Resonant tunneling of interacting electrons in a one-dimensional wire
Yu.V Nazarov1 and L.I. Glazman2

arXiv:cond-mat/0209090v2 [cond-mat.mes-hall] 10 Feb 2003

1

Department of Nanoscience, Delft University of Technology, 2628 CJ Delft, the Nethrelands
2
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455

We consider the conductance of a one-dimensional wire interrupted by a double-barrier structure
allowing for a resonant level. Using the electron-electron interaction strength as a small parameter,
we are able to build a non-perturbative analytical theory of the conductance valid in a broad region
of temperatures and for a variety of the barrier parameters. We find that the conductance may
have a non-monotonic crossover dependence on temperature, specific for a resonant tunneling in an
interacting electron system.
PACS numbers: 73.63.-b, 73.23.Hk, 73.21.Hb

and the conventional behavior of resonant tunneling
prescribed by Breit-Wigner formula may be restored.
Consequently, the conductance G(T ) may increase with
the temperature being lowered. How to match these
two opposite tendencies? We answer this question below
by finding the proper non-monotonic crossover function
G(T ) for an arbitrary asymmetry of the barrier and arbitrary position of the resonant level with respect to the
Fermi level, in the limit of weak interaction. The asymptotes of G(T ) agree with those found in Ref. [7,8] in the
context of the Luttinger liquid theory. The universality
of latter results was recently doubted in the theoretical
part of [6]. We find no ground for such doubts.
We will use an analogue of the renormalization method
developed in [9]. Within this method, the complicated
picture of many-electron transport is considered within
the traditional Landauer-B¨ ttiker elastic scattering foru
malism. The role of the interaction is to renormalize
the elastic scattering amplitudes. The renormalization
brings about an extra energy dependence of these amplitudes. It was shown in [9] that in the limit of weak interaction the most divergent terms in perturbation theory
indeed correspond to the purely elastic processes, thus
justifying the method. The advantage of the method
is that it allows one to investigate quantitatively the
crossover between the limits of weak tunneling and full
transmission across the barrier.
The original formulation of the method [9] disregarded
the energy dependence of scattering amplitudes in the
absence of interaction. While valid for a generic case of
a single scatterer, this assumption obviously fails to describe the resonant tunneling. To circumvent this, we extend the method to arbitrary energy dependence of scattering amplitudes. First step is to derive the first-order
interaction correction to scattering amplitudes. This can
be readily done along the lines of Ref. [9]. The correction
to transmission amplitude reads

The phenomenon of resonant tunneling is well-known
in the context of electron transport physics [1]. The hybridization of a discrete state localized in the barrier with
the extended states outside the barrier may strongly enhance the transmission coefficient for electrons incident
on the barrier with energy matching the energy of the
localized state. For a single electron, the transmission
coefficient at energies close to the resonance is given by
the Breit-Wigner formula [1]. However, if the barrier
carrying the resonant level separates conductors which
in equilibrium have a finite density of mobile electrons,
the problem of resonant tunneling becomes more complex
due to the electron-electron interaction. Manifestation of
resonant tunneling in the conductance of a solid-state device is inevitably sensitive to this interaction.
Some of the effects of electron-electron interaction do
not depend on the dimensionality d of the conductors–
leads separated by the barrier. For instance, the on-site
repulsion together with the hybridization of the localized
state with the states of continua lead to the Kondo effect
in the transmission across the barrier [1,2] at any d. The
Fermi-edge singularity also strongly affects the resonant
tunneling [3,4] in any dimension. The electron-electron
interaction within the leads, however, does not have a
strong effect if d > 1, and if the leads are not disordered.
In contrast, resonant tunneling across a barrier interrupting a one-dimensional (1d) wire is modified drastically by
the interaction within the wire. The importance of such
a setting is emphasized by the recent transport experiments with nanotubes and nanowires containing a quantum dot [5,6].
The electron-electron interaction enhances the
backscattering off the barrier for electrons with energy
close to the Fermi level [8]. We find that if the discrete level is not perfectly aligned with the Fermi level
in the leads, or the barrier structure has even slight
geometrical asymmetry, then the low-temperature linear conductance decreases to zero with the temperature,
G(T → 0) → 0. At sufficiently high electron energies
(measured from the Fermi level) the enhancement of
backscattering due to the interaction should get weaker,

δt(ǫ) =

t(ǫ)
2

0
−∞

dǫ′
∗
∗
[αL rL (ǫ)rL (ǫ′ ) + αR rR (ǫ′ )rR (ǫ)].
−ǫ

ǫ′

(1)
1

To describe resonant tunneling, we consider a compound scatterer made of two tunnel barriers with tunnel amplitudes t1,2 ≪ 1 separated by a distance πvF /δ.
This gives rise to a system of equidistant transmission
resonances separated by energy δ. We assume that one
of the resonances is anomalously close to Fermi energy
and concentrate on this one disregarding the others. The
scattering amplitudes in the absence of interaction are
then given by common Breit-Wigner relations:

Here the rL(R) are the reflection amplitudes for electrons
incoming from the left (right), and the coefficients αL(R)
represent the interaction within the left(right) part of
the 1d wire; energies ǫ and ǫ′ are measured from the
Fermi level. Transmission and reflection amplitudes rL,R
∗
satisfy the unitarity relation: rR t∗ = −rL t. The coefficients α are related to the Fourier components V (k) of
the corresponding electron-electron interaction potential
by α = (V (0) − V (2kF ))/2πvF ). In the limit of weak
interaction, these coefficients determine the exponents in
the edge density of states [8] for each part of the channel,
ν(ǫ) ∝ ǫα .
The integration over ǫ′ in the first-order correction Eq. (1), in general, yields a logarithmic divergence
at ǫ → 0. This indicates that the perturbation series
in the interaction potential can be re-summed with the
renormalization method. To account for the most divergent term in each order of the perturbation theory in α,
we proceed with the renormalization in a usual way [10].
On each step of the renormalization, we concentrate on
the electron states in a narrow energy strip around −E,
with E > 0 being the running cut-off. We evaluate the
interaction correction due to the electrons in these states
to the scattering amplitudes at energies ǫ close to Fermi
level, |ǫ| < E. These amplitudes are thus functions of
both ǫ and E. We correct those amplitudes according
to Eq. 1, reduce the running cut-off by the width of the
energy strip, and repeat the procedure. This yields the
following renormalization equation:
t(ǫ, E)
∂t(ǫ, E)
∗
=
[αL rL (ǫ, E)rL (−E)+
∂ ln E
2
∗
+ αR rR (−E)rR (ǫ, E)] ,

√
i ΓL ΓR
t(ǫ, Λ) =
,
(ΓL + ΓR )/2 − i(ǫ − ∆)
(−ΓL + ΓR )/2 − i(ǫ − ∆)
,
rL (ǫ, Λ) =
(ΓL + ΓR )/2 − i(ǫ − ∆)
where ΓL,R = |t1,2 |2 δ/2π are the level widths with respect to the electron decay into the left(right) lead and
∆ is the energy shift of the resonance with respect to the
Fermi Level; we assume here ∆ ≪ δ. We disregard possible energy dependence of t1,2 that could be relevant at
higher energies, which allows us to take the upper cut-off
Λ to be of the order of δ. The corresponding transmission
probability before the renormalizations,
|t(ǫ, Λ)|2 =

is the usual Lorentzian function of energy. The interaction corrections to ∆ and ΓL,R which come from bigger
energy scales, δ < E < EF , are assumed to be included
in the definitions of these quantities.
The next step is to solve the renormalization equations (2) and (4). To stay within the accuracy of the
method, in the solution we need to retain the terms
∝ αn [ln(Λ/ǫ)]n while same-order terms with a lower exponent of the logarithmic factor should be disregarded.
This allows for a substantial simplification. We proceed
by solving Eqs. (2) and (4) at higher energy (far from
the resonance), where the reflection from the compound
scatterer is almost perfect. In this case, we approximate
|rL,R (−E)| ≈ 1. It is possible to see that in this case the
renormalization of the tunnel amplitudes t1,2 of each constituent of our compound scatterer occurs separate from
each other, d ln t1,2 /d ln ǫ = αL,R /2. This renormalization can be incorporated into the energy dependence of
the effective level widths, ΓR,L (ǫ) = ΓR,L (ǫ/Λ)αR,L . The
result for |t(ǫ)|2 thus reads

(2)

provided that |ǫ| < E. We abbreviate here r(ǫ) ≡ r(ǫ, |ǫ|)
(and similar for t) indicating that the renormalization
of scattering amplitudes stops when the running cut-off
approaches |ǫ|. The initial conditions for this differential equation are set at upper cut-off energy Λ. If the
ǫ–dependence of the transmission amplitude in the absence of interaction, t(ǫ, Λ), can be disregarded, then all
the energy dependence of renormalized amplitudes comes
about as a result of the renormalization procedure. The
corresponding simplification of Eq. (2) then reads
∂|t(ǫ)|2
= (αR + αL )|t(ǫ)|2 (1 − |t(ǫ)|2 ),
∂ ln ǫ

(3)

and contains the transmission probabilities only. This
coincides with the results of Ref. [9]. However, the above
simplification is not possible in the more general case
we consider here. One can not even deal with a single
equation: the equation (2) shall be supplemented with a
similar equation for one of the reflection amplitudes,
1
∂rL (ǫ, E)
2
∗
= {αL [−rL (−E) + rL (ǫ, E)rL (−E)]
∂ ln E
2
∗
+αR rR (−E)t2 (ǫ, E)}.

ΓL ΓR
,
(ǫ − ∆)2 + (ΓL + ΓR )2 /4

|t(ǫ)|2 =

ΓL (ǫ)ΓR (ǫ)
.
(ǫ − ∆)2 + [ΓL (ǫ) + ΓR (ǫ)]2 /4

(5)

The above approximation of the integrand in Eq. (2)
becomes invalid at lower energies, where the transmission coefficient may become of the order of unity. The
energy scale ǫ at which this occurs can be evaluated from
˜

(4)

2

Eq. (5), and is given by the solution of equation 2˜ =
ǫ
ΓL (˜) + ΓR (˜). In the simplest case of αL = αR ≡ α ≪ 1,
ǫ
ǫ
it is 2˜ = (ΓL + ΓR )((ΓL + ΓR )/2Λ)α . At energies below
ǫ
ǫ, the reflection amplitudes in the integrand can be ap˜
proximated as r(ǫ′ ) ≈ r(ǫ). Under this assumption, we
immediately recover Eq.(3). At |ǫ| < ǫ, its solution yields
˜
|t(ǫ)|2 =

one finds G(T ) ∝ 1/T at temperatures T ≫ Γ, ∆; in the
limit T → 0, the conductance saturates at a finite value,
which reaches (1−β 2 )GQ if the Fermi level is tuned to the
resonance (∆ = 0). Interaction changes this picture noticeably. Let us start the discussion with the case ∆ = 0.
At high temperatures, T > ˜, the conductance can be
∼ ǫ
estimated as

˜
˜
ΓL (ǫ)ΓR (ǫ)
˜
˜
(ǫ − ∆)2 + ΓL (ǫ)ΓR (ǫ) + [ΓL (˜) − ΓR (˜)]2 /4
ǫ
ǫ

α−1

G∆=0 (T )/GQ = [π(1 − β 2 )/4] (T /˜)
ǫ

The temperature dependence can be thus ascribed to
interaction-induced power-law temperature dependence
of Γ. Furthermore, the low-temperature behavior differs strikingly for symmetric (β = 0) and asymmetric
(β = 0) resonance. For symmetric resonance, the conductance saturates at the ideal value of GQ . For β = 0
the conductance reaches at T ≈ ǫ its maximum value,
˜
which is smaller than (1 − β 2 )GQ , and drops to zero with
the further decrease of temperature,

(6)

with
˜
ΓL,R (ǫ) = ΓL,R (˜) (|ǫ|/˜)
ǫ
ǫ

αR +αL
2

.

(7)

Relations (6) and (7) determine the full crossover function for the resonant tunneling between the interacting
1d electron systems, if ǫ > |∆|.
˜∼
In the opposite case of a resonance distant from the
Fermi level, |∆| > ˜, we shall change the approximation
∼ǫ
at ǫ = |∆|. The answer is thus given by the equations
(6), (7) with ˜ being replaced by |∆|.
ǫ
A typical energy dependence of the transmission probability is sketched in the insert of Fig. 1. It combines an
overall Lorentz-like shape with a sharp dip at the Fermi
level. Note, that the transmission is not suppressed at
low energies for a perfectly symmetric barrier with the
resonant level tuned to coincide with the Fermi level, in
full agreement with Ref. [7].
It may seem that the numerical factors in the definition of the crossover energy ˜ and in the condition |∆| = ˜
ǫ
ǫ
of the crossover between low energy cut-offs are chosen
in a rather arbitrary fashion. Indeed, these two definitions could contain any other numerical factors of the
order of 1. The point is that fixing the numerical factors
with a greater precision would exceed the accuracy of our
renormalization method. In other words, the energy dependence of Γ in all above relations is assumed to be very
slow. It is this slow dependence that, in the limit α ≪ 1,
gives us the luxury of arbitrary choice of those numerical
factors.
To present quantitative conclusions, we discuss the linear conductance G(T ) in the case of αR = αL ≡ α.
Within the Landauer formalism, the conductance is given
by
∞

G(T ) = GQ
−∞

dǫ
|t(ǫ)|2 ,
4T cosh2 (ǫ/2T )

≃ Γ(T )/T. (9)

G∆=0 (T )/GQ = 1/β 2 − 1 (T /˜)
ǫ

2α

T < ˜.
∼ǫ

,

(10)

The temperature exponents at T < ǫ in both cases agree
∼˜
with those obtained in Refs. [7,8]. The exponent at β = 0
is the same as for a single tunnel barrier interrupting the
1d channel. It indicates that at low energies the electrons
get over the compound scatterer in a single quantum
transition. The high-temperature exponent arises from
the separate renormalization of each barrier, which signals the “sequential mechanism” of tunneling: the electron tunnels across one barrier first and waits a while
(≃ ¯ /Γ(ǫ)) before tunneling across another one. It is imh
portant to recognize though that no energy relaxation or
decoherence takes place during this waiting time. This
is especially clear from our calculation based on the Landauer formula: there are no inelastic processes included
which could provide for the relaxation or decoherence.
The increase of ∆ leads to a decrease of the conductance. For non-interacting electrons, the conductance
stays at a level of the order of its maximal value, G∆=0
for ∆ less than ΓL + ΓR , which determines the width of
the resonance in G(∆) at T < ΓL + ΓR . At higher tem∼
peratures, the effective resonance width is w ≃ T . Let
us discuss now the temperature dependence w(T ) and
the shape of the resonance G(∆) at fixed temperature
in the presence of interaction. For T ≫ ǫ, the width
˜
w ≃ T does not reveal any anomalous exponent. The
shape of the resonance in this regime is mainly determined by the thermal-activated exponential contribution
G(∆) ≃ exp(−|∆|/T )Γ(T )/T in Eq. (8). However, at
large ∆ ≫ w the power-law ”cotunneling” tail

(8)

where the conductance quantum unit for one fermion
mode is GQ = e2 /2π¯ . The results strongly depend on
h
the ratio of ΓR and ΓL . We will characterize this ratio
by the asymmetry parameter β ≡ |ΓL − ΓR |/(ΓR + ΓL )
which ranges from 0 to 1 and does not depend on energy,
provided that αR = αL . To emphasize the effect of interaction, let us recall that in the case of free electrons

2α 2

Gtail (∆) = GQ (1 − β 2 ) (T /˜)
ǫ

ǫ /∆2 ,
˜

replaces that exponential dependence [11]. The crossover
occurs at ∆ ≃ T ln(GQ /G∆=0 ) and corresponds to the
conductance Gcross ≃ G2 /GQ , this being much smaller
∆=0
than G∆=0 .
3

tures, respectively. The two high-temperature curves
(the smallest values of G∆=0 ) are hardly distinguishable from each other, and correspond to the resonance
width w ≃ T . Both medium-temperature curves show
a more narrow resonant peak with increased conductivity G∆=0 , and are still similar to each other, apart from
the scale. The real difference becomes visible for the
low-temperature curves. In the case of non-symmetric
resonance, the low-temperature curve is just reduced in
height with no noticeable change of the shape. This is
in contrast to the symmetric resonance, where the resonance peak gets taller and thinner. The symmetric resonance does not seem to be very realistic since any randomness in the barriers and/or in nanowire would cause
asymmetry. Provided symmetry is achieved, the symmetric resonance can be easily identified by its ideal conductance GQ .
To conclude, we have investigated the transmission
resonances of interacting electrons in 1d wires. For a
weak electron-electron interaction the transmission can
be considered as an elastic process, which allowed us to
build a comprehensive theory of the resonances, valid in
a broad range of temperature and parameters of the resonant level. The temperature dependence of the maximum conductance in general is not monotonic, and reveals important differences between symmetric and nonsymmetric resonances. The obtained quantitative results
present a comprehensive and consistent picture of the effect. It assures us in the qualitative validity of the picture at an arbitrary interaction strength. Although we
are not able to come up with an explicit expression for
the crossover function G(T ) in this case, such function,
with known high- and low-temperature asymptotes, does
exist by virtue of the renormalizability.
We acknowledge a useful communication with D.
Polyakov and I. Gornyi [12] which enabled us to present
Eqs. (2) and (4) in a correct and comprehensible way.
One of the authors appreciates many stimulating discussions with M. Grifoni and M. Thorwat. This work has
been initiated in the framework of 2001 Aspen Summer
Program and was partially done during a workshop at Institute of Theoretical Physics, UCSB. The hospitality of
both organizations is gladly acknowledged. This research
was sponsored by the NSF Grants DMR 97-31756, DMR
02-37296, EIA 02-10736, and the grants of FOM.

At T ≪ ǫ the apparent width of the non-symmetric
˜
resonance saturates at w ≃ ˜. The conductance thus
ǫ
drops uniformly at any ∆ following the power law (10).
The symmetric resonance presents an exception. In this
case, the width shrinks with the decreasing temperature,
α

w(T ) ≃ (T /˜) ˜,
ǫ ǫ
and G(T, ∆) acquires the scaling form, G(T, ∆) =
GQ /{1 + [∆/w(T )]2 }, in agreement with Ref. [7].
We further illustrate our results by a numerical evaluation of Eq. (8), see Figs 1–2. For this calculation, we
choose α = 0.2. By virtue of our approach, the relative
accuracy of the results is expected to be of the order of α.
The dependence G(T ) is not monotonic, and in the limit
T → 0 the conductance drops to zero at any β = 0, although for small β this is noticeable only at very low temperatures ( Fig. 1). The temperature dependence w(T )
of the width of the resonance G(∆) is shown in the left
panel of Fig. 2. If β = 0, this dependence saturates at
some value w(0) = 0.
1

10
10
10
10

-1

-2

1
-3

2

G/GQ

10

|t|

10

-4

-5

0
-2

-6

10

-5

-4

-3

~
ε/ε

0

10 10 10

-2

10

-1

4
1

2

10 10 10

3

~
Τ/ε

FIG. 1. Temperature dependence of resonant (∆ = 0) tunneling conductance. The asymmetry parameter β = 0 (top
curve), 0.2, 0.4, 0.6, and 0.8 (bottom curve). For symmetric
resonance (β = 0), the conductance saturates at T = 0. Inset:
The typical energy dependence of transmission coefficient.

1

100

G/GQ

0.8

~

w/ ε

10

0.6
0.4
0.2
0

0

1

∆/ ε

2

~

4

2

∆/ ε
~

4

0.6

G/GQ

-1

10

0.4

0.2
-2

10

0
-5

-4

-3

-2

-1

10 10 10 10 10 1 10 10
~

T/ ε

2

0

FIG. 2. Left: Half width at half maximum w vs.
temperature T for the values of asymmetry parameter
β = 0, 0.2, 0.4, 0.6, 0.8 (bottom to top curve). With the
decreasing temperature, the half width saturates for a
non-symmetric resonance, and continuously decreases for the
symmetric one. Right: The conductance dependence on the
position of the resonant level with respect to the Fermi level,
G(∆), for symmetric (top) and non-symmetric with β = 0.5
(bottom) resonances at three temperatures T /˜ = 0.04, 0.2, 1.
ǫ

[1] Tunneling Phenomena in Solids, edited by E. Burstein
and S. Lundquist (Plenum, New York, 1969), 579p.
[2] For a review on Kondo effect in quantum dots, see, e.g.,
I.L. Aleiner, P.W. Brouwer, L.I. Glazman, Physics Reports, 358, 309 (2002).
[3] K.A. Matveev and A.I. Larkin, Phys. Rev. B 46, 15337
(1992).
[4] A.K. Geim et al, Phys. Rev. Lett. 72, 2061 (1994).

The differences and similarities of symmetric and nonsymmetric resonances are further illustrated in the right
panels of Fig. 2. The three pairs of line shapes there
correspond to ”high”, ”medium”, and ”low” tempera4

Lett. 71, 3351 (1993); D. Yue, K.A. Matveev, and L.I.
Glazman, Phys. Rev. B 49, 1966 (1994).
[10] K. Wilson, Rev. Mod. Phys. 47, 773 (1975).
[11] For arbitrary interaction strength, these two limiting
cases were discussed in: A. Furusaki, Phys. Rev. B 57,
7141 (1998).
[12] D. G. Polyakov and I. V. Gornyi, cond-mat/0212355.

[5] O.M. Auslaender et al, Phys. Rev. Lett. 84, 1764 (2000).
[6] H.W.Ch.Postma et al, Science 293, issue 5527, 76 (2001).
[7] C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46, 7268
(1992); ibid., 46, 15233-15262 (1992).
[8] C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 12201223 (1992).
[9] K.A. Matveev, D. Yue, and L.I. Glazman, Phys. Rev.

5