Scaling and interaction-assisted transport in graphene with one-dimensional defects
M. Kindermann1

arXiv:1003.2414v1 [cond-mat.mes-hall] 11 Mar 2010

1

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
(Dated: March 2010)

We analyze the scattering from one-dimensional defects in intrinsic graphene. The Coulomb
repulsion between electrons is found to be able to induce singularities of such scattering at zero
temperature as in one-dimensional conductors. In striking contrast to electrons in one space dimension, however, repulsive interactions here can enhance transport. We present explicit calculations
for the scattering from vector potentials that appear when strips of the material are under strain.
There the predicted eﬀects are exponentially large for strong scatterers.
PACS numbers: 72.80.Vp, 71.45.Gm, 72.10.Fk, 73.63.-b

The Coulomb repulsion between electrons can have
profound consequences for the scattering of electrons in
conductors. In one-dimensional (1D) conductors it dramatically suppresses the conductance through impurities,
with a singularity at temperature T = 0. This is one of
the hallmarks of the Luttinger liquid state of interacting
electrons in 1D [1]. A particularly intuitive understanding of this eﬀect is due to Matveev, Yue, and Glazman
[2, 3]. In their approach interactions cause extra electron scattering from Friedel oscillations, which inhibits
transport. Friedel oscillations are density modulations
that form when electron waves incident on an impurity
interfere with backscattered waves. In 1D at T = 0 the
Friedel oscillation around an impurity is not integrable
over space and the induced scattering amplitude diverges
logarithmically in Born approximation.
Friedel oscillations in two dimensions (2D) at nonzero
chemical potential can be integrated and have much
weaker eﬀects for both, point-like [4–6] and 1D defects [7, 8]. Interactions play a more important role
in graphene [9–11] at zero chemical potential (“intrinsic graphene”), which forms a so-called marginal Fermi
liquid [12]. There is, for instance, experimental evidence
[13] of a singular renormalization of the Fermi velocity
[12, 14]. Nevertheless, also in intrinsic graphene scattering from point-like impurities of both, scalar and vector
potential character, does not receive any singular corrections from the Coulomb interaction [15, 16] [23].
In this Letter we show that the Coulomb interaction
in intrinsic graphene can have a singular impact on scatterers that are extended in one space dimension. A qualitative analysis of the Friedel oscillations at such scatterers reveals the reason: the Friedel oscillation due to
an electron with wavevector k ′ scattering with amplitude r from a 1D defect ﬁlls a strip of width ∼ 1/k ′ .
The corresponding exchange potential has support in the
same strip, implying a factor ∼ r/k ′ in the induced Born
scattering amplitude δrex . A second factor 1/k ′ comes
from the Coulomb interaction. For energy-independent r
the exchange interaction with electrons in all ﬁlled states
thus results in δrex ∝ r ﬁlled d2 k ′ /k ′2 . Since in intrinsic
graphene at T = 0 the entire valence band is ﬁlled, δrex

is logarithmically divergent. This suggests that when the
Coulomb interaction does not cause a Hartree potential it
can have the same drastic eﬀects for 1D scatterers in 2D
intrinsic graphene as for point defects in 1D conductors.
In contrast to the familiar situation in 1D [1], however, we demonstrate that the Coulomb interaction in
graphene can enhance electron transport. This intriguing
eﬀect is due to bound states that straddle defects. The
exchange interaction of transport electrons with electrons
in such bound states opens an additional transport path
across the defect, which increases the electric current.

FIG. 1: A vector potential Ay in a narrow strip (dark blue)
forms a scattering barrier in a sheet of otherwise ballistic
graphene (light blue with contacts in green). Ay may be induced by strain or, alternatively, by a pair of current-carrying
wires (brown).

We exemplify the above eﬀects with the scattering barrier shown in Fig. 1, formed by a vector potential in a
narrow strip of an otherwise ballistic sheet of intrinsic
graphene. Such vector potentials are induced by strain
[17, 18] or by nearby electric currents, see Fig. 1. By
symmetry vector potentials in graphene do not produce a
Hartree potential. As suggested by the above argument,
the Coulomb interaction therefore induces extra scattering whose amplitude diverges at low T , when the electron wavelength is much longer than the width a of the
barrier and scattering is energy-independent. The eﬀect
turns out to be exponentially enhanced for strong barriers, where it strongly impacts transport even at moderate T . In addition, the scatterer of Fig. 1 hosts bound

2
states that enhance transport through electron exchange
as motivated above. As a consequence, the barrier of
Fig. 1 becomes entirely transparent at T = 0, in striking
contrast to impurities in interacting 1D conductors that
completely block the electric current at T = 0.
Model: In the setup of Fig. 1 a vector potential
A = (0, Ay ) ; Ay =

a
a
χ
˜
Θ x+
−Θ x−
ea
2
2

(1)

[Θ(x) is the step function] is applied to a sheet of clean,
intrinsic graphene that we ﬁrst assume to be inﬁnite, L →
∞. The noninteracting Hamiltonian in valley γ thus is
Hγ = vσγ · (p − eA) .

(2)

Here, σ γ = (σx , γσy ) is a vector of Pauli matrices, p is
the electron momentum, v the electron velocity, and −e
the electron charge. The above model has parity symmetry, PHP = H, where Pψ(x, y) = σy ψ(−x, y), and
particle-hole (PH) symmetry, σz Hσz = −H. In intrinsic
graphene both symmetries are also respected by electronelectron interactions [24]. Without restriction we assume
χ > 0. Transport through the barrier at χ < 0 is ob˜
˜
tained through reﬂection on the x-axis. The above model
has been analyzed in Ref. [17] on a noninteracting level,
where A suppresses linear transport exponentially in χ if
˜
kT ≪ χv/a (we set h = 1). Below we study the eﬀects of
˜
¯
the Coulomb repulsion between electrons on this transport problem for short barriers (Λa ≫ 1, but ln Λa > 1,
∼
where Λ is the bandwidth of graphene).
We assume that the graphene sheet is suspended or
lying on an insulating substrate, leaving the Coulomb interaction unscreened with interaction potential VC (q) =
2π˜s v/|q| at interaction parameter rs and wavevector q
r
˜

F (χ) =

χ = χ − rs sinh χF (χ) ln
˜ ˜
˜ ˜

v
− χ + O(1)
˜
kT a

4e2 kT W ln 2
χ
1 + sinh χ tanh χ ln tanh
. (5)
h
πv
2

In G, the logarithmic temperature dependence of the interaction correction to χ competes with the factor T due
to the linear density of states. The predicted non-analytic
scale-dependence of χ is thus more easily observed in the
normalized conductance Gr = G/(G|χ=0 ) [27].
˜
Due to its logarithmic T -dependence the ﬁrst order
correction to χ may grow large even at rs ≪ 1. At the
˜
˜
corresponding low temperatures, when rs ln(v/kT a) > 1,
˜
∼

(3)

with the positive function

coth χ(cosh 3χ − 3 cosh χ) arcsin(tanh χ) + 2 cosh 2χ
− cosh χ sinh2 χ.
2π

The ﬁrst order result Eq. (3) manifests itself in
a non-analytic temperature-dependence of the conductance through the barrier, which takes the form
G=

[25]. In the weakly interacting limit rs ≪ 1 that we take
˜
henceforth the main eﬀect of electron-electron interactions is additional elastic scattering, as in the thoroughly
studied 1D problem [2, 3, 19]. Also like in 1D, such scattering is induced by the Hartree-Fock potential V HF due
to Friedel oscillations at the barrier. In addition, A here
induces bound states with an extra contribution to V HF .
The potential V HF is strongly constrained by the symmetries of the setup. Most importantly, a Hartree potential is precluded by PH-symmetry. Also, the exchange potential V ex is spin- and valley-diagonal. Transport in both valleys is identical since they are related
†
as H−1 = Uv H1 Uv , where Uv = σx commutes with the
current σx through the barrier [26]. We thus set γ = 1.
The compound scatterer composed of A and V ex is
conveniently described by the transfer matrix of electrons at zero energy M = T (∞, −∞)|ε=0 [6]. The Pand PH-symmetries of our model imply σy M σy = M −1
and σz M σz = M , respectively. Together with current
conservation σx M † σx = M −1 they constrain M to take
the form M = exp(χσz ). Low energy transport in the
setup of Fig. 1 with weak many-body interactions is thus
characterized by a single parameter χ. This reduces our
analysis to a calculation of χ. At rs = 0 we have χ = χ.
˜
˜
First order in rs : To begin with, we calculate the in˜
teraction correction to χ at ﬁrst order in rs . Computing
˜
˜
the correction to M due to scattering from the exchange
potential created by the noninteracting electron states in
Born approximation we ﬁnd in the limit a ≪ v/kT

(4)

the above perturbative calculation looses validity, as in
the 1D case [2, 3]. The origin of the logarithmic divergence here is, however, diﬀerent. In 1D the divergence
stems from a Friedel oscillation density ∝ 1/x at T = 0.
In the present 2D case the T = 0 Friedel oscillation density is ∝ 1/x2 and integrable. But the non-locality of V ex
gives rise to the same logarithmic divergencies as in 1D.
RG-analysis: We extend our analysis into the regime
rs ≪ 1, but rs ln(v/kT a) > 1, by a renormalization group
˜
˜
∼
(RG) calculation that re-sums the perturbation series in
the “leading logarithm” approximation [2, 3]. To this end
we successively integrate out shells of wavevectors [k ′ , bk ′ ]
with b > 1 in the order of decreasing k ′ ≪ 1/a. For
each shell we compute the exchange potential Vkex due to
′ ,b

3
the corresponding states. Thereafter we renormalize the
transfer matrix for low energy electrons at wavevector k
by Vkex . The resulting χ characterizes the wavefunctions
′ ,b
in the subsequent shell [k, bk]. The logarithmic scaledependence of δχ, Eq. (3), allows us to assume k ≪ k ′ in
this process. Introducing l = − ln k ′ a we thus ﬁnd that

takes the form Gr = 8 exp(−2χ)/3. In analogy with
the Kondo problem we deﬁne the temperature scale T ∗
where the transmission across the barrier becomes of order unity, Gr ≃ 1. At rs ≪ 3π exp(−χ) we ﬁnd
˜
˜

dχ
= −rs sinh χF (χ),
dl

At wavevectors k ′ < κb the bound states do not renormalize χ anymore, but the propagating states continue to
do so and dχ/dy = 3/2 for κb ≫ k ′ ≫ kT /v. This now
decreases the conductance, by a factor (1+˜s χ|k′ =κb /4)−3
r
if χ|k′ =κb ≫ 1. In Fig. 2 we plot the normalized resistance G−1 for various parameter values. Some of them
r
are within the regime rs ≪ 3π exp(−χ), where Eq. (8) is
˜
˜
˜
rigorously justiﬁed, others at rs < 3π exp(−χ), where Eq.
˜ ∼
(8) may still be qualitatively correct. At rs > 3π exp(−χ)
˜
˜
Eq. (8) does not have a unique solution anymore. Clearly,
the predicted eﬀect is large even for moderate χ > 1 and
˜∼
ln κb a > 1. Its observation therefore does not require
∼
access to large temperature intervals.

(6)

where χ = χ at l = 0. Here, rs is a scale-dependent
˜
interaction parameter that is renormalized as drs /dl =
2
−rs /4 [12] with the solution rs = rs /(1 + rs l/4). Eq.
˜
˜
(6) with this l-dependent rs is solved best by introducing
y = ln(1 + rs l/4), such that dχ/dy = −4 sinh χF (χ).
˜
We start the detailed analysis of Eq. (6) assuming χ ≫
˜
1. As long as χ ≫ 1 we use limχ→∞ F (χ) = 2/3π to ﬁnd
4
dχ
= − eχ .
dy
3π

(7)

Eqs. (6) and (7) make two remarkable predictions:
First, the minus signs in both equations imply that the
Coulomb interaction reduces the barrier strength. It thus
enhances transport through the studied scatterer, in evident contrast with the familiar situation in 1D [1–3].
Second, that enhancement is exponentially large in χ
such that for suﬃciently large χ the Coulomb interac˜
tion strongly impacts electron transport even at rs l ≪ 1.
Both predictions have their origin in bound states. We
ﬁnd that the interaction with the Friedel oscillations due
to propagating states decreases the conductance as in
1D. That suppression, however, is overcompensated by
bound states which form inside the scattering barrier at
|x| < a/2, but leak into the region |x| > a/2. Electrons
in bound states have the same probability for being to
the left of the barrier as to the right, guaranteed by Psymmetry [28]. The bound states therefore straddle the
barrier and the exchange interaction with electrons in
those bound states opens an additional path for electrons
to traverse the barrier. This explains the minus signs in
Eqs. (6) and (7). The resulting enhancement of the barrier transmission t is of order rs l and can easily exceed the
exponentially small noninteracting t ∝ exp(−χ), which
˜
implies the exponential renormalization of χ, Eq. (7).
This exponential renormalization limits our RG analysis to rs ≪ 3π exp(−χ). Also, the bound state ener˜
˜
gies εb = ±vky sechχ are exponentially small at χ ≫ 1.
The exponential renormalization of χ is thus cut-oﬀ at
κb = kT cosh χ|k′ =κb /v and it requires exponentially low
temperatures kT ≪ exp(−χ)v/a. Eq. (7) is solved by
˜
˜
χ = − ln e−χ +

4
y ,
3π

(8)

where y is evaluated at κb , y = ln[1 − (˜s /4) ln κb a]. The
r
self-consistent solution of Eq. (8) with this χ-dependent y
determines the normalized conductance, which for χ ≫ 1

kT ∗ =

v −3π/˜s
r
.
e
a

(9)

Gr 1
rs

Π

Χ

3Π

rs

10 000

Χ

Χ

Π
rs
2

5000

1

2

l

3

FIG. 2: Normalized resistance G−1 at χ = 5 and kT <
˜
r
v sech χ/a, when it is renormalized by bound states, as a func˜
tion of ∆l = ln (v sech χ/kT a).
˜

As χ decreases the scaling behavior crosses over into a
second regime, where χ ≪ 1 and F (χ) may be expanded
around χ = 0. In that case we ﬁnd dχ/dy = −4(10/3π −
1)χ3 [29] such that, now assuming for simplicity χ ≪ 1,
˜
χ=

χ
˜
1+

8χ2 (10/3π
˜

− 1)y

.

(10)

We conclude that the backscattering from A disappears
asymptotically at low T , very diﬀerently from static scattering potentials v in 1D. The coupling χ, however, is
only marginally irrelevant here (dχ/dl ∝ χ3 [30]). It is
renormalized more slowly than v in 1D, which is relevant
in the RG sense (dv/dl ∝ v) [1]. The renormalization of
rs in graphene further slows down the RG ﬂow of χ.
Discussion: In reality the unscreened and ballistic segments of graphene around the barrier in Fig. 1 have a ﬁnite length L (typically limited by contacts or by the elastic mean free path). Our predictions then require that the

4
entire Friedel oscillation ﬁts into those segments, implying an additional cut-oﬀ κL = cothχ/L for the RG-ﬂow
˜
and necessitating L ≫ a. Due to the strong renormalization at χ ≫ 1, however, the observation of the eﬀects
shown in Fig. 2 does not require excessively long samples.
While inelastic processes are negligible in the above
analysis of the renormalization of χ at rs ≪ 1 [19],
they can signiﬁcantly aﬀect the transport between barrier and contacts [20]. There inelastic scattering can be
neglected only if the barrier is more resistive than the
2
adjacent ballistic regions, such that (kT Gr )−1 ≫ rs L/v.
L
The cut-oﬀ κ further limits the regime where our predictions can be observed directly in the conductance to
2
2
rs Gr ≪ v/kT L ≪ tanh χ. At rs Gr > v/kT L the obser˜
∼
−1
vation of Gr requires experimental diﬀerentiation from
the inelastic background resistance, e.g. by varying χ.
˜
The vector potential A is induced by strain in the xdirection if that is along the “armchair” direction of the
graphene lattice [17]. Strain, however, also generates a
scalar, ”deformation” potential V < χv/a [17]. This
∼ ˜
breaks PH-symmetry, a Hartree-potential appears, and
our predictions require V ≪ max{v coth χ/L, kT }/rs.
˜
Also the bound states are strongly aﬀected by V and
consequently Eq. (7) holds only if V ≪ χ exp(−χ)v/a, a
˜
˜
tight constraint when χ ≫ 1. We remark that Eq. (8)
˜
(strictly valid at χ ≫ 1, but possibly qualitatively cor˜
rect also at χ ≃ 1) predicts strong eﬀects like those of
˜
Fig. 2 down to χ ≃ 1, when this extra constraint dis˜
appears. To avoid a nonzero V , A can be generated
alternatively by a pair of wires at x = 0, but diﬀerent
distances d1 , d2 ≃ a from the graphene plane (see Fig.
1). Opposite currents I and −I through such wires create a vector potential that falls oﬀ as 1/x2 at x ≫ a and
that has integral χ = dx Ay (x) = µ0 I(d1 − d2 ). In our
˜
limit rs ≪ 1, but l ≫ 1 the deviations of this potential
from the square-well shape Eq. (1) have negligible eﬀects.
Conclusions: We have shown that the Coulomb repulsion between electrons can have profound and unconventional eﬀects on transport in intrinsic graphene. General
considerations predict singular transport across energyindependent 1D defects without Hartree potential. In addition we have shown that interactions in graphene can
enhance transport through static scatterers, in surprising
contrast with the familiar result for 1D conductors. We
have presented detailed calculations for vector potential
scatterers that are induced by strain. There the above
eﬀects have exponentially large experimental signatures.
The author thanks Yu. V. Nazarov for discussion and
the KITP at UCSB for hospitality, where this work was
supported in part by the NSF, Grant No. PHY05-51164.
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[24] In the case of PH symmetry this is seen easiest in second
quantized form, when H is invariant under the transformation c → σz c† , where c is the electron annihilation
operator. PH-symmetry requires vanishing chemical potential and interaction with the ion charge density.
[25] Screening by the electrons in the graphene sheet itself is
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[26] In case A is due to strain, χ has opposite sign in the two
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˜
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electric currents that can easily be switched on and oﬀ.
[28] The bound states are non-degenerate (apart from the
spin and valley degeneracies).
[29] This equation has corrections of order χk′ a. It is strictly
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˜
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˜
[30] The absence of a term ∝ χ follows also from Refs. [15, 16].