Theory of resonant tunneling in bilayer-graphene/hexagonal-boron-nitride heterostructures

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by  Sergio C. de la Barrera

Theory of resonant tunneling in bilayer-graphene/hexagonal-boron-nitride
heterostructures
Sergio C. de la Barrera and Randall M. Feenstra
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213,
USA
A theory is developed for calculating vertical tunneling current between two sheets of bilayer graphene sep-
arated by a thin, insulating layer of hexagonal boron nitride, neglecting many-body effects. Results are pre-
5
sented using physical parameters that enable comparison of the theory with recently reported experimental
1
results. Observedresonanttunnelingandnegativedifferentialresistanceinthecurrent–voltagecharacteristics
0
are explained in terms of the electrostatically-induced band gap, gate voltage modulation, density of states
2
nearthe bandedge, andresonanceswith the upper sub-band. These observationsare comparedto ones from
r
a similar heterostructures formed with monolayer graphene.
M
2 In contrast to the well-known linear dispersion of (a) (b) E
monolayergraphene (MLG), charge carriersnear the six
h-BN BLG
] corners of the Brillouin zone in an isolated graphene bi-
l
l layer are described by a quadratic energy dispersion.1,2 h-BN substrate
a
An even more intriguing distinction with MLG is that,
h
- under the influence of external fields, the band struc- SiO2 dielectric
s ture of bilayer graphene (BLG) near the charge neutral- + Si back gate
e
m ity point becomes quartic, changing from semi-metallic k
to semiconducting as a small band gap is induced.3–5
.
t This tunability of the band gap can be exploited by in-
a (c) (d)
m troducing gates, doping, and interactions with substrate
materials in electronic devices based on BLG.6–8 In this
-
d paper,weconsidertheseeffects andothersina2Dto2D
n resonanttunnelingdevicecomposedoftwosheetsofBLG
o separated by a thin, insulating layer of hexagonal boron
c nitride (h-BN). In a vertical configurationwith an inter-
[
layer bias, tunneling between two-dimensional electron

2 gasesisconstrainedbysimultaneousenergyandmomen-
v tum conservation, leading to resonances in the current–
6 voltage(I–V)characteristicandthus regionsofnegative
4 differential resistance (NDR).9 Such devices were orig-
6 FIG.1. (a)Devicestructure,withdoubleblacklinesindicat-
4 inally proposed for conventional 2D quantum wells,10 ingeachgraphenebilayer(BLG)andthegroupoforangelines
0 but the theory was recently treated for MLG,11–18 and representing4to6layersofh-BN.(b)Alignmentofelectronic
. NDR was observed experimentally in high-quality de- bands at an off-resonant interlayer bias voltage; blue (solid)
1 vicesshortlythereafter.9,16,19Thetheorydiscussedinthe curves for one bilayer and red (dashed) for the other; (c) at
0
present work is particularly relevant to the recent obser- thevoltagewhichyieldstheprimarytunnelingresonance;(d)
5
1 vations of Fallahazad et al.9 atahighervoltagewhichalignsthelowerbandsofonebilayer
with the higher sub-bands of the other. The largest contri-
: For a given interlayervoltageand for bilayersthat are
v bution to tunneling current occurs near the states where the
i in crytallographic alignment, the electronic bands of the two bands intersect. Bands represent energy as a function of
X
top and bottom bilayer of graphene will overlapfor par- in-planecrystalmomentumnearoneofthesixcornersofthe
r ticular sets of states with equal energy and crystal mo- Brillouin zone.
a
mentum(Fig.1). Awayfromtheresonancevoltage,only
the states near the intersecting ring(s) can contribute to
thetunnelingcurrent(Fig.1b). However,foroneparticu- strate interactions, resonance can be observed for both
larvoltage,theelectrostaticpotentialbetweenthebands bothpositiveandnegativebiasvoltagesastheback-gate
will be zero, allowing all states between the two Fermi voltage(V )issweptfromonesigntotheother(Fig.2).
BG
levels to tunnel simultaneously (Fig. 1c). The shape and Recently, Fallahazad et al. have observed resonances
position of the resulting resonant peak in the I–V char- with precisely this behavior in devices fabricated with
acteristicdependsonthequantityandsignofchargecar- exfoliated BLG/h-BN/BLGon a h-BN/SiO substrate.9
2
riers in eachbilayer,and therefore indirectly on external The width and amplitude of each resonant peak relative
fields (gate voltages) and the electrostatic doping condi- tothebackground(non-resonant)currentaredetermined
tions. by the degree of coherence between tunneling wavefunc-
For example, in the absense of strong doping or sub- tions, as is discussed in detail in Ref. 17.
2
(a) (b) (c)
0.2
0.1
µS 0.0
−0.1
µD
−0.2
−0.3
ρS(E) ρD(E)
FIG. 3. (a) Electronic bands in the source (left) and drain
(right) electrodes with the tunneling barrier (band gap of
boron nitride) in between at a small positive bias. Dashed
linesindicatetheFermilevelsineach bilayer,µi =−eVi;not
to scale. Density of states corresponding to (b) the source
electrode and (c) the drain electrode in the same bias condi-
tion as panel (a); energy axes in units of eV. The alignment
of the divergences in the density of states near the valence
FIG. 2. Calculated tunnel current density versus interlayer band edge of each bilayer produces a large overlap of states
bias for undoped graphene bilayers for a range of gate volt- and thusa spikein tunnelingcurrent.
ages. Upper inset: a similar computation for a larger volt-
age range highlighting secondary resonances from the higher
sub-bands. Lower inset: a closer view of the VBG = 0 case,
gapis a ring of states concentric with the K-point. This
varyingcoherencelength,adisorderparameterinthetheory,
arrangement of states causes divergences in the DOS at
from 50nm (solid, light) to 10nm (solid, dark).
the conduction and valence band edges, which can yield
additionalspikesinthetunnelingcurrentforcertainelec-
trostatic arrangements. Whereas the primary feature in
In addition to the primary resonance, the higher sub- the tunneling current occurs when the electrostatic po-
band of one bilayer can also come into alignment with tentials in the source φ and drain φ electrodes are
S D
the lower sub-band of the second bilayer causing a simi- aligned, ∆φ = φ −φ = 0, other features due to over-
S D
lar spike in the tunneling current. Secondary resonances lapofthelargeDOSnearthebandedgescanoccurwhen
aswellasanincreaseinbackgroundcurrentfromtheup- one of the four conditions ∆φ±E /2±E /2 = 0 is
g,S g,D
per bands entering the tunneling energy window can be satisfied (where E is the band gap in each bilayer), as
g,i
observedatlargervoltagesasshownintheupperinsetof inFigs.3and4. ThesefeaturesintheI–V characteristic
Fig.2. Interactionswiththeuppersub-bandsaredistinct are distinct from those caused directly by momentum-
from monolayer devices, and may provide opportunities conserving effects with complete band alignment (as in
for multi-state logic devices. Fig.1)andarelesssensitivetothe relaxationofmomen-
Atasmallervoltagescale,andespeciallyatlowertem- tum conservation(decoherence),but may be observedin
peratures, it is possible to observe additional features tandem with the latter. In MLG there are no equivalent
due to the tunable band gap in BLG. The presence of band edges, and thus these additional sharpfeatures are
a transverse electric field across a graphene bilayer in- absent in monolayer vertical tunneling devices.
duces a potential difference between the two individual Weuseatight-bindingmodelforthedispersionofBLG
layers of graphene. This broken layer symmetry causes withnearest-neighborhoppingenergyγ ≈3.1eV,inter-
0
a small band gap to open up around the charge neu- layer hopping energy γ ≈ 0.4eV, and interlayer poten-
1
trality point which increases with the magnitude of the tial asymmetry U.20 Higher order considerationssuch as
potential difference across the bilayer. In the tunnel- the trigonal warping of the bands (azimuthal asymme-
ing device modeled here, the interlayer and gate volt- try) were found to have a negligible impact on the tun-
ages modulate the separate potential difference across neling and thus were excluded. The occupation of levels
each individual bilayer in a coupled system.8 As a re- andbandgapineachelectrodevaries with the set ofap-
sult, the band gaps in both bilayers vary with voltage plied voltages, and thus the electrostatic potentials are
(typically at different rates), which affects the overall requiredtocalculatethetunnelingcurrent. Thesepoten-
tunneling current. For non-zero band gap, the precise tials are determined by first solving a matrix equation
form of the energy dispersion is quartic near the gap, as q = C V , treating each monolayer of graphene sepa-
i ij j
in Figs. 1 and 3(a). Moreover, the location of the band rately,to obtainthe transversefields acrosseachbilayer.
3
contained in the matrix element
(1) (2) (3)
¯h2 dΨ dΨ∗
M = dS Ψ∗ β −Ψ α , (2)
αβ 2mZ (cid:18) α dz β dz (cid:19)
which is evaluated in a similar way as for MLG in Refs.
3 4 11 and 17. We calculate the surface integral in Eq. 2
(4)
2 over a region defined by the length scale of wavefunc-
tion coherence in the device, a parameter we call the
characteristic coherence length, L. This is a disorder
parameter which defines the degree of momentum con-
1 servation and thus controls the width and amplitude of
resonantfeaturesintheI–V characteristic. Themomen-
tum (wavevector) conservation, chiral (angular) terms,
and crystallographic misorientation are encapsulated in
the matrix elements, while energy conservation is con-
tained in the δ-function that appears in Eq. 1.11,17
In contrast to the theory for MLG devices, for BLG
this δ-function must be evaluated using the quartic dis-
persionrelationinorderto capture band-gapandhigher
sub-bandeffects. Convertingthe sums overstates in Eq.
1 to integrals over k, we can evaluate the δ-function by
changing variables from E to k,
δ(k−k )
δ[E(k )−E(k )+∆φ]→ i (3)
S D |f′(k )|
Xi i
forallcombinationsofbandsbetweeneachbilayer,where
f is equal to the original argument of the δ-function, k
i
′
are the zeros of this argument, and f is the derivative
with respect to k. This procedure allows us to remove
onek-integrationandproceedto calculatingthe current.
FIG. 4. Low-voltage tunneling current for a device with a
Asmallamountofbroadeningisintroducedtohandlethe
small amount of doping on the top (drain) bilayer at 10K
singularitiesthatarisenearthebandedges(animaginary
showing a number of small features due to the alignment of
′
various band-edges, as explained in Fig. 3. I–V curves are term iǫ is added to the |f | denominator, with epsilon
shifted vertically for clarity. Numbered insets show theband typically equal to 10−2¯hvF).
alignmentforeachofthefourlabeledpointsalongtheVBG = Comparingourtheorywiththeexperimentalresultsof
40V curve. Arrows indicate electron current that produces Fallahazadetal.9,wefindfortheundopeddeviceatroom
thesharp feature in each case. temperature (Fig. 2) verygoodagreementboth interms
ofthe peakshapesandthegate-voltagedependence. For
the low-temperature results of Fig. 4, small peaks as-
We then use those fields to solve a second matrix equa- sociated with DOS features become prominent, super-
tion treating each bilayer with the local DOS for each imposed on a broad momentum-conserving background
layer within the bilayers. This method can accomodate current. We believe the situation found in experiment
both top and bottom gates, thoughwe chose to focus on at low temperature is the same, showing a similar sharp
matching with devices with only one gate in the present peak superimposed on a smooth background current.9
work. Net charges are calculated using full Fermi inte- The interpretation offered in Ref. 9 associates the sharp
gralsqi =e(ni−pi)−e(n0,i−p0,i),ni =e dEρ(E)f(E) peak itself with a momentum-conservingresonanteffect,
toaccountforquantumcapacitanceandthRermaloccupa- but no origin for the broad background is provided. Al-
tion,withenvironmentaldopingdensitiesn0,i. Wecalcu- ternatively, in our interpretation, both features can be
latethetunnelingcurrentbysummingoverthetransition well understood. The data for the undoped device at
rates between all states in the source and drain bilayers, room temperature (Fig. 2) can be similarly understood
withinthe sameframework. SharpDOSfeaturesarenot
2πe
2
I =g g |M | [f (E )−f (E )]δ(E −E ), seen for the latter, either in theory or experiment, since
s v αβ S α D β α β
¯h
Xα,β the higher temperature leads to a reduction in the am-
(1) plitude of the sharp peaks (at elevated temperature the
with spin and valley degeneracies g , g and state labels tunnel current includes contributions from nearby states
s v
α and β in the source and drain bilayers.11 The over- that are thermally occupied, leading to a reduction in
lap integrals between states in the source and drain are strength of the sharp peaks). This distinction between
4
DOS versus momentum-conserving effects, as provided ACKNOWLEDGEMENTS
by our theory, provides an expanded interpretation of
the experimental results.9 We would like to thank Emanuel Tutuc for useful dis-
While resonant tunneling in MLG heterostructures is cussions. Thisworkwassupportedinpartbythe Center
novel and intriguing, the additional sub-bands in BLG forLowEnergySystemsTechnology(LEAST)one ofsix
as well as its unusual behavior in the presence of trans- centers of STARnet, a Semiconductor Research Corpo-
verse fields provides many additional channels for inter- ration program supported by MARCO and DARPA.
esting tunneling phenomena. Although the results pre-
sented here were calculated with zero angular misorien-
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