Magnetoconductance of carbon nanotube p-n junctions
A. V. Andreev

arXiv:0706.0735v1 [cond-mat.mes-hall] 5 Jun 2007

Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA
(Dated: June 5, 2007)
The magnetoconductance of p-n junctions formed in clean single wall carbon nanotubes is studied
in the noninteracting electron approximation and perturbatively in electron-electron interaction, in
the geometry where a magnetic ﬁeld is along the tube axis. For long junctions the low temperature magnetoconductance is anomalously large: the relative change in the conductance becomes
of order unity even when the ﬂux through the tube is much smaller than the ﬂux quantum. The
magnetoconductance is negative for metallic tubes. For semiconducting and small gap tubes the
magnetoconductance is nonmonotonic; positive at small and negative at large ﬁelds.
PACS numbers: 75.47.Jn, 73.23.Ad, 73.63.Fg

Magnetoconductance arises from the orbital and Zeeman coupling of electrons to the external magnetic ﬁeld,
H. As long as the ﬂux through the crystalline unit
cell is much smaller than the ﬂux quantum Φ0 = hc/e
magnetotransport may be described in the semiclassical
approximation [1] ignoring the band structure changes
due to the presence of H. In most crystals this condition holds at all experimentally realizable ﬁelds. Recently much attention was focused on magnetotransport properties of carbon nanotube devices. Because
of their large radius the magnetic ﬁeld aﬀects the onedimensional electron spectrum [2] even at relatively weak
ﬁelds. This leads to interesting magnetotransport phenomena [3, 4, 5, 6, 7, 8, 9, 10].
In this paper we show that the magnetic ﬁeld dependence of the one-dimensional band structure results in a
peculiar mechanism of magnetoconductance of p-n junctions in carbon nanotubes. This mechanism is relevant to
the magnetoresistance of nominally undoped metallic and
small gap nanotubes placed on an insulating substrate.
In this case the long range disorder potential caused by
charged impurities in the substrate creates p- and n- regions in the tube, and backscattering of electrons arises
mainly from the gaps between p- and n- regions, where
the semiclassical description of electron transport fails.
We study the magnetoconductance of a p-n junction
formed in a clean single wall carbon nanotube for magnetic ﬁelds parallel to the tube axis. The device is depicted in Fig. 1 a). The p- and n- regions can be formed
by appropriately biasing the top gates. Such devices were
recently used to measure [11] thermodynamic properties
of electron liquid in carbon nanotubes. It is shown below that for realistic device parameters similar to those
of Ref. 11 the low temperature magnetoconductance becomes of order unity while the ﬂux Φ through the tube
cross-section is much smaller than Φ0 .
Before proceeding to detailed calculations let us qualitatively discuss the origin of strong magnetoconductance. We choose the x- and y- axes to be respectively along the tube axis and along its circumference.
The one-dimensional electron sub-band spectrum is de-

FIG. 1: a) Device sketch: a nanotube rests on an insulating
substrate. The n- and p- regions are created by biasing the
top gates. b) Band diagram of the device. Tilted solid lines
represent the bottom of the conduction and the top of the
valence bands. The width of the center region is 2L. The
width of the classically forbidden region is 2xo = 2∆/eE.
Inset: one dimensional spectrum is obtained by dissecting
the graphene Brillouin zone by the py = 0 line (dashed line).
The low energy states lie in the vicinity of K and K points.

termined by intersections of the graphene Brillouin zone
with the ky = (Φ/Φ0 + m)/R lines, see the inset in
Fig. 1 b). Here R is the tube radius and m an integer. The electron spectrum near the K and K points
becomes ± ∆2 + ( vpx )2 , where v ≈ 8 × 105 m/s is
the electron velocity in graphene, and ∆ is half the energy gap between the valence and conduction sub-bands.
In the center region between the p- and n- banks the
external potential is assumed to be approximately linear, U (x) = eEx, where e is the electron charge and E
the electric ﬁeld. The tilting of electron energy bands in
the external potential produces a spatial region of width
2x0 = 2∆/eE, where electron motion is classically forbidden, see Fig. 1 b). The device conductance is governed
by the Landau-Zener tunneling across this region. With
exponential accuracy the tunneling probability may be

2
found in the WKB approximation. The electron momentum px along the tube depends on the position as px =
(eEx − )2 − ∆2 /v. This gives the transmission probability T = exp 2 Im px dx = exp[−π∆2 /( veE)].
The rigorous calculation given below shows that the preexponential factor is equal to unity. The magnetoconductance arises from the ﬂux dependence of the band
gap [2], ∆ = ∆0 ± vΦ/RΦ0 , where ∆0 is half-the band
gap at zero ﬂux and the ± sign corresponds to the diﬀerent valleys, K and K . Accounting for electron spin and
the two valleys and neglecting the Zeeman splitting one
obtains for the device conductance
G0 =

2e2
h

exp −
j=1,2

π v
eER2

∆0 R (−1)j Φ
+
v
Φ0

2

.

(1)
If the electric ﬁeld is not too strong, eER
v/R,
the magnetoconductance becomes of order unity while
the ﬂux is still small, Φ
Φ0 . In the case of metallic tubes, ∆0 = 0, the magnetoconductance is negative.
For semiconducting and small gap tubes the magnetoconductance is nonmonotonic; positive at small ﬁelds and
negative at large ones. The conductance maximum is attained at Φmax ≈ Φ0 ∆0 R/ v. For semiconducting tubes,
∆0 ≈ v/3R, this gives Φmax ≈ Φ0 /3. For small gap
tubes the zero ﬂux gap arises only due to curvature effects and is rather small, ∆0
v/R. In this case the
conductance maximum is achieved at Φmax
Φ0 . A
nonmonotonic magnetoresistance was recently observed
in Ref. 8 in nanotube devices with a diﬀerent geometry.
In the noninteracting electron approximation the magnetoconductance only weakly depends on the temperature T as long as the latter is smaller than the Fermi
energy in the banks. In this regime the energies of
electrons participating in transport lie in the narrow
band of width T around the chemical potential. In this
energy range deviations of electric potential from the
linear form U (x) ≈ eEx are negligible. This results
in energy-independent transmission coeﬃcient and thus
temperature-independent conductance.
In the presence of electron-electron and electronphonon interactions electrons can be transferred between
the p- and n- regions at ﬁnite temperature by thermal
activation. At T
|∆ − ∆0 | = v Φ/RΦ0 the rate of
inelastic processes is practically independent of the magnetic ﬁeld and magnetoconductance arises mainly from
the tunneling mechanism discussed above. In this regime
inelastic transfers shunt the tunneling mechanism and
suppress magnetoconductance. A crude estimate of the
characteristic temperature T ∗ , above which the magnetoconductance suppression becomes signiﬁcant, can be obtained in the tunneling regime by equating the activation
∞
T
rate, ∼ 0 dx exp[−(∆ + eEx)/T ] = eE in exp(−∆/T ),
in
with in being the inelastic mean free path, to the tunnelπ∆2
ing rate, ∼ exp − eE v . According to this estimate the

noninteracting electron result, Eq. (1) provides a good
description of the conductance for T < eER.
At zero temperature the ﬁnite reﬂection amplitude at
the p-n contact leads to the appearance of Friedel oscillations in the electron density. The additional scattering
of electrons from the Friedel oscillations in the presence
of electron-electron interactions gives a correction [12]
to the noninteracting result for the device conductance,
Eq. (1). This correction is evaluated below to ﬁrst order
in electron-electron interaction. It is given by Eq. (15)
and is plotted in Fig. 2 b). It remains small even if the
interaction constant, e2 / v is of order unity.
The device conductance in the noninteracting electron
approximation, Eq. (1), immediately follows from the results of Cheianov and Falko [13] for a graphene p-n junction that were obtained using transfer matrices. Below
we present a consideration in terms of wave functions
that is more convenient for the treatment of the interaction correction to magnetoconductance. Electron eigenstates of energy ε obey the Dirac equation, which by an
appropriate basis choice can be cast in the form
[U (x) − ε − i v σz ∂x + ∆σy ]ψ = 0,
where σi are Pauli matrices. Introducing the dimension√
√
less coordinate ξ = eEx/ eE v, energy = ε/ eE v,
√
and momenta q = ∆/ v eE and k(ξ) = (U (ξ) −
√
ε)/ v eE we rewrite this equation as
k(ξ) − i∂ξ
−iq
iq
k(ξ) + i∂ξ

u
v

= 0.

(2)

The dimensionless momentum k(ξ) changes from −kf at
ξ → −∞ to +kf at ξ → +∞, where kf
1 is the dimensionless Fermi momentum in the banks. In the center
region between the banks the coordinate-dependent momentum k(ξ) is linear in ξ, k(ξ) ≈ ξ − , and the spinor
amplitudes u and v satisfy the diﬀerential equation,
2
(∂z + a + z 2 )f = 0.

Here a = i − q 2 for u, and a = −i − q 2 for v, and we
introduced the diﬀerence coordinate, z = ξ − . The independent solutions of this equation are parabolic cylinder functions [14] that can be expressed in terms of the
conﬂuent hypergeometric function F (α, γ, z),
z2
1 ia 1 2
F
+ , , iz ,
2
4
4 2
2
z
3 ia 3 2
fo (z) = z exp −i
F
+ , , iz .
2
4
4 2
fe (z) = exp −i

Two linearly independent solutions of Eq. (2) are
ψ1 = e−

πq 2
4

ue
−u∗
o

,

ψ2 = e−

πq 2
4

−uo
u∗
e

,

(4)

3
where

0.1

q2 1
F −i , , iz 2 ,
(5a)
4 2
1
q2 3
z2
F
− i , , iz 2 . (5b)
uo (z) = q z exp −i
2
2
4 2

0.008

z2
ue (z) = exp −i
2

0.05
0.006

-4

-2

2

4

0.004
0.002

-0.05
0.2

0.4

0.6

0.8

1

1.2

1.4

-0.1

Equation (2) conserves the current along the tube axis,
Ix = ψ † σz ψ. From the form of the current operator it
is clear that the top/bottom components of the pseudospinor ψ represent the amplitudes of the right-/left- moving waves. Thus the scattering states incident from the
left, ψL , and right, ψR , can be found by requiring that the
bottom/top component of the spinor vanish at ξ → ±∞.
Using Eqs. (4) and (5) and the large distance asymptotics
of the conﬂuent hypergeometric function [14],
Γ(γ) z α−γ
Γ(γ)
(−z)−α +
e z
, (6)
Γ(γ − α)
Γ(α)

F (α, γ, z → ∞) ≈

we obtain for the scattering states incident from the right
and left,
ψL =

uL
vL

=

ψR =

uR
vR

=

ψ1 + α ψ2

,
1 + |α|2
ψ2 − α ∗ ψ1
1 + |α|2

(7a)
.

(7b)

Here α is given by
2

α=e

−i π
4

q
1
q Γ 2 −i 4
,
2 Γ 1 − i q2
4

|α|2 = tanh

πq 2
.
4

(8)

FIG. 2: a) Dimensionless electron density as a function of ξ for
q = 0.3, q = 0.6, and q = 0.9. b) The ratio of the correction
to the transmission coeﬃcient to the interaction constant λ,
Eq. (15), as a function of q.

Next we evaluate the ﬁrst interaction correction to
Eq. (1) at zero temperature. To ﬁrst order in interaction the correction to the device conductance can be obtained by considering the change in the transmission amplitude for a particle at the Fermi level that arises from
the additional scattering from the Hartree-Fock potential induced by the electron density [12]. The induced
Hartree-Fock potential has two qualitatively diﬀerent effects on the transmission amplitude: i) By enhancing the
eﬀective electric ﬁeld inside the classically forbidden region it increases the tunneling amplitude, and ii) It causes
additional backscattering from the Friedel oscillations in
the classically allowed region. The analysis below shows
that repulsive interaction increases the transmission amplitude.
Using Eqs. (5) and (6) it is easy to show that the spinor
wave functions in Eq. (4) are normalized to a δ-function
of the dimensionless energy,
∞
−∞

The z → ±∞ asymptotics of the right-moving wave
(the top spinor component) in the scattering states ψL
and ψR are;
√
uL (z) ≈

uR (z) ≈ −

2
− πq
8

πe

2

[1 − sgn(z)|α| ] |z|
1 + |α|2 Γ

q
2

√

2
− πq
8

iπe

2
i q2

1
2

[1 + sgn(z)] |z|

1+

, (9a)

2

1+

e

2
−i z2

2
i q4

.(9b)

From Eq. (9a) one ﬁnds the transmission amplitude,
uL (ξ)
1 − |α|2
πq 2
=
= exp −
ξ→+∞ uL (−ξ)
1 + |α|2
2

t0 ≡ lim

.

This gives the transmission coeﬃcient is T0 = exp(−πq 2 ),
in agreement with Ref. 13. Expressing q in terms of
the system parameters using expressions presented in the
text above Eq.(2), and taking into account the magnetic
ﬁeld dependence of the energy gap one obtains the device
conductance, Eq. (1).

(10)

Therefore the electron density, upon subtraction of the
uniform ion background, is
∞

n(ξ) =

2
−i z2

+ i q4

2
i q2

|α|2 Γ

e

†
dξψi (ξ − )ψj (ξ − ) = 2πδ( − )δij .

i=1,2

−∞

sign(− )d †
ψi (ξ − )ψi (ξ − ).
4π

(11)

In this equation the energy is measured from the Fermi
level and the electron hole symmetry of the problem was
used. Plots of electron density, Eq. (11), for diﬀerent
values of q are presented in Fig. 2 a). The Friedel oscillations in charge density appear only at ﬁnite refection
amplitude and fall oﬀ as 1/ξ 2 . The extra power of 1/ξ
in comparison with the usual one-dimensional Fermi gas
arises from the linearly growing external potential. Because of the fast decay of the oscillations the correction
to the transmission amplitude is free from infrared diverv/eE.
gences and arises from distances x ∼
Below we consider the case of a short range interaction.
This should be a reasonable approximation because the
Fourier transform of the Coulomb interaction depends on
the transferred momentum only logarithmically. Since
the characteristic scattering momentum is
eE/v the

4
dimensionless interaction constant can be estimated as
2
2
1
λ ∼ 1+κ e v ln eER , where κ is the dielectric constant of
v
the substrate. We restrict the consideration to metallic tubes, for which electron spectra in the presence of
the ﬂux remain degenerate in the two valleys. Then the
Hartree-Fock potential is V (ξ) = 3λ n(ξ), where the factor 3 = 4 − 1 arises from the spin and valley degeneracy.
In the presence of the perturbation potential the wave
incident from the left at the Fermi level, = 0, can be
written as ψL (ξ) + χ(ξ). The correction, χ(ξ), to the
ˆ
wave function satisﬁes the equation [H0 + V (ξ)]χ(ξ) =
ˆ 0 = ξ − i∂ξ σz + qσy being the unper−V (ξ)ψL (ξ) with H
turbed Hamiltonian. To ﬁrst order in perturbation the
solution of this equation is
dξ GR (ξ, ξ )V (ξ )ψL (ξ ),

χ(ξ) =

(12)

ˆ
where GR (ξ, ξ ) = (H0 + iη)−1 is the Green’s function
that can be expressed in terms of the spinors in Eq. (4),
∞

GR (ξ, ξ ) =
i=1,2

−∞

†
d ψi (ξ − )ψi (ξ − )
.
2π
+ iη

(13)

In order to ﬁnd the correction to the transmission amplitude one needs only the large distance asymptotics of
χ(ξ). Therefore only the on-shell part of the Green’s
function will contribute to Eq. (12) at ξ → ∞,
χ(ξ) = −

i
2

ψi (ξ)

†
dξ ψi (ξ )V (ξ )ψL (ξ ).

(14)

i

Since the wave functions ψ1,2 are related to the scattering
states ψL,R by the unitary transformation Eq. (7) the
sum over i in Eq. (14) can be understood to run over
i = L, R. Next we notice that for i = L the integral
in Eq. (14) is purely real. Therefore this term will only
change the phase of the transmission amplitude, but not
its modulus. Thus to compute the ﬁrst correction to the
transmission amplitude we may replace χ(ξ) by
i
χ(ξ) = − ψR (ξ)
˜
2

†
dξ ψR (ξ )V (ξ )ψL (ξ ).

The transmission amplitude can be found from the equa˜
tion t = t0 +limξ→+∞ (1,0)·χ(ξ) . Using Eq. (9b) we obtain
uL (−ξ)
for the correction δT to the transmission coeﬃcient,
πq 2

3e− 2
δT
=−
λ
1 + |α|2

low temperature conductance of metallic devices. In the
tunneling regime, q
1, the relative correction to the
transmission amplitude is expected to be strong because
of the exponential dependence of the tunneling amplitude on the eﬀective electric ﬁeld. Therefore the method
described above becomes inapplicable.
In summary, the low temperature magnetoconductance of p-n junctions in clean single wall carbon nanotubes was studied in the geometry where the magnetic
ﬁeld is along the tube axis. For weak band-tilting ﬁeld
E
v/eR2 the magnetoconductance of long, L
R,
junctions becomes of order unity while the ﬂux through
the tube is much smaller than the ﬂux quantum. In the
noninteracting electron approximation the device conductance is given by Eq. (1). The magnetoconductance
is positive for metallic tubes and nonmonotonic for semiconducting and small gap tubes. The interaction correction to the zero temperature magnetoconductance was
studied to ﬁrst order in perturbation theory. It arises
due to the change in the eﬀective electric ﬁeld in the
gap between the p- and n- regions and due to the scattering from the Friedel oscillations. In contrast to the
one-dimensional Fermi gas, the Friedel oscillations in the
present geometry fall oﬀ as 1/x2 , which leads to the absence of infrared divergence in the correction to the tunneling amplitude. The net correction to the tunneling
probability is positive. It is given by Eq. (15) and is
plotted in Fig. 2 b). If the reﬂection coeﬃcient is not
too strong the noninteracting electron result, Eq. (1), is
rather accurate even if the coupling constant is of order
unity. At ﬁnite temperatures, in addition to tunneling
across the classically forbidden region electrons can be
transferred between p- and n- regions by being promoted
across the band gap due to inelastic electron-electron and
electron phonon scattering. The estimate of activation
transfer rate shows that tunneling processes described
by Eq. (1) dominate the transport for T < eER.
I would like to thank D. Cobden, E. Mishchenko, T. D.
Son and B. Spivak and for useful discussions.

†
dξ n(ξ )Im[α∗ ψR (ξ )ψL (ξ )].

(15)

It may be evaluated using Eqs. (7), (4), (5) and (11), and
is plotted as a function of q in Fig. 2 b). At weak reﬂection, q < 1, the correction to transmission coeﬃcient is
small even if the interaction constant is of order unity.
This is because the Friedel oscillations amplitude is proportional to the reﬂection amplitude. Thus the noninteracting result, Eq. (1), provides a good description for the

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