Zero bias anomaly out of equilibrium
Zero bias anomaly out of equilibrium
D. B. Gutman1, Yuval Gefen2, and A. D. Mirlin3,4,∗
Dept. of Physics, University of Florida, Gainesville, FL 32611, USA
Dept. of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Institut für Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany
Inst. für Theorie der kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
(Dated: August 7, 2021)
The non-equilibrium zero bias anomaly (ZBA) in the tunneling density of states of a diffusive
metallic film is studied. An effective action describing virtual fluctuations out-of-equilibrium is
derived. The singular behavior of the equilibrium ZBA is smoothed out by real processes of inelastic
scattering.
PACS numbers: 73.23.-b, 73.40.Gk, 73.50.Td
The suppression of tunneling current at low bias due
to electron-electron interaction is known as the zero bias
anomaly (ZBA). The theory of ZBA for disordered metals
at thermal equilibrium has been developed, on a pertur-
bative level, by Altshuler and Aronov [1, 2]. The non-
perturbative generalization of this theory was achieved
by Finkelstein [3]. Measurements of the tunneling density
of states (DOS) in biased quasi-one-dimensional wires [4]
call for an extension of the theory to non-equilibrium
setups. In this work we study the ZBA for disordered
metallic films out of equilibrium, in both the perturba-
tive and the non-perturbative (in interaction) limits.
Besides the experimental motivation, the problem of
ZBA in a non-equilibrium system is of fundamental the-
oretical interest. At equilibrium, the distribution of elec-
trons in phase space has a single edge at the Fermi sur-
face. The Coulomb interaction between the tunneling
electron and the electrons in the Fermi sea excites vir-
tual particle-hole pairs around the Fermi edge, leading
to the suppression of the tunneling DOS, similarly to the
Debye-Waller factor. The suppression gets stronger when
the electron energy approaches the Fermi energy. Out of
equilibrium, the distribution of particles may have sev-
eral sharp edges rather than a single one at the Fermi
surface, which poses important questions addressed in
this work: How will the excitation of electron-hole pairs
in this situation affect the tunneling DOS? Will there
be an interpaly between the two edges? We show that
the two edges are not independent: one edge affects the
ZBA near the other via real interaction-induced scatter-
ing processes governing the dephasing of electrons in the
non-equlibrium regime. From this point of view the prob-
lem we are considering is a representive of a class of phe-
nomena that involve renormalization away from thermal
equilibrium, such as the Fermi edge singularity [5] and
the Kondo effect [6].
What makes the ZBA particularly interesting is
its deep connection to various conceptually impor-
tant phenomenological ideas. At equilibrium, the non-
perturbative results [3] have been reproduced by quan-
tum hydrodynamical methods [7], and, within the frame-
work of the theory of dissipation [8], by methods that
rely on the fluctuation-dissipation theorem. Our work
circumvents this restriction. Starting with the Keldysh
non-linear σ-model, we derive an effective action that ac-
counts for virtual fluctuations in disordered metals away
from equilibrium. This action is complementary to the
one for kinetics of real fluctuations (such as noise) devel-
oped earlier [11, 12, 13, 14]. Further, we discuss a connec-
tion between our theory and phenomenological methods
[7, 8, 9, 10]. As a central application of our theory, we an-
alyze the ZBA problem and calculate the tunneling DOS
for a two-dimensional (2D) diffusive metallic film subject
to external bias.
Consider an electron which tunnels from a tip into a
metal, subject to an external bias. The Coulomb inter-
action U(q) (U(q) = 2πe2/q, d = 2) causes electrons in
the metal to readjust their position, such that they try
to screen the added charge. Their motion is impeded by
static disorder. For a sufficiently good metal, character-
ized by the dimensionless conductance g = ǫF τ ≫ 1, (ǫF
is the Fermi energy and τ – the elastic mean free time)
the electron motion is diffusive. Low energy excitations
are accounted for by the non-linear Keldysh σ-model [15]
iS[Q,φ] = iTr{φT (U−1 + ν0)σ1φ}
Tr{D(∇Q)2 − 4∂tQ} − iπν0Tr{φαγ
αQ}, (1)
where Q(r, t, t′) is a 2 × 2 matrix field (in the Keldysh
space) describing slow electronic modes, φ(r, t) is a 2-
vector in the Keldysh space representing the Coulomb
interaction, and Tr denotes the trace both over Keldysh
indices (TrK), spatial, and temporal coordinates. The Q
field is subject to a non-linear constraint,
dt1Q(r, t, t1)Q(r, t1, t
′) = δ(t− t′)σ0, (2)
γ1 ≡ σ0 is a 2×2 unit matrix, γ2 ≡ σ1 the first Pauli ma-
trix, ν0 is the DOS at the Fermi energy in the absence of
interaction, and D is the diffusion coefficient. The theory
can be rewritten in terms of the field Q only. Integrating
the φ field out, we derive the following action:
iS[Q] = −
Tr{D(∇Q)2 − 4∂tQ}
i(πν0)
TrK{σ1Q}
U−1 + ν0
TrK{Q}. (3)
http://arxiv.org/abs/0704.0728v1
Physical observables can be found by differentiating the
generating functional
Z[ϕ1, ϕ2] = exp (iS[Q]) exp
1 + ν0U
2ϕ1ϕ2 − πϕ1TrK{Q} − πϕ2TrK{σ1Q}
over the source fields ϕ1 and ϕ2.
So far all low energy excitation in the problem have
been kept indiscriminately. The price one pays for this is
an extreme technical complexity of the theory. In some
cases, this description is excessive and the theory can be
simplified by singling out a particular subspace of the
Q matrices. The resulting theory is less general, but
more suitable for tackling a specific class of problems.
The noise statistics in disordered systems is a remark-
able example of such a simplification. An extension of
the Boltzman-Langevin approach [16] to high order cor-
relators is achieved on the subspace of matrices that are
diagonal in the Wigner representation,
Q(r, ǫ, t) = e
f̄(r,ǫ,t)
1 2− 4f(r, ǫ, t)
f̄(r,ǫ,t).
The theory [11, 13, 14] resulting from the σ-model (3),
reduced to the subspace (5), accounts for real fluctuations
in agreement with the cascade idea of Nagaev [17].
Below we use a similar ideology, “projecting” the σ-
model on the subspace appropriate for the ZBA prob-
lem. According to works on ZBA at equilibrium, these
are virtual fluctuations of gauge fields that dominate the
suppression of the tunneling DOS [3, 15, 18]. This sug-
gests that gauge-type fluctuations constituting local-in-
time rotations of the saddle point Λt−t′ ,
Q(r, t, t′) = e
f̄(r,t)+ i
f(r,t)Λ
r,t−t′e
f(r,t′)−
f̄(r,t′),
r,t−t′ =
1 2− 4n(r, t− t′)
, (6)
where n(r, t− t′) is the Fourier transform of the local dis-
tribution function n(r, ǫ), are to be retained. Plugging
(6) into the action (3) and expanding up to quadratic
order in f and f̄ , we derive an effective theory of fluctu-
ations in an interacting diffusive conductor.
iS[f, f̄ ] =
(dω)(dq)
f̄(−ω,−q) ω
−Dq2 +
1 + ν0U(q)
f(ω, q)
−Dq2T (ω)f̄(−ω,−q)f̄(ω, q)
. (7)
Here the effective temperature T (ω) is defined as
T (ω) =
dǫ n(ǫ) [2− n(ǫ − ω)− n(ǫ+ ω)] , (8)
and n(ǫ) is a distribution function determined by the
Boltzmann equation.
To demonstrate the consistency of our approach, we
first show that the effective action (7) reproduces cor-
rectly known density correlation functions. The sym-
metrized density correlation function is given by
ρ̂(r, t), ρ̂(0, 0)
〉ω,q = −
∂φ2(ω, q)∂φ2(−ω,−q)
1 + ν0U
ω2〈ff〉ω,q . (9)
Similarly, the response function can be expressed as
i〈[ρ̂(r, t), ρ̂(0, 0)]−θ(t)〉ω,q =
∂φ1(ω, q)∂φ2(−ω,−q)
1 + ν0U
ω2〈f f̄〉ω,q +
1 + ν0U
. (10)
Calculating the correlation functions entering Eqs. (9),
(10) using the action (7),
〈ff〉ω,q =
4Dq2T (ω)
Dq2 + iω
1+ν0V (q)
〈fω,qf̄−ω,−q〉 ≡ 〈f f̄〉ω,q =
Dq2 − iω
1+ν0U(q)
) ,(11)
we reproduce the known results for for the spectral func-
tion of density fluctuations
ρ̂(r, t), ρ̂(0, 0)
〉ω,q =
2T (ω)
|Dq2[1 + ν0U(q)] + iω|2
and the density-density response function,
i〈[ρ̂(r, t), ρ̂(0, 0)]−θ(t)〉ω,q =
Dq2[1 + ν0U(q)]− iω
It is worth noting the analogy between our effective ac-
tion and phenomenological theories that describe ZBA at
equilibrium within an effective environment model [9, 10].
The latter explain the ZBA as the influence of virtual
fluctuations of an electromagnetic field at equilibrium [8]
on the electron tunneling. In view of the fluctuation-
dissipation theorem, fluctuations of the electromagnetic
field are determined by the complex impedance of the
system. In the zero dimensional case, modes of the elec-
tromagnetic field can be considered as independent quan-
tum harmonic oscillators. Being suddenly shaken by the
incoming electron, they move away from the equilibrium
position, reducing the overlap with their original config-
uration hence suppressing electron tunneling amplitude.
The action (7) describes similar processes in a 2D diffu-
sive system and without the assumption of thermal equi-
librium, thus keeping the information about both real
and virtual processes in a non-equilibrium state.
Now we are ready to apply the theory to the problem
of ZBA out of equilibrium. The tunneling DOS
ν(ǫ) =
GR(ǫ, p)−GA(ǫ, p)
can be rewritten in terms of Q matrices as
ν(ǫ) =
dt e−iǫ(t−t
′)〈TrK Q(r, t
′, t)σz〉Q . (13)
Plugging Eq. (6) into Eq.(13), we find
= 1+2i
dt n(t) eiǫt−
Iff (t) sin
Iff̄ (t)
, (14)
where we used the notations
Iff (t) =
(dω)(dq) (1 − cosωt) 〈〈ff〉〉ω,q ,
Iff̄ (t) =
(dω)(dq) sinωt 〈〈f f̄〉〉ω,q . (15)
There is a subtle point related to the definition of the cor-
relator 〈〈ff〉〉 and 〈〈f f̄〉〉 in (15). The ZBA should vanish
in the absence of interaction (U = 0). This physically ob-
vious statement is valid to all orders of the diagrammatic
calculations (all potentially singular contributions vanish
after the time integration over the Keldysh contour) and
is satisfied by the non-linear σ model. However, it ceases
to be an exact feature of the theory when the full space
of Q matrices is reduced to the subspace Eq. (6). This
problem is easily cured. Since we are only interested in
summing up the leading interaction-induced ln2 ǫ contri-
butions to the tunneling DOS (to all orders), these are
the interaction-induced parts of 〈ff〉 and 〈f f̄〉 that we
have to keep in Eq. (15) [19]:
〈〈f f̄〉〉ω,q ≡ 〈f f̄〉ω,q − 〈f f̄〉ω,q,U=0
(Dq2 − iω)[Dq2(1 + ν0U)− iω]
. (16)
Analogously,
〈〈ff〉〉ω,q =
4Dq2T (ω)(2 + ν0U)U
|Dq2 − iω|2|Dq2(1 + ν0U)− iω|2
. (17)
Equations (14)–(17) constitute our general result for the
non-equilibrium ZBA. At equilibrium they reproduce the
known results for the ZBA in the perturbative [1, 2] and
non-perturbative [3] regimes.
To illustrate the effect of non-equilibrium conditions on
ZBA we consider a diffusive film of length L connected to
two zero-temperature reservoirs with voltage difference
V . We assume that the energy relaxation in the film
can be neglected, i.e. that L is shorter than the energy
relaxation length, a condition met at not too high bias.
In this case the solution of the Boltzmann equation is a
double-step function
n(ǫ, x) = an0(ǫ− eV/2) + (1 − a)n0(ǫ+ eV/2) , (18)
where n0(ǫ) is the Fermi distribution function, a = x/L,
and x is the distance from the reservoir. This distribution
corresponds to the effective temperature
T (ω) = [a2 + (1− a)2]Teq(ω)
+a(1− a)[Teq(ω + eV ) + Teq(ω − eV )] , (19)
where Teq(ω) is the equilibrium value
Teq(ω) = ω coth(ω/T ) →
|ω| . (20)
Substituting Eqs. (16), (17) into Eqs.(15),( 14), we find
the tunneling DOS out of equilibrium:
= a exp
8π2ν0D
log(D2κ4τt−)
+(1− a) exp
8π2ν0D
log(D2κ4τt+)
. (21)
This result is valid in both the perturbative (when the
exponentials can be expanded up to the first non-trivial
term) and non-perturbative regimes. The DOS consists
of two terms corresponding to electrons coming from the
left and right reservoir. The energy scales governing the
argument of the logarithm in these two terms are
= max
a(1− a)eV log
4πν0D
, (22)
respectively (κ = 2πe2ν).
The evolution of the ZBA in the tunneling DOS with
decreasing conductance (i.e. from the perturbative to
the non-perturbative regime) is illustrated in Fig. 1. The
DOS has a two-dip shape with minima reached at the
energies where the distribution function exhibits discon-
tinuous jumps (i.e. at the positions of the Fermi edge
in the left and right leads, ǫ = ± eV
). Away from the
minima, the DOS is controlled by the energy measured
from the corresponding edge. As this energy decreases
(we get closer to one of the Fermi edges), the singularity
near this edge gets affected by the presence of the other
edge. The broadening of the ZBA singularity takes place
on a new energy scale,
(V ) =
a(1 − a)
4πν0D
eV log
. (23)
The notation introduced in Eq. (23) stresses the analogy
with the equilibrium dephasing rate τ−1
(T ) governing
the infrared cutoff of the interference phenomena such
as the weak localization [2]. The emergence of τ−1
shows that inelastic scattering processes (responsible for
dephasing) lead to smearing of the ZBA singularity.
Qualitatively the results can be explained as follows.
Virtual fluctuations of the gauge fields act on a quasipar-
ticle and suppress the tunneling DOS. Their influence is
limited by the quasiparticle life time. An external bias
-0.001 -0.0005 0 0.0005 0.001
( a )
-0.001 -0.0005 0 0.0005 0.001
FIG. 1: Tunneling DOS vs. energy for the double-step dis-
tribution function (18) with a = 0.5. The dimensionless con-
ductance is, from top to bottom, g = 100, 10 (a) and 1 (b).
enhances fluctuations of the electromagnetic field, which
in turn dephase the tunneling quasiparticle. This results
in a competition between the virtual processes establish-
ing the ZBA and the real fluctuations (noise) cutting it
Let us stress the crucial difference with the equilib-
rium situation. At equilibrium, the ZBA singularity
in 2D is cut off by temperature (which enters via the
distribution function); specifically, the dephasing rate
(T ) does not affect the ZBA in any essential way,
since τ−1
(T ) ≪ T . It is thus a distinct feature of the
strongly non-equilibrium regime that the ZBA is smeared
by the inelastic scattering (dephasing) rate.
In conclusion, we have developed an effective theory
for virtual fluctuations in disordered metals out of equi-
librium. Using this theory, we studied the ZBA in a 2D
metallic film biased by external voltage. We have found
that out of equilibrium the tunneling DOS has a double-
dip structure with minima reached at the ”edges” of the
particle distribution. The ZBA near any of the ”edges” is
influenced by the other one. The suppression of DOS is
smoothed out by the real processes of inelastic scattering
(dephasing) with characteristic energy scale τφ(V ). Fur-
ther applications and extensions of our theory include, in
particular, the ZBA in quasi-one-dimensional and strictly
one-dimensional (Luttinger liquid) wires [20] out of equi-
librium; these results will be reported elsewhere.
We thank D. Bagrets, N. Birge, A. Finkelstein, D.
Maslov, A. Shnirman, and A. Shytov for useful discus-
sions. This work was supported by NSF-DMR-0308377
(DG), US-Israel BSF, ISF of the Israel Academy of Sci-
ences, and DFG SPP 1285 (YG), DFG Center for Func-
tional Nanostructures, EC Transnational Access Pro-
gram at the WIS Braun Submicron Center (ADM), and
Einstein Minerva Center.
[*] Also at Petersburg Nuclear Physics Institute, 188300
St. Petersburg, Russia.
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