Transverse field effect in graphene ribbons
arXiv:0704.1052v2  [cond-mat.mes-hall]  2 Aug 2007
Transverse field effect in graphene ribbons
D. S. Novikov
W. I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
(Dated: July 31, 2007)
It is shown that a graphene ribbon, a ballistic strip of carbon monolayer, may serve as a quan-
tum wire whose electronic properties can be continuously and reversibly controlled by an externally
applied transverse voltage. The electron bands of armchair-edge ribbons undergo dramatic transfor-
mations: The Fermi surface fractures, Fermi velocity and effective mass change sign, and excitation
gaps are reduced by the transverse field. These effects are manifest in the conductance plateaus, van
Hove singularities, thermopower, and activated transport. The control over one-dimensional bands
may help enhance effects of electron correlations, and be utilized in device applications.
PACS numbers: 73.23.-b 73.63.-b 72.80.Rj
Building nanoscale systems with pre-determined prop-
erties has long been the focus of basic and applied re-
search. Progress in this field is tied to the recent ad-
vancements in the synthesis of quantum nanowires [1]
and quantum dots [2] via control of the growth process,
as well as in the growth and selection of carbon nanotubes
[3]. The characteristics of these devices, however, are set
by design and are typically difficult to modify. Ideally,
one would like to be able to tune the system’s properties
reversibly after synthesis.
In the present work we suggest that a graphene ribbon
(GR), a ballistic strip of recently discovered [4] carbon
monolayer, may serve as a quantum wire whose electronic
properties can be continuously and reversibly controlled
by the external transverse voltage. The setup makes use
of the massless relativistic electron dispersion in graphene
[4–6], with the valence and conduction bands touching at
a conical Dirac point [3, 7].
Electron dispersion in GRs varies depending on their
chirality, as the transverse confinement of Dirac fermions
is sensitive to the boundary conditions [8–16]. This
has prompted proposals for GR applications as field-
effect transistiors [17] and valley filters [18]. Further-
more, GRs have been suggested as a host of interesting
many-body phenomena, including spin polarization on
the edges [19, 20], and as a basis for building coupled
electron spin qubits [21]. Recently spectral gaps in GRs
have been measured [22], scaling approximately inversely
with the ribbon width.
The basic idea of the present proposal is that the prop-
erties at the Dirac point are fragile and can be affected by
external fields [23]. Not suprisingly, the proposed strong
electric field effect is similar to that considered for carbon
nanotubes [24–26]. Unfortunately, small radiusR ∼ 1 nm
of single-walled tubes requires very large transverse fields
E[MV/cm] ≃ 25/R2[nm]; this has so far hindered observation
of band transformation. Remarkably, with GRs, the rib-
bon width (that plays the role of the tube circumference)
may vary in a broad range, L ∼ 10−200 nm [22], and the
effects of strong band transformation become realistic.
In the setup shown in Fig. 1, electrons in a GR are con-
fined along the x axis, while the longitudinal momentum
ky ≡ k is conserved. The effect of the applied transverse
voltage V is to induce the potential
U(x) = −eE(x− L/2) . (1)
The acting field E ∝ Eext ∝ V can be assumed uni-
form and proportional to the external field Eext as long
as the bands are not strongly mixed (as described be-
low); e is the unit charge. We subtracted the average,
setting
dxU(x) = 0 (the subtracted constant adds to
the chemical potential controlled by the gate voltage Vg).
The natural units for the transverse voltage and energy
u = eEL/∆L , ∆L = h̄v/L ≃ 0.7 eV/L[nm] , (2)
where v ≃ 106m/s is graphene’s Fermi velocity. The
ballistic limit of transport is implied.
We now give an overview of the results. The electron
band transformation is shown in Figs. 2 and 3. The lon-
gitudinal electron bands change qualitatively when the
FIG. 1: (color online). The setup. Left and right electrodes
carry the voltage ±V/2, producing the external field Eext. The
carrier concentration is controlled by the back gate voltage Vg.
http://arxiv.org/abs/0704.1052v2
−10 −5 0 5 10
Wave vector  kL along the ribbon
10 20 30
FIG. 2: (color online). Transverse field effect in metallic GRs.
Inset: Velocity reversals in metallic GRs occur at the zeros
of the function g(u), Eq. (9); first reversal voltage u1 ≈ 9.2.
Fine lines show the u ≪ 1 and u ≫ 1 asymptotic behavior
of g(u). Top: The voltage u = 15 above the reversal value
u1. The Fermi surface acquires a pair of small pockets. Small
gaps at k = 0 are due to imperfect boundaries [20]. Bottom:
Landauer conductance G (bold integers and colors) in the
units of G0 = 2e
2/h, as the number of transverse modes at the
Fermi energy Vg. Dashed cut corresponds to the top panel.
dimensionless transverse voltage approaches u ∼ 10. A
number of effects follow:
(i) The Landauer conductance [27] is quantized in the
units of G0 = 2e
2/h, similar to that in point contacts in
GaAs [28]. The crucial difference is that now the posi-
tions and widths of the plateaus can be controlled by the
transverse voltage. The sharp steps in Figs. 2 and 3 in
real systems will be smoothened out by finite tempera-
ture or weak disorder, while the conductance values on
the plateaus will remain equal to the quantized values.
−4 −3 −2 −1 0 1 2 3 4
Wave vector  kL  along the ribbon
2 ∆(u) 
5 10 15
FIG. 3: (color online). Transverse field effect in semiconduct-
ing GRs. For voltages u > ut ≈ 4.5 above threshold the
effective mass at k = 0 is negative. Top: Bands transforma-
tion, starting from u = 0 (dotted), to just above threshold
u = 6 (solid), to above threshold, u = 10 (dashed, only lowest
subbands shown), where the gap is minimal at k 6= 0. In-
set: Gap suppression occurs above the threshold ut. Bottom:
Landauer conductance plateaus in the units of G0 = 2e
(ii) The thermopower S ∝ −∂ lnG/∂Vg, being propor-
tional to the conductance derivative [29], peaks at the
borders between the domains in Figs. 2 and 3 (bottom).
(iii) The Fermi velocity in metallic GRs is reduced by
the field, v → vg(u) [Fig. 2 inset and Eq. (9) below]. As
a consequence, the one-dimensional density of states at
the band center ν(0) = 2/{πh̄v|g(u)|} increases [factor 2
is due to spin degeneracy]. This increase magnifies the
effects of electron interactions. The latter may manifest
themselves via the increase of the Luttinger liquid expo-
nent in a sufficiently long ribbon, and through excitonic
instabilities (resulting in interaction-induced gaps).
(iv) The Fermi velocity changes sign for the field val-
ues corresponding to zeroes of g(u), causing strong van
Hove singularities in metallic GRs. The Fermi surface
fractures, with each sign change adding a pair of small
pockets to the Fermi surface (Fig. 2, top). This effect
produces extra conductance plateaus (Fig. 2, bottom).
(v) There is a threshold voltage ut ≃ 4.5 above which
the effective mass of the lowest energy subband in semi-
conducting GRs changes sign, so that the longitudinal
electron dispersion acquires symmetric minima at small
but nonzero k (Fig. 3). The excitation gap is then re-
duced by the field (Fig. 3 inset). This effect can be de-
tected in the shift of the conductance plateaus (Fig. 3,
bottom), and in the activated transport measurements.
(vi) The band structure remains electron-hole symmet-
ric at any field for both metallic and semiconducting GRs
due to the Dirac symmetry of the problem. Thus the
conductance plots of Figs. 2 and 3 are independent of
the polarity of the gate and transverse voltages.
Turning to possible applications, the setup may serve
as a field-effect transistor with a tunable working point,
in which the “transverse”, V , and the “normal”, Vg, field
effects can be utilized separately. Furthermore, one may
selectively amplify combinations α(V −V 0)+β(Vg−V 0g ),
α = ∂G/∂V |V and β = ∂G/∂Vg|V by choosing an appro-
priate working point V = (V 0, V 0g ) on the edge of the
conductance plateau. Tuning the parameters to achieve
a large gain for say, V − Vg, combined with the device’s
large input and low output impedance, is reminiscent of
an operational amplifier. By the same token, strong con-
ductance nonlinarity in both inputs V and Vg may render
this setup into a few-nm size signal multiplier, or even
into a logic gate.
We now outline the details of the calculation. At the π-
band center (ǫ = 0), the electron dispersion is determined
by the two inequivalent Dirac points in the Brillouin zone.
The low-energy states Ψ(r) = eiKxψ+(r) + e
−iKxψ−(r)
are represented [3] in terms of the smoothly varying en-
velope ψ = {ψ+, ψ−} that consists of the pair of the two-
component spinors ψ+ and ψ− with values on the two
sublattices of the honeycomb lattice [here K = −4π/3a0,
where a0 =
3acc is the graphene lattice constant, and
acc = 0.144 nm is the Carbon bond length]. The dy-
namics of the envelope is governed by the Dirac equation
Hψ = ǫψ, with the effective Hamiltonian
, H± = ±ih̄vσ1∂x − h̄vkσ2 + U(x) ,
where σ1,2 are the Pauli matrices. The boundary condi-
tions Ψ(r)|x=0,L = 0 at the armchair edges dictate [14]
ψ+(0) + ψ−(0) = 0 , ψ+(L) + e
iφnψ−(L) = 0 , (4)
where the phase φn = KL = − 2π3 (n + 1 − δ), and
L = 1
(n+1− δ)a0 is the effective ribbon width (the dis-
tance between the sites on which Ψ vanishes). The phase
φn may incorporate corrections coming from imperfect
edges, similar to the curvature-induced corrections in
nanotubes [30]. (For example, the δt/t ≈ 0.12 change
in the hopping amplitude at the edges due to the passi-
vated bonds [20] reduces the effective width L and the
boundary phase φn by the amount ∝ δ = 3
≈ 0.20.)
The system (3) and (4) is solved numerically (Figs. 2
and 3) via the transfer matrix approach similar to that
of Refs. [24, 25, 31]. Eq. (3) is equivalent to
∂xψ± = ±Pψ± , P(x) = kσ3 + iσ1(U − ǫ)/h̄v . (5)
The armchair boundary conditions (4) require tr (SS̃) =
2 cosφn for the product of the transfer matrices
S = Txe
P(x)dx , S̃ = T̃xe
P(x)dx , (6)
where Tx and T̃x symbolize the “chronological” and “anti-
chronological” orderings of the operators P(x) that do
not commute for different x.
In the absence of the field, the GR spectrum consists of
one-dimensional Dirac bands with |ǫk=0| = ∆L× π3 |n+1−
3p− δ|, p = 0,±1,±2, ... . Thus GRs with n = 3p− 1 are
metallic (with small gap ∝ δ originating from imperfect
boundaries [20]), in which case the lowest energy mode is
non-degenerate, and the rest are doubly-degenerate (the
latter degeneracy is lifted by the finite δ). The ribbons
with n = 3p and n = 3p − 2 are semiconducting, with
non-degenerate bands, and excitation gaps |ǫk=0| = ∆L×
(1∓ δ) correspondingly.
To study the transverse field effect it is convenient to
employ the chiral gauge transformation [24, 25]
ψ± = e
±iσ1ϕ(x)ψ̃± , ϕ =
U(x′)dx′/h̄v , (7)
that preserves the boundary conditions (4) and trans-
forms the system (3), H± → H̃±,
H̃± = h̄v
−kσ2e±2iσ1ϕ(x) ± iσ1∂x
. (8)
The transformation (7) shows that the spectrum at k = 0
is unaffected by the field. For the metallic GRs with
ideal edges (eiφn ≡ 1), the two degenerate k = 0, ǫ = 0
eigenstates, each consisting of a pair {ψ̃+, ψ̃−}, are
|1〉 =
and |2〉 =
Projecting the Hamiltonian (8) onto these states, H̃ →
h̄vkg(u)σ2, we find the spectrum around k = 0
ǫ = ±h̄v|kg(u)| , g =
dξ cos [uξ(1− ξ)] . (9)
The function g(u) is plotted in Fig. 2 inset. For |u| ≪ 1,
g ≃ 1 − u2/60. Its |u| ≫ 1 form g ≃
π/|u| cos[(|u| −
π)/4] determines the successive voltages un ≈ ±(3+4n)π,
n = 0, 1, 2, ..., where the k = 0 velocity changes sign. At
those voltages the dispersion ǫ ∼ k3 at k = 0, causing
the van Hove singularity ν(ǫ) ∼ |ǫ|−2/3 in the density
of states at ǫ = 0, and an additional pair of pockets
of Fermi surface emerges. In the |u| ≫ 1 limit, such
pockets appear at the zeroes of g for both metallic and
semiconducting GRs.
Electron interactions in graphene result in the RPA
screening of the external field Eext. The screening is
scale-invariant, E = Eext/κ, for an infinite sheet [32],
κ = 1 + 2πe2/4h̄v ≃ 5. The depolarization problem
in nanotubes [24–26, 33–36] also yields κ ≃ 5 practi-
cally independent of the tube radius and chirality. This
linear-screening estimate will remain valid in GRs as long
as the subbands are not strongly mixed. In the oppo-
site case the field on the ribbon edges, estimated in the
Thomas-Fermi fashion, develops an algebraic singular-
ity [37], which corresponds to filling the Fermi-surface
pockets. As a result, the uniform field model (1) is jus-
tified for weak to moderate fields. For the acting field
u = 10, the required external field uext ≃ 50 is achieved
at EextL ≃ 1V across the ribbon width L = 30 nm.
To conclude, the transverse voltage applied across a
graphene nanoribbon dramatically affects its longitudinal
electronic dispersion. The Fermi surface breaks up into
pockets for the metallic ribbons, and the excitation gap
closes for the semiconducting ones. The strong field effect
can lead to interesting physical phenomena as well as be
utilized in carbon-based electronic devices.
This work has benefited from illuminating discussions
with M. Fogler, L. Glazman and L. Levitov. The research
was sponsored by NSF grants DMR 02-37296 and DMR
04-39026.
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