharedeterminedbythedropsofvoltagesappliedto
ǫ(h¯v)2
three- or four-terminal structures, see Fig. 1a. Before F(x)=sign(x) |x|, ε = . (11)
d 4e2d
study of the current-voltage characteristics, we consider
p
the dependencies of the total current I on these three Here”+”and”−”standforG -andG -layers,ε >0
t b Ft,b
energies. In Fig. 2 we plot the tunneling current, which or < 0 correspond to electron or hole doping, and the
is determined by Eqs. (9) and (7), versus ∆ for different dielectric permittivity ǫ≃4 is the same for BN and SiO
2
4
FIG. 3: (Color online) Tunneling current I versus V at
tb
V =0 for structures with 4- and 6-layer BN barriers (black
bg
andred curves,respectively;squaresand triangles areexper-
imental points [8]). Solid, dotted, and dashed curves corre-
spondtoχ=1. 0.5,and0. Insetsshow∆andconcentrations
nt or nb (solid or dashed lines) versus Vtb.
FIG. 4: (Color online) (a) Current density I versus V for
tb
layers, see Refs. 7 and 8. For the double-gated struc- 4-layer BN barrier at V =0 (red), -25 V (blue), and -50 V
bg
turewithtopgateseparatedbyBNinsulatorofthickness (black). ExperimentalpointsareshownforV =0(triangles)
bg
d , one obtains similar expressions for ε and ∆ after and-50V(hexagons). Insetshows∆versusV . (b)Thesame
t Ft,b tb
the replacement eV ε → eV ε −eV ε , see voltages as in panel(a) for 6-layerBN barrier.
tb d tb d tg dt
shown in Fig. 1a. The approach (11) is valid for the
heavily-doped graphene layers, so that the fields V /d,
tb
V /d , and V /d should be strong enough. On the
bg s tg t in Fig. 4 for N =4, 6 and χ = 1, 0. Once again, there
other hand, these fields are restricted by the breakdown
is a reasonable agreement of I(V ,V ) with the avail-
tb bg
condition for BN layer, when these fields should be ≪7
able experimental data for N =4 at V < 0. A visible
tb
MV/cm. [17]
asymmetry of I takes place if V →−V but deviations
tb tb
from experimental data increase. The dependencies for
N =6 are similar but I is ∼2-times weaker and the res-
III. CURRENT-VOLTAGE CHARACTERISTICS onant dissipative contribution is absent for Vbg =−25 V
because ∆6=0, see inset.
In this section we analyze the current-voltage charac- An enhancement of the resonant dissipative tunnel-
teristics for the back- and double-gated structures. Be- ing contributions takes place in the double-gated (four-
low, the tunneling current density is determined by Eqs. terminal)structure,whenI dependsonVtb,Vtg,andVbg.
(7) and (9) with the Fermi energies and the level split- These dependencies are shown in Fig. 5 for the struc-
ting written through the voltages applied according to ture with 10-layer BN cover layer (dt =3.4 nm; with a
Eqs. (11). The parameters described both the elastic negligible tunneling, if dt ≫2 nm [19]) at different top-
scattering processes and the interlayer tunnel coupling andback-gatevoltages(Vtg =0correspondstothethree-
are chosen the same as for the above calculations shown terminalstructure). Sincethecondition∆=0isrealized
in Fig. 2. now at higher energies, the resonant dissipative tunnel-
Fortheback-gatedstructurewithN-layerBNbarriers ingpeaksare>10timesgreaterthanthebackgroundcur-
(N =4 and 6) and g =0.3 the dependencies of I versus rent. Thisisthecentralresultwhichshouldbeimportant
V at fixed V are shown in Fig. 3 at V = 0, when for the transistor applications. Note, that for ∆ ∼0 at
tb bg bg
Ia−reVt∝b =V−IaVntbd. ∆[18is] Tthheeseulmectorofnlin(heaorle)ancdonscqeunatrrae-trioonost pVetbak∼0is(stuhpepcruersvseed3. inBeFyiogn.d5ao)f tthheisrnesaornroawntrdeigsisoipna,ttihvee
tb
functions, see insets. The current-voltage characteristics resonant condition ∆(Vtb)≈0 should not be suppressed
I(V ) are in reasonable agreement with the experimen- by lateral inhomogeneities and the peaks caused by the
tb
tal data of Ref. 8 if we used the scale factor J with the resonant dissipative tunneling should not be smeared.
T
above-estimated T and T . The dependencies on χ are Overall, the consideration presented here gives an ad-
s
weak enough (≤25% for N =4 and ≤10% for N =6, not equate theoreticaldescriptionofthe tunneling transistor
shown). If ∆ ∼0, when V ≃-0.55 V for 6-layer barrier, effect which is in reasonable agreement with the exper-
tb
a negative resonant contribution due to dissipative tun- imental data. [8] The analysis performed opens a way
neling gives ∼30% variations of the I-V characteristic. for the verificationofscattering mechanismsand tunnel-
SuchapeculiaritywasnotfoundinRef. 8;probably,itis ing parameters in G/BN/G structures. In addition, the
duetoalateralredistributionsofchargesoralarge-scale double-gated (four-terminal) structure was not consid-
inhomogeneities of the samples used. The I-V charac- ered before and this structure show a great (ten times)
teristics of back-gated structures at V 6= 0 are plotted enhancement of tunneling current tunability.
bg
5
andthe results obtainedcan be applied for characteriza-
tion of scattering mechanisms and tunneling parameters
in the tunnel-coupled graphene structures. More impor-
tantly, that these results open a way for improvement of
tunneling transistor, a new type of graphene-based de-
vice. We believe that our study will stimulate a further
investigation of these device applications.
Appendix: Tunneling Hamiltonian
Below we describe the tunnel-coupled states in
G/BN/G structure using the 6-column wave function
[ψ ,φ,ψ ] which is written through the spinors ψ cor-
t b t,b
respond to the G and G graphene layers. These layers
t b
are connected through the spinor φ described the sin-
gle BN-layer. Within the tight-binding approach, [15]
FIG. 5: (Color online) (a) Dependencies I(Vtb) for double- the eigenstate of energy E is determined by the problem
gatedstructurewith4-layerBNbarrierand10-layerBNcover
written through the 6×6 Hamiltonian
layer under Vbg = −50 V and Vtg = 1.5 V (1), 0.75 V (2), -
0a.n7d5dVas(h3e)d, acnudrv-e1s.5coVrr(e4sp).onIdnsetot sχho=w0s ∆andve1r,sursesVptebc.tiSvoelliyd. hˆGt −E hˆGBN 0 ψt
hˆ+ hˆ −E ˆh φ =0, (A.1)
(b) The same as in upperpanel for Vbg =0 V. (cid:12)(cid:12)(cid:12) G0BN BhˆN+BNG hˆGBbN−GE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ψb (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
wher(cid:12)e hˆ and hˆ are the 2 × 2(cid:12)H(cid:12)ami(cid:12)ltonians of G ,
IV. CONCLUSION Gt Gb t
G ,andBN-layers,whilehˆ andhˆ describeweak
b GBN BNG
interlayer coupling of G and G sheets with BN layer.
t b
We have adopted the theory of low-dimensional tun- Under a transverse voltage applied, hˆ = ˆh ±∆/2,
neling [4, 6] to the case of the tunneling charge trans- wherehˆ istheHamiltonianofsinglegGrat,pbheneGlayerand
fer in independently contacted graphene structures with G
∆ is a splitting energy between the cross-points in G -
multi-layer BN barrier taking into account the two ba- t
and G -layers. For the low-energy (|E| <1 eV) states, if
sic mechanisms of charge transfer: resonant dissipative b
|E| ≪ |ε | where ε ∼3.4 eV and ε ∼-1.4 eV are the
tunneling and non-dissipative tunneling. Both the non- c,v c v
c- and v-band extrema energies in BN, we can eliminate
dissipative tunneling currentand the non-resonantdissi-
the spinor φ from the system (A.1). As a result, the
pativetunnelingprocessesareresponsibleforthecurrent-
eigenstate problem is determined by the 4×4 effective
voltage characteristics of back-gated structures. [8] The
tunneling Hamiltonian
resonant dissipative tunneling regime is achieved for the
double-gated structures and these devices demonstrate hˆ +τˆ −E τˆ ψ
aimtpeonrttaimntesfoernthraanncseismtoerntapopflIic-Vaticohnasr.acteristics which is (cid:12) G τˆ+t hˆG+τˆb−E (cid:12)(cid:12)ψbt (cid:12)=0. (A.2)
(cid:12) (cid:12)(cid:12) (cid:12)
Further, we discuss the assumptions used. Because of Here t(cid:12)(cid:12)he 2×2 matrix τˆ describes(cid:12)(cid:12)tu(cid:12)(cid:12)nne(cid:12)(cid:12)ling through BN
a luck of data on stacking order in G/BN/G structures insulator, while τˆ and τˆ correspond to the tunneling
t b
the tunneling energiesT andT wereestimatedfromthe renormalizationof t- and b-states:
s
current-voltage characteristics of Ref. 8 and this result
τˆ≈−hˆ hˆ−1 hˆ , (A.3)
is in agreement with the tight-binding model. [15] More GBN BN BNG
accurate estimates for T and Ts should be based on ad- τˆt ≈−hˆGBNhˆ−B1Nhˆ+GBN, τˆb ≈−hˆ+BNGhˆ−B1NhˆBNG.
ditional structural measurements. The simplified elec-
trostatics description, given by Eq. (11), fails for the Thus, we arrive to the Hamiltonian (1) with the renor-
low-doped G - or G -layers, under weak interlayer fields malizationcontributionsτ includedtohˆ ;these cor-
t b t,b Gt,b
applied. Inthesenarrowregions,amorecomplicatedde- rections are negligible for the weak tunneling regime.
scription, which involves the quantum capacitance effect For the case of N-layer BN insulator, we consider
andthe contactphenomena,shouldbe applied. The rest the 2(N + 2)-column wave function with N-spinors
ofassumptions(themodelofelasticscattering,[16]iden- φ ,...,φ described the BN layers. These states are
1 N
ticalscatteringinGt,b-layers,weaknessoflong-rangedis- coupled by the interlayer hopping matrices hˆBNBN and
order, and the single-particle approach) are rather stan- hˆ+ which are placed to upper and lower sub-
BNBN
dard. diagonals of 2(N +2)×2(N +2) Hamiltonian. [15] Af-
Toconclude,webelievethatthe descriptionoftunnel- tereliminationsofthe spinorsφ ,...,φ fromthetight-
1 N
ing processes is an essential part of physics of graphene bindingeigenstateproblem,wearrivetoEq. (A.3)where
6
the non-diagonal matrix τˆ is replaced by orderofε−1 where ε is determinedbyε . The hop-
BN BN c,v
ping matrices hˆ , hˆ , and hˆ are estimated
τˆN ≈−hˆGBN(hˆ−B1NhˆBNBN)N−1hˆ−B1NhˆBNG. (A.4) bytheineterlayerGoBvNeerlapBNinGtegralγ aBnNdBtNheyarestrongly
dependent on a stacking order of G/BN/G structure.
For the numerical estimates of T and T given by Eq.
s
(8) we assume that the diagonal matrix ˆh−1 is of the
BN
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