Vibrational effects on low-temperature properties of molecular conductors
Jernej Mravlje a,∗, Anton Ramšak b,a, Rok Žitko a
aJožef Stefan Institute, Jamova 39, Si-1000, Ljubljana, Slovenija
bFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Si-1000, Ljubljana, Slovenija
Abstract
We calculate characteristic correlation functions for the Anderson model with additional phonon-assisted coupling to the odd
conduction channel. This model describes, for example, the behavior of a molecule embedded between two electrodes in linear
transport experiments where the position of the molecule with respect to the leads affects the tunneling amplitudes. We use
variational projection-operator method and numerical renormalization group (NRG) method. The spin is Kondo screened either
by even or odd conduction channel depending on the gate voltage and electron-phonon coupling. However, in all regimes the gate-
voltage dependence of the zero temperature conductance is found to be qualitatively the same as in the model with no coupling
to the vibrational mode.
Key words: Kondo effect, molecular conductors, dynamic symmetry breaking
PACS: 72.15.Qm,73.23.-b,73.22.-f
In recent years the studies of quantum impurity sys-
tems have undergone a considerable revival due to improve-
ments in experimental techniques for measuring the elec-
tron transport through quantum dots and single molecules,
as well as due to the development of the DMFT technique
which maps interacting lattice problems to quantum impu-
rity problems with an additional self-consistency condition.
The prototype model for this class of problems is the An-
derson model for a single impurity in a metallic host with
Himp = Un↑n↓ + ǫn, where n = n↑ + n↓ is the number
of electrons occupying the impurity orbital with energy ǫ
relative to the chemical potential, and with the Coulomb
repulsion U due to the double occupancy of the impurity
orbital.
An important class of quantum impurity models include
coupling to the bosonic degrees of freedom describing the
vibrational modes of the molecule or phonons in the bulk.
There are two basic types of the electron-phonon coupling,
(i) the Holstein coupling of form nx, where n is the elec-
tron density and x the oscillator displacement, and (ii) the
coupling to hopping term of form vx, where v is the hop-
ping (hybridization) operator that couples the impurity to
the conduction band. While type (i) is more relevant when
the oscillation is related to the change in volume to which
the electron is confined (breathing modes), type (ii) is rel-
∗ Corresponding author. E-mail address: jernej.mravlje@ijs.si,
Fax: +38661 477-3724.
evant whenever the displacement modulates the hopping
probability.
The addition of the Holstein term to the AndersonHamil-
tonian effectively reduces the Coulomb repulsion and the
hybridization Γ. The effect of the electrons on the phonon
propagator is also interesting: when effective U changes
sign, a peak in the phonon propagator at reduced frequen-
cies (the ’soft mode’) emerges [1,2]. The soft mode is re-
lated to the charge susceptibility, which is increased in this
regime [3].
Very recently, similar behavior was found also in the case
where the electron-phonon coupling term is of the form
Hel−ph = gxvodd, where vodd describes the hopping from
impurity orbital to the odd conduction channel (antisym-
metric combination of the orbitals of the noninteracting
part of the Hamiltonian) [4]. The model without phonons
consist of veven, which couples the impurity only to the even
conduction channel (symmetric combination of orbitals of
the noninteracting part of Hamiltonian).
The same model (but for finite U instead of U → ∞
treatment of Ref. [4]) was analyzed also with the varia-
tional projection-operator method [5]. In this method the
ground state is expressed in terms of the ground state of
an auxiliary noninteracting Hamiltonian [6]. Several vari-
ants were tested, with parameters chosen so as to allow for
coupling, (i) only to even channel, (ii) only to odd channel,
(iii) a combination of both. Variational method applied to
Preprint submitted to Elsevier October 26, 2018
http://arxiv.org/abs/0704.1973v1
0 0.5 1
(iii)
0 0.25 0.5
(c) (d)
x (iii)
Figure 1. Variational results [U/Γ = 5,Γ/D = 0.04,Ω = 2.5Γ; Ω
is the phonon frequency, D the bandwidth. δ = 0 (left pannels),
g/U = 0.36 (right pannels)]: (a, b) Variational ground state energy.
(c) Displacement fluctuations and displacement for (iii). In (i) and
(ii) the parity symmetry is retained, hence the displacement vanishes.
(d) The conductance.
variant (iii) leads in certain parameter regimes to a ground
state of broken parity symmetry (see Fig. 1(a,b)), marked
by non-vanishing expectation value of displacement (Fig.
1(c), thin-dotted) and consequently [5] considerably re-
duced conductance through a molecule ( plotted as a func-
tion of departure from particle-hole symmetric point δ =
ǫ+U/2 in Fig. 1(d)). As discussed below, the ground state
should have a well defined parity, therefore only the vari-
ants (i) and (ii) of the variational procedure correspond to
the ground state of correct symmetry. While it would be in-
structive to implement the variational method in a manner
which correctly took into account the tunneling between
the classically degenerate minima of the oscillator poten-
tial [4], it currently appears that the implementation would
require calculating matrix elements between two distinct
Hartree-Fock vacua, which precludes the use of Wick’s the-
orem upon which our current implementation of the varia-
tional procedure is based [6].
We thus present preliminary results obtained with the
NRG method, which does not suffer from this problem.
The parity in NRG results is not broken; the expectation
values of the displacement and of the hopping term vodd
thus vanish. In Fig. 2(a) we compare the fluctuations of
hopping to even and odd channels as a measure of the ’ac-
tivity’ of corresponding channels. For g large enough the
latter are larger, corresponding to increased fluctuations of
the displacement and the emergence of the soft mode. The
ground state of the system, as seen from the NRG renor-
malization flow (not shown here) corresponds to the Fermi
liquid ground state of the single-channel Kondo problem
[7] with the characteristic quasi-particle scattering phase
shift δq.p ∼ π/2 in the even or odd channel, depending on
whether the effective phonon mediated coupling to the odd
channel is smaller or larger than the direct coupling to the
even channel. When the couplings match (marked in Fig.
2(b) by vertical lines), an unstable fixed-point of the two-
channel Kondo model type is found.
0 0.5 1 1.5
0 0.5 1 1.5
Ω/U=0.5
Ω/U=0.75
Ω/U=1
Figure 2. NRG results [U/Γ = 25, D = U ]: (a) fluctuations of hop-
ping to even (full) and odd (dashed) channel, (b) fluctuations of
displacement.
0 0.2 0.4 0.6 0.8 1
 even
Figure 3. NRG results [U/Γ = 25, D = U,Ω/U = 1, g/Ω = 1.3]: Zero
temperature quasiparticle scattering phase shifts in even (circles) and
odd (squares) channel plotted versus departure from p-h symmetry
calculated using NRG. Corresponding conductance is also plotted
(dashed). Note that the phase shifts are defined modulo π.
In Fig. 3 the scattering phase-shifts are plotted as a
function of δ. The coupling to the phonon-assisted channel
is chosen so that in the ground state the impurity spin is
screened by the odd channel for small δ, while for larger δ it
is screened by the even channel, as seen from δq.p.;even,odd ∼
π/2, respectively. Further away from the symmetric point
the model is tuned into the valence fluctuating regime
where the Kondo effect is suppressed. When the model is
used to describe a molecule (or a quantum dot) embedded
between two leads, the scattering phase shifts directly de-
termine the differential conductance (Fig. 3 dashed). The
conductance curve shown is qualitatively equal to that of
the generic one-electron transistor in the single-channel
Kondo regime, despite the fact that the Kondo effects
occurs in different channels as the gate voltage is swept.
The work was supported by SRA under grant Pl-0044.
References
[1] A. C. Hewson and D. Meyer, J. Phys.: Condens. Matter 14, 427
(2002).
[2] G. S. Jeon, T. Park, and H. Choi, Phys. Rev. B 68, 045106
(2003).
[3] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 72,
121403(R) (2005).
[4] C. A. Balseiro, P. S. Cornaglia, and D. R. Grempel, Phys. Rev.
B 74, 235409 (2006).
[5] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 74, 205320
(2006).
[6] T. Rejec and A. Ramšak, Phys. Rev. B 68, 033306 (2003).
[7] For a review of the Kondo physics see: A. C. Hewson, The Kondo
Problem to Heavy Fermions, University Press, Cambridge, 1993.
	References

  We calculate characteristic correlation functions for the Anderson model with
additional phonon-assisted coupling to the odd conduction channel. This model
describes, for example, the behavior of a molecule embedded between two
electrodes in linear transport experiments where the position of the molecule
with respect to the leads affects the tunneling amplitudes. We use variational
projection-operator method and numerical renormalization group (NRG) method.
The spin is Kondo screened either by even or odd conduction channel depending
on the gate voltage and electron-phonon coupling. However, in all regimes the
gate-voltage dependence of the zero temperature conductance is found to be
qualitatively the same as in the model with no coupling to the vibrational
mode.

Vibrational effects on low-temperature properties of molecular conductors
Jernej Mravlje a,∗, Anton Ramšak b,a, Rok Žitko a
aJožef Stefan Institute, Jamova 39, Si-1000, Ljubljana, Slovenija
bFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Si-1000, Ljubljana, Slovenija
Abstract
We calculate characteristic correlation functions for the Anderson model with additional phonon-assisted coupling to the odd
conduction channel. This model describes, for example, the behavior of a molecule embedded between two electrodes in linear
transport experiments where the position of the molecule with respect to the leads affects the tunneling amplitudes. We use
variational projection-operator method and numerical renormalization group (NRG) method. The spin is Kondo screened either
by even or odd conduction channel depending on the gate voltage and electron-phonon coupling. However, in all regimes the gate-
voltage dependence of the zero temperature conductance is found to be qualitatively the same as in the model with no coupling
to the vibrational mode.
Key words: Kondo effect, molecular conductors, dynamic symmetry breaking
PACS: 72.15.Qm,73.23.-b,73.22.-f
In recent years the studies of quantum impurity sys-
tems have undergone a considerable revival due to improve-
ments in experimental techniques for measuring the elec-
tron transport through quantum dots and single molecules,
as well as due to the development of the DMFT technique
which maps interacting lattice problems to quantum impu-
rity problems with an additional self-consistency condition.
The prototype model for this class of problems is the An-
derson model for a single impurity in a metallic host with
Himp = Un↑n↓ + ǫn, where n = n↑ + n↓ is the number
of electrons occupying the impurity orbital with energy ǫ
relative to the chemical potential, and with the Coulomb
repulsion U due to the double occupancy of the impurity
orbital.
An important class of quantum impurity models include
coupling to the bosonic degrees of freedom describing the
vibrational modes of the molecule or phonons in the bulk.
There are two basic types of the electron-phonon coupling,
(i) the Holstein coupling of form nx, where n is the elec-
tron density and x the oscillator displacement, and (ii) the
coupling to hopping term of form vx, where v is the hop-
ping (hybridization) operator that couples the impurity to
the conduction band. While type (i) is more relevant when
the oscillation is related to the change in volume to which
the electron is confined (breathing modes), type (ii) is rel-
∗ Corresponding author. E-mail address: jernej.mravlje@ijs.si,
Fax: +38661 477-3724.
evant whenever the displacement modulates the hopping
probability.
The addition of the Holstein term to the AndersonHamil-
tonian effectively reduces the Coulomb repulsion and the
hybridization Γ. The effect of the electrons on the phonon
propagator is also interesting: when effective U changes
sign, a peak in the phonon propagator at reduced frequen-
cies (the ’soft mode’) emerges [1,2]. The soft mode is re-
lated to the charge susceptibility, which is increased in this
regime [3].
Very recently, similar behavior was found also in the case
where the electron-phonon coupling term is of the form
Hel−ph = gxvodd, where vodd describes the hopping from
impurity orbital to the odd conduction channel (antisym-
metric combination of the orbitals of the noninteracting
part of the Hamiltonian) [4]. The model without phonons
consist of veven, which couples the impurity only to the even
conduction channel (symmetric combination of orbitals of
the noninteracting part of Hamiltonian).
The same model (but for finite U instead of U → ∞
treatment of Ref. [4]) was analyzed also with the varia-
tional projection-operator method [5]. In this method the
ground state is expressed in terms of the ground state of
an auxiliary noninteracting Hamiltonian [6]. Several vari-
ants were tested, with parameters chosen so as to allow for
coupling, (i) only to even channel, (ii) only to odd channel,
(iii) a combination of both. Variational method applied to
Preprint submitted to Elsevier October 26, 2018
http://arxiv.org/abs/0704.1973v1
0 0.5 1
(iii)
0 0.25 0.5
(c) (d)
x (iii)
Figure 1. Variational results [U/Γ = 5,Γ/D = 0.04,Ω = 2.5Γ; Ω
is the phonon frequency, D the bandwidth. δ = 0 (left pannels),
g/U = 0.36 (right pannels)]: (a, b) Variational ground state energy.
(c) Displacement fluctuations and displacement for (iii). In (i) and
(ii) the parity symmetry is retained, hence the displacement vanishes.
(d) The conductance.
variant (iii) leads in certain parameter regimes to a ground
state of broken parity symmetry (see Fig. 1(a,b)), marked
by non-vanishing expectation value of displacement (Fig.
1(c), thin-dotted) and consequently [5] considerably re-
duced conductance through a molecule ( plotted as a func-
tion of departure from particle-hole symmetric point δ =
ǫ+U/2 in Fig. 1(d)). As discussed below, the ground state
should have a well defined parity, therefore only the vari-
ants (i) and (ii) of the variational procedure correspond to
the ground state of correct symmetry. While it would be in-
structive to implement the variational method in a manner
which correctly took into account the tunneling between
the classically degenerate minima of the oscillator poten-
tial [4], it currently appears that the implementation would
require calculating matrix elements between two distinct
Hartree-Fock vacua, which precludes the use of Wick’s the-
orem upon which our current implementation of the varia-
tional procedure is based [6].
We thus present preliminary results obtained with the
NRG method, which does not suffer from this problem.
The parity in NRG results is not broken; the expectation
values of the displacement and of the hopping term vodd
thus vanish. In Fig. 2(a) we compare the fluctuations of
hopping to even and odd channels as a measure of the ’ac-
tivity’ of corresponding channels. For g large enough the
latter are larger, corresponding to increased fluctuations of
the displacement and the emergence of the soft mode. The
ground state of the system, as seen from the NRG renor-
malization flow (not shown here) corresponds to the Fermi
liquid ground state of the single-channel Kondo problem
[7] with the characteristic quasi-particle scattering phase
shift δq.p ∼ π/2 in the even or odd channel, depending on
whether the effective phonon mediated coupling to the odd
channel is smaller or larger than the direct coupling to the
even channel. When the couplings match (marked in Fig.
2(b) by vertical lines), an unstable fixed-point of the two-
channel Kondo model type is found.
0 0.5 1 1.5
0 0.5 1 1.5
Ω/U=0.5
Ω/U=0.75
Ω/U=1
Figure 2. NRG results [U/Γ = 25, D = U ]: (a) fluctuations of hop-
ping to even (full) and odd (dashed) channel, (b) fluctuations of
displacement.
0 0.2 0.4 0.6 0.8 1
 even
Figure 3. NRG results [U/Γ = 25, D = U,Ω/U = 1, g/Ω = 1.3]: Zero
temperature quasiparticle scattering phase shifts in even (circles) and
odd (squares) channel plotted versus departure from p-h symmetry
calculated using NRG. Corresponding conductance is also plotted
(dashed). Note that the phase shifts are defined modulo π.
In Fig. 3 the scattering phase-shifts are plotted as a
function of δ. The coupling to the phonon-assisted channel
is chosen so that in the ground state the impurity spin is
screened by the odd channel for small δ, while for larger δ it
is screened by the even channel, as seen from δq.p.;even,odd ∼
π/2, respectively. Further away from the symmetric point
the model is tuned into the valence fluctuating regime
where the Kondo effect is suppressed. When the model is
used to describe a molecule (or a quantum dot) embedded
between two leads, the scattering phase shifts directly de-
termine the differential conductance (Fig. 3 dashed). The
conductance curve shown is qualitatively equal to that of
the generic one-electron transistor in the single-channel
Kondo regime, despite the fact that the Kondo effects
occurs in different channels as the gate voltage is swept.
The work was supported by SRA under grant Pl-0044.
References
[1] A. C. Hewson and D. Meyer, J. Phys.: Condens. Matter 14, 427
(2002).
[2] G. S. Jeon, T. Park, and H. Choi, Phys. Rev. B 68, 045106
(2003).
[3] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 72,
121403(R) (2005).
[4] C. A. Balseiro, P. S. Cornaglia, and D. R. Grempel, Phys. Rev.
B 74, 235409 (2006).
[5] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 74, 205320
(2006).
[6] T. Rejec and A. Ramšak, Phys. Rev. B 68, 033306 (2003).
[7] For a review of the Kondo physics see: A. C. Hewson, The Kondo
Problem to Heavy Fermions, University Press, Cambridge, 1993.
	References
