Electronic spin working mechanically (Review Article)
A single-electron tunneling (SET) device with a nanoscale central island that can move with respect to the bulk source- and drain electrodes allows for a nanoelectromechanical (NEM) coupling between the electrical current through the device and mechanical vibrations of the island. Although an elec...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1195362025-02-09T10:26:02Z Electronic spin working mechanically (Review Article) Shekhter, R.I. Gorelik, L.Y. Krive, I.V. Kiselev, M.N. Kulinich, S.I. Parafilo, A.V. Kikoin, K. Jonson, M. К 100-летию со дня рождения А.А. Галкина A single-electron tunneling (SET) device with a nanoscale central island that can move with respect to the bulk source- and drain electrodes allows for a nanoelectromechanical (NEM) coupling between the electrical current through the device and mechanical vibrations of the island. Although an electromechanical “shuttle” instability and the associated phenomenon of single-electron shuttling were predicted more than 15 years ago, both theoretical and experimental studies of NEM-SET structures are still carried out. New functionalities based on quantum coherence, Coulomb correlations and coherent electron-spin dynamics are of particular current interest. In this article we present a short review of recent activities in this area. Financial support from the Swedish VR, and the Korean WCU program funded by MEST/NFR (R31-2008-000- 10057-0) is gratefully acknowledged. This research was supported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10. I.V.K. and A.V.P. acknowledge financial support from the National Academy of Sciences of Ukraine (grant No. 4/13-N). I.V.K. thanks the Department of Physics at the University of Gothenburg for hospitality. 2014 Article Electronic spin working mechanically (Review Article) / R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, M. Jonson // Физика низких температур. — 2014. — Т. 40, № 7. — С. 775-792. — Бібліогр.: 83 назв. — англ. 0132-6414 PACS 81.07.Oj, 73.23.Hk https://nasplib.isofts.kiev.ua/handle/123456789/119536 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 100-летию со дня рождения А.А. Галкина К 100-летию со дня рождения А.А. Галкина |
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К 100-летию со дня рождения А.А. Галкина К 100-летию со дня рождения А.А. Галкина Shekhter, R.I. Gorelik, L.Y. Krive, I.V. Kiselev, M.N. Kulinich, S.I. Parafilo, A.V. Kikoin, K. Jonson, M. Electronic spin working mechanically (Review Article) Физика низких температур |
| description |
A single-electron tunneling (SET) device with a nanoscale central island that can move with respect to the
bulk source- and drain electrodes allows for a nanoelectromechanical (NEM) coupling between the electrical
current through the device and mechanical vibrations of the island. Although an electromechanical “shuttle” instability
and the associated phenomenon of single-electron shuttling were predicted more than 15 years ago, both
theoretical and experimental studies of NEM-SET structures are still carried out. New functionalities based on
quantum coherence, Coulomb correlations and coherent electron-spin dynamics are of particular current interest.
In this article we present a short review of recent activities in this area. |
| format |
Article |
| author |
Shekhter, R.I. Gorelik, L.Y. Krive, I.V. Kiselev, M.N. Kulinich, S.I. Parafilo, A.V. Kikoin, K. Jonson, M. |
| author_facet |
Shekhter, R.I. Gorelik, L.Y. Krive, I.V. Kiselev, M.N. Kulinich, S.I. Parafilo, A.V. Kikoin, K. Jonson, M. |
| author_sort |
Shekhter, R.I. |
| title |
Electronic spin working mechanically (Review Article) |
| title_short |
Electronic spin working mechanically (Review Article) |
| title_full |
Electronic spin working mechanically (Review Article) |
| title_fullStr |
Electronic spin working mechanically (Review Article) |
| title_full_unstemmed |
Electronic spin working mechanically (Review Article) |
| title_sort |
electronic spin working mechanically (review article) |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2014 |
| topic_facet |
К 100-летию со дня рождения А.А. Галкина |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119536 |
| citation_txt |
Electronic spin working mechanically (Review Article) / R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, M. Jonson // Физика низких температур. — 2014. — Т. 40, № 7. — С. 775-792. — Бібліогр.: 83 назв. — англ. |
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Физика низких температур |
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2025-11-25T20:29:57Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7, pp. 775–792
Electronic spin working mechanically
(Review Article)
R.I. Shekhter1, L.Y. Gorelik2, I.V. Krive3,4, M.N. Kiselev5, S.I. Kulinich3,
A.V. Parafilo3, K. Kikoin6, and M. Jonson1,7,8
1Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden
E-mail: shekhter@physics.gu.se
2Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
3B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
4Physical Department, V.N. Karazin National University, Kharkov 61077, Ukraine
5The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste 1-34151, Italy
6School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
7SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK
8Department of Physics, Division of Quantum Phases and Devices, Konkuk University, Seoul 143-701, Korea
Received March 4, 2014, revised March 27, 2014, published online May 21, 2014
A single-electron tunneling (SET) device with a nanoscale central island that can move with respect to the
bulk source- and drain electrodes allows for a nanoelectromechanical (NEM) coupling between the electrical
current through the device and mechanical vibrations of the island. Although an electromechanical “shuttle” in-
stability and the associated phenomenon of single-electron shuttling were predicted more than 15 years ago, both
theoretical and experimental studies of NEM-SET structures are still carried out. New functionalities based on
quantum coherence, Coulomb correlations and coherent electron-spin dynamics are of particular current interest.
In this article we present a short review of recent activities in this area.
PACS: 81.07.Oj Nanoelectromechanical systems;
73.23.Hk Single-electron tunneling.
Keywords: electron-spin dynamics, single-electron shuttle, Kondo effect.
Contents
1. Introduction ......................................................................................................................................... 776
2. Shuttling of single electrons ................................................................................................................ 776
2.1. Shuttle instability in the quantum regime of Coulomb blockade ................................................. 776
3. Electro- and spintro-mechanics of magnetic shuttle devices ............................................................... 778
3.1. Spin-controlled shuttling of electric charge ................................................................................. 779
3.2. Spintro-mechanics of magnetic shuttle devices ........................................................................... 780
3.3. Spintronics of shuttles ................................................................................................................. 782
3.4. Electron shuttle based on electron spin ....................................................................................... 783
3.5. Mechanically assisted magnetic coupling between nanomagnets ............................................... 785
4. Resonance spin-scattering effects. Spin shuttle as a “mobile quantum impurity” . .............................. 786
5. Conclusions ......................................................................................................................................... 790
References ............................................................................................................................................... 790
© R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson, 2014
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
1. Introduction
Electric weak links play a crucial role in modern
nanoelectronics since they offer a natural way to inject
electrons into small conducting areas. At the same time
weak links of nanometer size offer new functionality due
to the mesoscopic properties of the small conductors that
form such links. Coulomb blockade of tunneling, resonant
tunneling, quantum spin coherence, spin-dependent tunnel-
ing and weak superconductivity are just examples of new
phenomena (compared to bulk transport phenomena) that
lead to new physics in nanometer sized weak electric links.
Special interest is focused on the nonequilibrium evolution
of “hot” electrons with voltage-controllable excess energy.
Point-contact spectroscopy of elementary excitations and
nanoelectromechanical (NEM) shuttle instabilities are the
brightest examples of functionalities based on properties of
accelerated electrons in point contacts. The nonequilibrium
nature of an electronic system is most prominently mani-
fested if excitation modes, which are spatially localized in
the vicinity of a weak link, interact with the “hot” elec-
trons. Then even a low level of energy transfer from the
electrons does not prevent these excitations from accumu-
lating a significant amount of energy, with the energized
electrons acting as power supply.
Single-electron tunneling (SET) transistors are nano-
devices with particularly prominent mesoscopic features.
Here, the Coulomb blockade of single-electron tunneling at
low voltage bias and temperature [1] makes Ohm’s law for
the electrical conductance invalid in the sense that the elec-
trical current is not necessarily proportional to the voltage
drop across the device. Instead, the current is due to a tem-
porally discrete set of events where electrons tunnel quan-
tum-mechanically one-by-one from a source to a drain
electrode via a nanometer size island (a “quantum dot”).
This is why the properties of a single electronic quantum
state are crucial for the operation of the entire device.
Since the probability for quantum mechanical tunneling
is exponentially sensitive to the tunneling distance, it fol-
lows that the position of the quantum dot relative to the
electrodes is crucial. On the other hand the strong Cou-
lomb forces that accompany the discrete nanoscale charge
fluctuations, which are a necessary consequence of a cur-
rent flow through the SET device, might cause a signifi-
cant deformation of the device and move the dot, hence
giving rise to a strong electromechanical coupling. This
unique feature makes the so-called nanoelectromechanical
SET (NEM-SET) devices, where mechanical deformation
can be achieved along with electronic operations, to be one
of the best nanoscale realizations of electromechanical
transduction.
In this review we will discuss some of the latest achieve-
ments in the nanoelectromechanics of NEM-SET devices
focusing on the new functionality that exploits the coher-
ence of quantum charge and spin subsystems in their inter-
play with mechanical subsystem. By choosing magnets as
components of the device one may take advantage of a ma-
croscopic ordering of electrons with respect to their spin.
We will discuss how the electronic spin contribute to elec-
tromechanical and mechano-electrical transduction in a
NEM-SET device. New effects appear also due to many-
body reconstruction of the electron spectrum in the metal-
lic leads related to exchange interaction with spin localized
in the moving shuttle. This interaction opens a new chan-
nel of Kondo resonance tunneling between the shuttle and
the leads, which contributes to specific “Kondo-nanome-
chanics”.
This review is an update of our earlier reviews of shut-
tling [2–4]. Other aspects of nanoelectromechanics are
only briefly discussed here. We refer readers to the well-
known reviews of Refs. 5–9 on nanoelectromechanical
systems for additional information.
2. Shuttling of single electrons
A single-electron shuttle can be considered as the ulti-
mate miniaturization of a classical electric pendulum capa-
ble of transferring macroscopic amounts of charge between
two metal plates. In both cases the electric force acting on
a charged “ball” that is free to move in a potential well
between two metal electrodes kept at different electro-
chemical potentials, = L ReV µ − µ , results in self-oscil-
lations of the ball. Two distinct physical phenomena,
namely the quantum mechanical tunneling mechanism for
charge loading (unloading) of the ball (in this case more
properly referred to as a grain) and the Coulomb blockade
of tunneling, distinguish the nanoelectromechanical device
known as a single-electron shuttle [10] (see also [11]) from
its classical textbook analog. The regime of Coulomb
blockade realized at bias voltages and temperatures
, CeV T E (where 2= /2CE e C is the charging energy, C is
the grain’s electrical capacitance) allows one to consider
single-electron transport through the grain. Electron tun-
neling, being extremely sensitive to the position of the
grain relative to the bulk electrodes, leads to a shuttle in-
stability — the absence of any equilibrium position of an
initially neutral grain in the gap between the electrodes.
2.1. Shuttle instability in the quantum regime of Coulomb
blockade
First, we consider the single-electron shuttle effect in
the simplest model [12] where the grain is modeled as a
single-level quantum dot (QD) that is weakly coupled (via
a tunnel Hamiltonian) to the electrodes (see Fig. 1). The
Hamiltonian corresponding to this model reads
( ) ( )
tot
= , = ,
= ,j j
QD v tl
j L R j L R
H H H H H+ + +∑ ∑ (1)
776 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
where the Hamiltonian
( ) †= ( )j
kj j kjl kj
k
H a aε − µ∑ (2)
describes noninteracting electrons in the left ( =j L ) and
right ( =j R) leads, which are kept at different chemical
potential jµ and have a constant density states jν ; † ( )kjkja a
creates (annihilates) an electron with momentum k in lead
j . The quantum dot is described by two parts. It is single-
electron level Hamiltonian and Hamiltonian of harmonic
potential in which QD vibrates
† †
0= ,QDH c c dxc cε − (3)
2 21= ( ),
2vH x p+ (4)
where † ( )c c is the creation (annihilation) operator for
an electron at the dot, 0ε is the energy of the resonant level,
x is the dimensionless coordinate operator (normalized
by the amplitude 0x of zero-point fluctuations, 0x =
0/M= ω , M is the mass of QD), p is the correspond-
ing momentum operator ([ , ] =x p i), 0ω is the frequency of
vibrons, 2
0 0/( )d eE M x= ω is the characteristic electrome-
chanical interaction constant. For convenience we use di-
mensionless variables. The physical meaning of the second
term in Eq. (3) for usual shuttle systems is the interaction
energy due to the coupling of the electron charge density
on the dot with the electric field (E ) in the gap between
electrodes. Here, for convenience, all energies measure in
units 0ω , time in units of 1
0
−ω . Note, that in general the
mechanism of electromechanical interaction could have
different nature (electrostatic interaction charge on the dot
with gate electrode, interaction in magnetic field due the
Lorentz force, due exchange force between electrons with
spin and spin polarized leads, see next sections).
The tunneling Hamiltonian ( )j
tH in Eq. (1) has the form
( ) †
0= exp ( / ) h.c.j
t j kj
k
H t jx a cλ +∑ (5)
Here =j ± for /L R electrodes, 0 jt is the bare tunneling
amplitude, which corresponds to a weak dot-electrode
coupling, λ is the characteristic tunneling length. The ex-
plicit coordinate dependence in the tunneling Hamiltonian
indicates sensitivity of tunnel matrix elements to a shift of
the quantum dot center-of-mass coordinate with respect to
its equilibrium ( = 0cmx ) position. The x-dependence in
Eq. (5) represents also additional interaction with vibronic
degree of freedom.
Even in such a simple formulation the single-electron
shuttle problem is quite complex. In this section we review
some main results of electron shuttling (without involving
the spin degree of freedom) and present the basic idea of
the methods of solution based on the equation of motion
for the matrix density. The advantage of this method is that
it is possible to explicitly consider the quantum dot dynam-
ics in quantum regime and take into account the coherent
dynamics of spin electron states in a magnetic field, see the
next section.
The time evolution of the system is obtained from the
Liouville–von Neumann equation for the total density ma-
trix
ˆ ˆ( ) = [ , ( )] .ti t H t∂ σ σ (6)
In order to consider the dynamics of the electronic state in
the dot and the vibronic degrees of freedom we reduce the
total density operator by tracing over all electronic states in
the leads, leads( ) = Tr { ( )}t tρ σ . We assume that electrons in
the leads are in equilibrium and that they are not affected
by the coupling to the dot. So, we factorize the density ma-
trix, leads( ) ( )t tσ ≈ ρ ⊗ σ (this approximation is always valid
for 2
0= 2 | | exp [ / ] 1j j jt xΓ πν λ ). After shifting the x
axis by / 2d we get the system of equation of motion for
the diagonal elements of density matrix 0 = 0 | | 0ρ 〈 ρ 〉 and
1 = 1| |1ρ 〈 ρ 〉 , where †|1 = | 0d〉 〉, as
0 0 0 1
1= , { ( ), } ( ) ( ),
2 2
L R Rt v
di H x x x x ∂ ρ − + ρ − Γ ρ + Γ ρ Γ
(7)
1 1 1 0
1= , { ( ), } ( ) ( ),
2 2
R L Lt v
di H x x x x ∂ ρ − − ρ − Γ ρ + Γ ρ Γ
(8)
where ( ) = ( / 2)j jx x dΓ Γ + . The off-diagonal density ma-
trix elements are decoupled from the equation of motion of
the diagonal elements. It is easy to take into account dissi-
pation of the system. The corresponding dissipation term is
= ( / 2)[ ,{ , }] ( / 2)[ ,[ , ]]L i x p x x pγ ρ − γ ρ − γ (γ is the dissi-
pation rate).
Now we find the condition under which the vibrational
ground state of the oscillator becomes unstable. For this we
consider the time evolution of the expectation value of the
Fig. 1. Model system consisting of a movable quantum dot placed
between two leads. An effective elastic force acting on the dot
due to its connections to the leads is described by a parabolic
potential. Only one single electron state is available in the dot and
the noninteracting electrons in the leads are assumed to have a
constant density of states. Reprinted with permission from [12],
D. Fedorets et al., Europhys. Lett. 58, 99 (2002). 2002, EDP
Sciences.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 777
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
coordinate, ( ) = Tr { }x t x +ρ , and the momentum operators,
( ) = Tr{ }p t p +ρ , of the island (here 0 1+ρ ≡ ρ + ρ ). To the first
order by 1−λ , for symmetric tunneling couplings (0)LΓ =
(0) = /2R= Γ Γ and in the high bias voltage limit ( L Rµ − µ =
)eV= → ∞ the equations of motion for the first vibrational
moments become closed, so that [13]
= , = ,
2
dx p p p x n−−γ − − (9)
2= ,n n x− −
Γ
−Γ +
λ
where 1= 1 2 Trn− − ρ . The solution of Eq. (9) for the quan-
tum dot displacement is ( ) e cosrtx t A t≈ , where r =
thr1/ 2( )= γ − γ is the increment of the shuttle instability.
If the dissipation rate γ is below the threshold value
2
thr = / [ ( ) 1]dγ Γ λ Γ + , then the expectation value of the
dot coordinate grows exponentially in time and the vib-
rational ground state is unstable. It was shown [13] that
this exponential increase of the displacement drives the
system into the nonlinear regime of the vibration dyna-
mics, where the system reaches a stable steady state of de-
veloped shuttle motion.
In order to analyze this stable state (i.e., the solution of
the system Eqs. (7), (8)) it is convenient to use the Wigner
function representation [13,14]. The Wigner distribution
function for the density operator +ρ is defined as
1( , ) e /2 | | /2 .
2
ipW x p d x x
+∞
− ξ
+ +
−∞
≡ ξ + ξ ρ − ξ
π ∫ (10)
The dynamics of the oscillating QD is characterized by its
trajectory (distribution) in the phase space ( ,x p) for
2 2/2 /2 = constp x+ . Now we proceed to polar coordinates
( , )A ϕ , where = sinx A ϕ and = cosp A ϕ . An equation for
( , )W A+ ϕ is derived from Eqs. (7) and (8) after straightfor-
ward calculations (for details see [13]). In the leading order
of perturbation theory by the small parameters /d λ , 2−λ ,
and γ this equation takes the form of a stationary Fokker–
Planck equation for the zeroth Fourier component of the
Wigner function ( )W A+
0 1( ) ( ) ( ) = 0,D A D A W A
A A +
∂ ∂ − ∂ ∂
(11)
where 2
1 01 0= ( ), = ( )D A D A D AD A are drift- and diffu-
sion coefficients (analytical expression of this coefficients
will be presented in Sec. 3.4). The normalized solution of
Eq. (11) has the form of a Boltzman distribution,
11
00
( )= exp
( )
A D AW Z dA
D A
−
+
∫ . (12)
The stationary solution of the oscillating dot is localized in
the phase space around points where W + is maximal. From
Eq. (12) one can see that the maximum of the Wigner
function is determined by zeros of the drift coefficient
1( ) = 0mD A ( 1( ) < 0mD A′ ). In the vicinity of this point, W +
can be approximated by a Gaussian distribution function.
For the spinless shuttle problem it can be shown that W +
always has an extremum at = 0A : maximum for thr>γ γ
and minimum for thr<γ γ . So the vibrational ground state
is unstable when the dissipation is below threshold value as
has been shown by solving the equation system (9). The
function W + has also a maximum for the non-zero ampli-
tude CA , which corresponds to the stable limit cycle ampli-
tude of shuttle oscillations (for more details see [13]).
One can distinguish two regimes of “quantum” (for
4/d −λ λ ) and “quasiclassical” ( 4/d −λ λ ) shuttle
motion [15]. In the quasiclassical regime Gaussian distri-
bution is narrow and in quantum regime the width of dis-
tribution “bell”' is of the order of 1λ , i.e., the Wigner
function is smeared around classical phase trajectory. It is
interesting to note that there is a range of parameters where
both vibrational and shuttle regimes are present (a region
where the Wigner function has two maxima).
3. Electro- and spintro-mechanics of magnetic shuttle
devices
In this section we will explore new functionalities that
emerge when nanomechanical devices are partly or com-
pletely made of magnetic materials. The possibility of
magnetic ordering brings new degrees of freedom into play
in addition to the electronic and mechanical ones consid-
ered so far, opening up an exciting perspective towards
utilising magneto-electromechanical transduction for a
large variety of applications. Device dimensions in the
nanometer range mean that a number of mesoscopic phe-
nomena in the electronic, magnetic and mechanical subsys-
tems can be used for quantum coherent manipulations. In
comparison with the electromechanics of the nanodevices
considered above the prominent role of the electronic spin
in addition to the electric charge should be taken into ac-
count.
The ability to manipulate and control spins via electri-
cal [16–18], magnetic [19] and optical [20,21] means has
generated numerous applications in metrology [22] in re-
cent years. A promising alternative method for spin ma-
nipulation employs a mechanical resonator coupled to the
magnetic dipole moment of the spin(s), a method which
could enable scalable quantum information architectures
[23] and sensitive nanoscale magnetometry [24–26]. Mag-
netic resonance force microscopy (MRFM) was suggested
as a means to improve spin detection to the level of a single
spin and thus enable three dimensional imaging of macro-
molecules with atomic resolution. In this technique a single
spin, driven by a resonant microwave magnetic field inter-
acts with a ferromagnetic particle. If the ferromagnetic
particle is attached to a cantilever tip, the spin changes the
cantilever vibration parameters [27]. The possibility to de-
tect [27] and monitor the coherent dynamics of a single
778 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
spin mechanically [28] has been demonstrated experimen-
tally. Several theoretical suggestions concerning the possi-
bility to test single-spin dynamics through an electronic
transport measurement were made recently [29–32]. Com-
plementary studies of the mechanics of a resonator coupled
to spin degrees of freedom by detecting the spin dynamics
and relaxation were suggested in [29–36] and carried out in
[37]. Electronic spin-orbit interaction in suspended nano-
wires was shown to be an efficient tool for detection and
cooling of bending-mode nanovibrations as well as for
manipulation of spin qubit and mechanical quantum vibra-
tions [38–40].
An obvious modification of the nanoelectromechanics
of magnetic shuttle devices originates from the spin-
splitting of electronic energy levels, which results in the
known phenomenon of spin-dependent tunneling. Spin-
controlled nanoelectromechanics which originates from
spin-controlled transport of electric charge in magnetic
NEM systems is represented by number of new magneto-
electromechanical phenomena.
Qualitatively new opportunities appear when magnetic
nanomechanical devices are used. They have to do with the
effect of the short-ranged magnetic exchange interaction
between the spin of electrons and magnetic parts of the
device. In this case the spin of the electron rather than its
electrical charge can be the main source of the mechanical
force acting on movable parts of the device. This leads to
new physics compared with the usual electromechanics of
nonmagnetic devices, for which we use the term spintro-
mechanics. In particular it becomes possible for a movable
central island to shuttle magnetization between two mag-
netic leads even without any charge transport between the
leads. The result of such a mechanical transportation of
magnetization is a magnetic coupling between nanomag-
nets with a strength and sign that are mechanically tunable.
In this section we will review some early results that
involve the phenomena mentioned above. These only
amount to a first step in the exploration of new opportuni-
ties caused by the interrelation between charge, spin and
mechanics on a nanometer length scale.
3.1. Spin-controlled shuttling of electric charge
By manipulating the interaction between the spin of
electrons and external magnetic fields and/or the internal
interaction in magnetic materials, spin-controlled nano-
electromechanics may be achieved.
A new functional principle — spin-dependent shuttling
of electrons — for low magnetic field sensing purposes
was proposed by Gorelik et al. in Ref. 41. This principle
may lead to a giant magnetoresistance effect in external
magnetic fields as low as 1–10 Oe in a magnetic shuttle
device if magnets with highly spin-polarized electrons
(half metals [42–46]) are used as leads in a magnetic shut-
tle device. The key idea is to use the external magnetic
field to manipulate the spin of shuttled electrons rather
than the magnetization of the leads. Since the electron
spends a relatively long time on the shuttle, where it is
decoupled from the magnetic environment, even a weak
magnetic can rotate its spin by a significant angle. Such a
rotation allows the spin of an electron that has been loaded
onto the shuttle from a spin-polarized source electrode to
be reoriented in order to allow the electron finally to tunnel
from the shuttle to the (differently) spin-polarized drain
lead. In this way the shuttle serves as a very sensitive
“magnetoresistor” device. The model employed in Ref. 41
assumes that the source and drain are fully polarized in
opposite directions. A mechanically movable quantum dot
(described by a time-dependent displacement ( )x t ), where
a single energy level is available for electrons, performs
driven harmonic oscillations between the leads. The exter-
nal magnetic field, H , is perpendicular to the orientations
of the magnetization in both leads and to the direction of
the mechanical motion.
The spin-dependent part of the Hamiltonian is speci-
fied as
† † † †
magn ( ) = ( )( ) ( ),
2
g HH t J t a a a a a a a a↓ ↓↑ ↑↓ ↓↑ ↑
µ
− − + (13)
where ( ) = ( ) ( )R LJ t J t J t− , ( ) ( )L RJ t are the molecular
fields induced by exchange interactions between the on-
grain electron and the left (right) lead, g is the gyromag-
netic ratio and µ is the Bohr magneton. The proper Liou-
ville–von Neumann equation for the density matrix is ana-
lyzed and an average electrical current is calculated for the
case of large bias voltage.
In the limit of weak exchange field, maxJ Hµ one
may neglect the influence of the magnetic leads on the on-
dot electron spin dynamics. The resulting current is
2
0
2 2
( /2) tanh ( /4)sin=
( /2) ( /4)sin tanh
e wI
w
ω ϑ
π ϑ +
, (14)
where w is the total tunneling probability during the con-
tact time 0t , while 0/g Hϑ π µ ω is the rotation angle of
the spin during the “free-motion” time.
The theory [41] predicts oscillations in the
magnetoresistance of the magnetic shuttle device with a
period pH∆ , which is determined from the equation
0 = (1 ) pg w Hω µ + ∆ . The physical meaning of this rela-
tion is simple: every time when 0 / = 1/ 2nω Ω +
( = /g HΩ µ is the spin precession frequency in a magnetic
field) the shuttled electron is able to flip fully its spin to
remove the “spin-blockade” of tunneling between spin
polarized leads having their magnetization in opposite di-
rections. This effect can be used for measuring the me-
chanical frequency thus providing dc spectroscopy of
nanomechanical vibrations.
Spin-dependent shuttling of electrons as discussed
above is a property of noninteracting electrons, in the sense
that tunneling of different electrons into (and out of) the
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 779
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
dot are independent events. The Coulomb blockade phe-
nomenon adds a strong correlation of tunneling events,
preventing fluctuations in the occupation of electronic
states on the dot. This effect crucially changes the physics
of spin-dependent tunneling in a magnetic NEM device.
One of the remarkable consequences is the Coulomb pro-
motion of spin-dependent tunneling predicted in Ref. 47.
In this work a strong voltage dependence of the spin-flip
relaxation rate on a quantum dot was demonstrated. Such
relaxation, being very sensitive to the occupation of spin-
up and spin-down states on the dot, can be controlled by
the Coulomb blockade phenomenon. It was shown in
Ref. 47 that by lifting the Coulomb blockade one stimu-
lates occupation of both spin-up and spin-down states thus
suppressing spin-flip relaxation on the dot. In magnetic
devices with highly spin-polarized electrons electronic
spin-flip can be the only mechanism providing charge
transport between oppositely magnetized leads. In this case
the onset of Coulomb blockade, by increasing the spin-flip
relaxation rate, stimulates charge transport through a mag-
netic SET device (Coulomb promotion of spin-dependent
tunneling). Spin-flip relaxation also modifies qualitatively
the noise characteristics of spin-dependent single-electron
transport. In Refs. 48, 49 it was shown that the low-
frequency shot noise in such structures diverges as the spin
relaxation rate goes to zero. This effect provides an effi-
cient tool for spectroscopy of extremely slow spin-flip re-
laxation in quantum dots. Mechanical transportation of a
spin-polarized dot in a magnetic shuttle device provides
new opportunities for studying spin-flip relaxation in quan-
tum dots. The reason can be traced to a spin-blockade of
the mechanically aided shuttle current that occurs in devic-
es with highly polarized and collinearly magnetized leads.
As was shown in Ref. 50 the above effect results in giant
peaks in the shot-noise spectral function, wherein the peak
heights are only limited by the rates of electronic spin flips.
This enables a nanomechanical spectroscopy of rare spin-
flip events, allowing spin-flip relaxation times as long as
10 µs to be detected.
The spin-dependence of electronic tunneling in magnet-
ic NEM devices permits an external magnetic field to be
used for manipulating not only electric transport but also
the mechanical performance of the device. This was
demonstrated in Refs. 51, 52. A theory of the quantum co-
herent dynamics of mechanical vibrations, electron charge
and spin was formulated and the possibility to trigger a
shuttle instability by a relatively weak magnetic field was
demonstrated. It was shown that the strength of the mag-
netic field required to control nanomechanical vibrations
decreases with an increasing tunnel resistance of the device
and can be as low as 10 Oe for gigaohm tunnel structures.
A new type of nanoelectromechanical self excitation
caused entirely by the spin splitting of electronic energy
levels in an external magnetic field was predicted in Ref. 54
for a suspended nanowire, where mechanical motion in a
magnetic field induces an electromotive coupling between
electronic and vibrational degrees of freedom. It was
shown that a strong correlation between the occupancy of
the spin-split electronic energy levels in the nanowire and
the velocity of flexural nanowire vibrations provides ener-
gy supply from the source of dc current, flowing through
the wire, to the mechanical vibrations thus making possible
stable, self-supporting bending vibrations. Estimations
made in Ref. 54 show that in a realistic case the vibration
amplitude of a suspended carbon nanotube (CNT) of the
order of 10 nm can be achieved if magnetic field of 10 T is
applied.
3.2. Spintro-mechanics of magnetic shuttle devices
New phenomena, qualitatively different from the electro-
mechanics of nonmagnetic shuttle systems, may appear in
magnetic shuttle devices in a situation when short-range
magnetic exchange forces become comparable in strength
to the long-range electrostatic forces between the charged
elements of the device [54]. There is convincing evidence
that the exchange field can be several tesla at a distance of
a few nanometers from the surface of a ferromagnet [55–58].
Because of the exponential decay of the field this means
that the force experienced by a single-electron spin in
the vicinity of magnetic electrodes can be very large. The-
se spin-dependent exchange forces can lead to various
“spintro-mechanical” phenomena.
Mechanical effects produced by a long-range electro-
static force and short-ranged exchange forces on a movable
quantum dot are illustrated in Fig. 2. The electrostatic force
acting on the dot, placed in the vicinity of a charged elec-
trode (Fig. 2(a)), is determined by the electric charge ac-
cumulated on the dot. In contrast, the exchange force in-
duced by a neighboring magnet depends on the net spin
accumulated on the dot. While the electrostatic force
Fig. 2. A movable quantum dot in a magnetic shuttle device can
be displaced in response to two types of force: (a) a long-range
electrostatic force causing an electromechanical response if the
dot has a net charge, and (b) a short-range magnetic exchange
force leading to “spintro-mechanical” response if the dot has a net
magnetization (spin). The direction of the force and displace-
ments depends on the relative signs of the charge and magnetiza-
tion, respectively. Reprinted with permission from [59], R.I.
Shekhter et al., Phys. Rev. B 86, 100404 (2012). 2012, Ameri-
can Physical Society.
780 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
changes its direction if the electric charge on the dot
changes its sign, the spin-dependent exchange force is in-
sensitive to the electric charge but it changes direction if
the electronic spin projection changes its sign. A very im-
portant difference between the two forces is that the elec-
trostatic force changes only as a result of injection of addi-
tional electrons into (out of) the dot while the spintronic
force can be changed due to the electron spin dynamics
even for a fixed number of electrons on the dot (as is the
case if the dot and the leads are insulators). In this case
interesting opportunities arise from the possibility of trans-
ducing the dynamical variations of electronic spin (in-
duced, e.g., by magnetic or microwave fields) to mechani-
cal displacements in the NEM device. In Ref. 59 a
particular spintro-mechanical effect was discussed — a
giant spin-filtering of the electron current (flowing through
the device) induced by the formation of what we shall call
a “spin-polaronic state”.
The Hamiltonian that describes the magnetic
nanomechanical SET device in Ref. 59 has the standard
form (its spin-dependent part depends now on the mechan-
ical displacement of the dot). Hence
lead tunnel dotH H H H= + + ,
where
†
leads
, ,
= ks ksks
k s
H a aσ σσ
σ
ε∑
describes electrons (labeled by wave vector k and spin
= ,σ ↑ ↓) in the two leads ( = ,s L R). Electron tunneling
between the leads and the dot is modeled as
†
tunnel
, ,
= ( ) h.c.s ks
k s
H T x a cσσ
σ
+∑ , (15)
where the matrix elements (0)( ) = exp ( / )s sT x T x λ (λ is
the characteristic tunneling length) depend on the dot posi-
tion x . The Hamiltonian of the movable single-level dot is
†
dot 0 0= [ sgn ( ) ( )] ,CH b b n J x U n nσ ↓↑
σ
ω + ε − σ +∑ (16)
where sgn ( , ) = 1↑ ↓ ± , CU is the Coulomb energy associat-
ed with double occupancy of the dot and the eigenvalues of
the electron number operators nσ is 0 or 1. The position
dependent magnitude ( )J x of the spin-dependent shift of
the electronic energy level on the dot is due to the ex-
change interaction with the magnetic leads. Here we ex-
pand ( )J x to linear order in x so that (0)( ) =J x J jx+ and
without loss of generality assume that (0) = 0J .
The modification of the exchange force, caused by
changing the spin accumulated on the dot, shifts the equi-
librium position of the dot with respect to the magnetic
leads of the device. Since the electron tunneling matrix
element is exponentially sensitive to the position of the dot
with respect to the source and drain electrodes one expects
a strong spin-dependent renormalization of the tunneling
probability, which exponentially discriminates between the
contributions to the total electrical current from electrons
with different spins. This spatial separation of dots with
opposite spins is illustrated in Fig. 3. While changing the
population of spin-up and spin-down levels on the dot (by
changing, e.g., the bias voltage applied to the device) one
shifts the spatial position x of the dot with respect to the
source/drain leads. It is important that the Coulomb block-
ade phenomenon prevents simultaneous population of both
spin states. If the Coulomb blockade is lifted the two spin
states become equally populated with a zero net spin on the
dot, = 0S . This removes the spin-polaronic deformation
and the dot is situated at the same place as a nonpopulated
one. In calculations a strong modification of the vibrational
states of the dot, which has to do with a shift of its equilib-
rium position, should be taken into account. This results in
a so-called Franck–Condon blockade of electronic tunnel-
ing [60,61]. The spintro-mechanical stimulation of a spin-
polarized current and the spin-polaronic Franck–Condon
blockade of electronic tunneling are in competition and
their interplay determines a nonmonotonic voltage depend-
ence of the giant spin-filtering effect.
To understand the above effects in more detail consider
the analytical results of Ref. 59. A solution of the problem
can be obtained by the standard sequential tunneling ap-
proximation and by solving a Liouville equation for the
density matrix for both the electronic and vibronic subsys-
tems. The spin-up and spin-down currents can be ex-
pressed in terms of transition rates (energy broadening of
the level) and the occupation probabilities for the dot elec-
tronic states. For simplicity we consider the case of a
strongly asymmetric tunneling device. At low bias voltage
and low temperature the partial spin current is
Fig. 3. Diagram showing how the equilibrium position of the
movable dot depends on its net charge and spin. The difference in
spatial displacements discriminates transport through a singly
occupied dot with respect to the electron spin. Reprinted with
permission from [59], R.I. Shekhter et al., Phys. Rev. B 86,
100404 (2012). 2012, American Physical Society.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 781
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
22
0 0
2
0
1exp sgn ( ) ,
2
L x xe
Iσ
Γ − − σ β ω λ
(17)
where 2
0 0= /xβ ω λ . In the high bias voltage (or tempera-
ture) regime, max { , } peV T E , where the polaronic blo-
ckade is lifted (but double occupancy of the dot is still pre-
vented by the Coulomb blockade), the current expression
takes the form
[ ]
2
0
2exp 2 1 2 sgn ( ) ,L
B
xe
I nσ
Γ
+ − σ β
λ
(18)
where Bn is Bose–Einstein distribution function. The scale
of the polaronic spin-filtering of the device is determined
by the ratio β of the polaronic shift of the equilibrium spa-
tial position of a spin-polarized dot and the electronic tun-
neling length. For typical values of the exchange interac-
tion and mechanical properties of suspended carbon
nanotubes this parameter is about 1–10. As was shown this
is enough for the spin filtering of the electrical current
through the device to be nearly 100% efficient. The tem-
perature and voltage dependence of the spin-filtering effect
is presented in Fig. 4. The spin-filtering effect and the
Franck–Condon blockade both occur at low voltages and
temperatures (on the scale of the polaronic energy; see
Fig. 4(a)). An increase of the voltage applied to the device
lifts the Franck–Condon blockade, which results in an ex-
ponential increase of both the current and the spin-filtering
efficiency of the device. This increase is blocked abruptly
at voltages for which the Coulomb blockade is lifted. At
this point a double occupation of the dot results in spin
cancellation and removal of the spin-polaronic segregation.
This leads to an exponential drop of both the total current
and the spin polarization of the tunnel current (Fig. 4(b)).
As one can see in Fig. 4(b) prominent spin filtering can be
achieved for realistic device parameters. The temperature
of operation of the spin-filtering device is restricted from
above by the Coulomb blockade energy. One may, however,
consider using functionalized nanotubes [62] or graphene
ribbons [63] with one or more nanometer-sized metal or
semiconductor nanocrystal attached. This may provide a
Coulomb blockade energy up to a few hundred kelvin,
making spin filtering a high-temperature effect [59].
3.3. Spintronics of shuttles
In this subsection we discuss the possibility to manipu-
late the spin of tunneling electrons by an external magnetic
field and how it can affect electron transport through a
nanoelectromechanical device. In the simplest model, we
assume that the left and right electrodes are fully spin po-
larized. The movable single level quantum dot (in the ab-
sence of a magnetic field) can vibrate in the gap between
two leads. A bias voltage is applied but electron transport
through the system is blocked since the source and drain
leads are fully spin polarized in opposite direction. An ex-
ternal magnetic field applied perpendicular to the direction
of the magnetization in the electrode leads to precession
of the electron spin of the quantum dot and as a conse-
quence the electron transport is unblocked. The Hamilto-
nian of the system has the form [52] of Eq. (1) with
†
leads = jk jk jkjkH c cΣ ε ( = , = ( , )j L R j→ ↑ ↓ ) and
† † † ††
0= ( ) ( ) ,
2QD
hH dx c c c c c c Uc c c cσ σ ↓ ↓↑ ↑↓ ↓↑ ↑
σ
ε − − + +∑
(19)
where 0= /Bh g Hµ ω is the dimensionless magnetic field.
To analyze this system we use the method described in
Sec. 2. A quantum master equation for the reduced density
matrix operator 0 0 | | 0ρ ≡ 〈 ρ 〉, | |↑ρ ≡ 〈↑ ρ ↑〉 , | |↓ρ ≡ 〈↓ ρ ↓〉 ,
and | |↑↓ρ ≡ 〈↑ ρ ↓〉 is obtained in analogy with the spinless
case
[ ]0 0
0
{ ( ), }
= , ( ) ( ),
2
L
R Rv
x
i H xd x x
t ↓
∂ρ Γ ρ
− + ρ − + Γ ρ Γ
∂
(20)
( ) { }1= , ( ), ,
2 2v
hi H i x
t
↓
+↓ ↓↑↓ ↑↓
∂ρ
− ρ + ρ − ρ − Γ ρ ∂
(21)
( )= ,
2v
hi H i
t
↑
↑ ↑↓ ↑↓
∂ρ
− ρ − ρ − ρ + ∂
0 2( ) ( ) ( ) ( ),L L R Rx x x x+ Γ ρ Γ + Γ ρ Γ (22)
1= , [ ] ( ),
2 2v
hi H i x
t
↑↓
+↓↑↓ ↑ ↑↓
∂ρ
− ρ + ρ − ρ − ρ Γ ∂
(23)
1= , [ ] ( ) ,
2 2v
hi H i x
t
↓↑
+↓↓↑ ↑ ↑↓
∂ρ
− ρ − ρ − ρ − Γ ρ ∂
(24)
[ ]2 2
2
{ ( ), }
= , ( ) ( ),
2
R
L Lv
x
i H xd x x
t ↑
∂ρ Γ ρ
− − ρ − + Γ ρ Γ
∂
(25)
where ( ) = ( ) ( )L Rx x x+Γ Γ + Γ . The set of Eqs. (20)–(25) is
derived in the high bias voltage limit:
0 0/2eV U− ε − ω .
In general, the problem can be solved in two limits with
and without the Coulomb blockade regime. In the Cou-
lomb blockade regime the second electron cannot tunnel
onto the quantum dot due to Coulomb repulsion. Hence the
Fig. 4. Spin polarization of the current through the model NEM-
SET device under discussion. Reprinted with permission from
[59], R.I. Shekhter et al., Phys. Rev. B 86, 100404 (2012). © 2012,
American Physical Society.
782 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
probability for double occupancy 2 0ρ → . First we focus
on the case without Coulomb blockade.
Here we repeat the analysis scheme for the evolution of
the stationary solution ( )W A+ for the probability of the
shuttle to vibrate with an amplitude A . Expanding the
function 1( )D A around = 0A one can get the condition for
the shuttle instability 2 2 2
thr< = (2 )/ ( )h d hγ γ Γ λ + Γ . As in
the case of spinless electron, the function W+ has a maxi-
mum at = 0A (stable point) when dissipation rate γ is
above the threshold value. In the opposite case the vibra-
tional ground state is unstable.
The positive bounded function
10 ( , ) = (2 ( ) ) /A h D A dβ − γ λ
has only one maximum and monotonically decreases for
large A . It was shown in [52] that if < 3h Γ, the function
0β has a maximum at = 0A , while for > 3h Γ, this func-
tion has a minimum at = 0A . The structure of the function
0β determines the behavior of the system in the parameter
space d h− (or hγ − ). There are several areas or phases. In
the first phase (vibronic), defined by 0/ < 1/ [max ( )]d h Aγλ β ,
the system is in the lowest vibrational state ( = 0A is a sta-
ble point). The shuttle phase is developed when thr<γ γ
and there is only one stable point at 0A ≠ . The third phase
is the mixed phase. It appears because the two above phas-
es become unstable if h exceeds the critical value 3Γ .
In the Coulomb blockade regime the same analysis
gives that 1( )D A is positive for all values of h if < 4 / 3Γ .
On the other hand, if > 4 / 3Γ , there is a range of magnetic
field strenghts where a shuttle instability does not occur.
In particular, when 1Γ this interval is 0 < < / 2h Γ .
This implies that in the adiabatic regime of charge trans-
port ( 1)Γ in weak magnetic field there is no instability
and the electrically driven electron shuttle is realized only
in strong magnetic fields.
3.4. Electron shuttle based on electron spin
In the previous subsection we studied the shuttle insta-
bility in the case of an electromechanical coupling between
the quantum dot and the leads. In the Coulomb blockade
regime a shuttle instability appears if an external magnetic
field h exceeds the critical value cr = 3h Γ . Here we will
study the shuttle instability in the case when the interaction
between the dot and the leads is due to a magnetic (ex-
change) coupling [53].
The Hamiltonian of the system is similar to the one con-
sidered in Sec. 3.3. The only difference is that the quantum
dot Hamiltonian reads
† †
0= ( )QDH a a a a↓↑ ↓↑ε + −
† † † †( )( ) ( )( )L RJ x a a a a J x a a a a↓ ↓↑ ↑↓ ↓↑ ↑− − − − −
† † † †( ) .
2
g H a a a a Ua a a a↓ ↓↑ ↑↓ ↓↑ ↑
µ
− + − (26)
In what follows we will consider the symmetrical case,
( ) = ( )R LJ x J x− and restrict ourselves to the Coulomb
blockade regime, 2
0/2 > | /2 |U e C eV − ε .
Following Ref. 52 one gets equations of motion for
the reduced density matrix operators 0 0 | | 0ρ ≡ 〈 ρ 〉,
| |↑ρ ≡ 〈↑ ρ ↑〉 , | |↓ρ ≡ 〈↓ ρ ↓〉 , and | |↑↓ρ ≡ 〈↑ ρ ↓〉:
0
0= [ , ]vi H
t
∂ρ
− ρ −
∂
0{ ( ), }/2 ( ) ( ) ,L R Rx x x↓− Γ ρ + Γ ρ Γ (27)
= [ , ] [ ( ), ]vi H i J x
t
↑
↑ ↑
∂ρ
− ρ + ρ −
∂
†
0( )/2 ( ) ( ) ,L Lih x x↑↓ ↑↓− ρ − ρ + Γ ρ Γ (28)
= [ , ] [ ( ), ]vi H i J x
t
↓
↓ ↓
∂ρ
− ρ − ρ +
∂
†( )/2 { ( ), }/2 ,Rih x ↓↑↓ ↑↓+ ρ − ρ − Γ ρ (29)
= [ , ] { ( ), }vi H i J x
t
↑↓
↑↓ ↑↓
∂ρ
− ρ + ρ +
∂
( )/2 ( )/2Rih x↓ ↑ ↑↓+ ρ − ρ − ρ Γ . (30)
In Eqs. (27)–(30) ( ) = exp ( 2 / )j x j xΓ Γ λ and ( )J x =
( ) ( )L RJ x J x= − . In what follows we assume a linear x-de-
pendence of ( )J x : ( ) ..., = 2 (0) > 0RJ x x J−α + α ′ .
The difference between our operator equations and the
corresponding equations in Ref. 52 (rewritten for the Cou-
lomb blockade case) is the appearance of terms induced by
the coordinate-dependent exchange interaction ( )J x . These
appear in Eqs. (27)–(30) as a commutator term for ↑ρ and
↓ρ and as an anti-commutator term for ↑↓ρ . In contrast to
the electrically driven shuttle, the driving force in our case
is strongly connected to the spin dynamics, which results
in a completely different dependence of the shuttle behav-
ior on magnetic field.
Both linear and nonlinear regimes of the shuttling dy-
namics can be conveniently analyzed by using the Wigner
function representation of the density operators [14]. This
approach allows one to calculate the Wigner distribution
function ( , )W x pρ for the vibrational degree of freedom to
lowest order in the small parameters α and 1/ λ for small
(compared to λ ) shuttle vibration amplitudes A . The rele-
vant Wigner function, (0) ( )W AΣ , averaged over the shuttle
phase ϕ ( = sinx A ϕ), solves the stationary Fokker–
Planck equation as in Eq. (11) with drift- and diffusion
coefficients containing the factors
2 3 2 2
1 2 2
0
3 3=
( , )3
h hD
Q hh
α Γ Γ + −
λ ΓΓ +
, (31)
2 22
1 0
0 2 2
0
( , ) ( , )
=
2 ( , )3
Q h Q hhD
Q hh
− α Γ + λ ΓΓ
ΓΓ +
, (32)
respectively, where
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 783
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
2
2 2 2 2 2 2
0 ( , ) = (1 2 ) ( 3 5) ,
4
Q h h hΓ
Γ − − Γ + Γ + − (33)
2 4
2 2
1
9 5( , ) = 1 (1 2 ) .
4 4
Q h h
Γ Γ
Γ + + + Γ −
(34)
In Eqs. (31)–(34) all energies are normalized with respect
to the energy quantum ω of the mechanical vibrations:
1ω → , /g H hµ ω → , ( )/ ( )J x J xω → , ( )/ ( )j jx xΓ ω → Γ
[ 2( ) = 2 | ( ) |j jx T xΓ πν are partial level widths].
For 1A the solution of Eq. (11) takes the form of a
Boltzmann distribution function, (0) exp ( )WΣ −β , where
2= /2A is the dot's vibrational energy, and 1/ β , where
2 2 2
2 2
1 0
2 3 3= ,
( , ) ( , )
h
Q h Q h−
αΓ − Γ −
β
λ α Γ + λ Γ
(35)
is an effective temperature. Since the functions 0Q and 1Q are
positive, the sign of the effective temperature is determined
by the relation between magnetic field, level width and vib-
ration quantum. In particular the effective temperature is ne-
gative at small magnetic fields, | | < cH H , where cg Hµ =
2 23( )= Γ + ω (reverting to dimensional variables).
A negative β implies that the static state of the dot
( = 0)A is unstable and that a shuttling regime of charge
transport ( 0A ≠ ) is realized. It is interesting to note that β
is finite even as 0h → . This apparent paradox may be re-
solved by considering the Fokker–Planck equation in its
time-dependent form and noting that the rate of change of
the oscillation amplitude at the instability is defined by the
coefficient 1D . This coefficient scales as 2
1( )D h h∝ as
0h → and therefore the shuttle phase is only realized for-
mally after an infinitely long time in this limit. As a function
of magnetic field 1D has a maximum, max 1
1 = 0.6( / )D −α λ Γ ,
at opt = 0.4h Γ . Therefore, optimal magnetic fields are in
the range 0.1 1− T if Γ =10–100 µeV. For high magnetic
fields, | | > cH H , there is no shuttling regime (at least not
with a small vibration amplitude, 1A ) and the vibronic
regime, corresponding to small fluctuations of the quantum
dot around its equilibrium position, is stable.
The amplitude of the shuttle vibrations that develop as
the result of an instability is still described by Eq. (11) for
the Wigner distribution function. However, for large am-
plitudes, 1A , the drift- and diffusion coefficients 2
1A D
and 0AD can no longer be evaluated analytically. Fortu-
nately, it is sufficient to know the amplitude- and magne-
tic-field dependence of 1D for a qualitative analysis. This is
because a positive value of the drift coefficient means that
energy is pumped into the dot vibrations, while a negative
value corresponds to damping (cooling) of the vibrations.
Therefore, magnetic fields for which 1( ) = 0D A and
1 ( ) < 0D A′ correspond to a stable stationary state of the dot
and a local maximum of the Wigner function. Based on this
picture one concludes (see Fig. 5) that at low magnetic
fields, 1< ch h , a shuttling regime with a large vibration am-
plitude is realized, while at high magnetic fields, 1> ch h ,
the situation is more complicated. Here one of two
1 2( < <c ch h h ; > ch h ) or three ( 2 < <c ch h h ) shuttling
regimes with different amplitudes can be stable depending
on the initial conditions. If the dot is initially in the static
state ( = 0A ) a stable shuttle regime only appears for
< ch h as already mentioned.
Thus the magnetic shuttle device acts in “opposite” way
as compared to electromechanical one. A particularly
transparent picture of how spintro-mechanics affects shut-
tle vibrations emerges in the limit of weak magnetic field
H and large electron tunneling rate ( )S DΓ between dot and
source- and drain electrodes. In order to explore this limit,
where 2( / ) /S DHΓ ω µ Γ and / 2ω π is the natural
vibration frequency of the dot, we focus first on the total
work done by the exchange force F as the dot vibrates
under the influence of an elastic force only. In the absence
of an external magnetic field the dot is in this case occu-
pied by a spin-up electron emanating from the source elec-
trode. This spin is a constant of motion and hence no elec-
trical current through the device is possible since only
spin-down states are available in the drain electrode. Dur-
ing the oscillatory motion of the dot the exchange force is
therefore always directed towards the source electrode
while its magnitude only depends on the position of the
dot, 0 ( )F F x= . As a result, no net work is done by the ex-
change force on the dot. This is because contributions are
positive or negative depending on the direction of the dot’s
motion and cancel when summed over one oscillation pe-
riod. A finite amount of work can only be done if the ex-
change force deviates from 0 ( )F x as a result of spin-flip
processes induced by the external magnetic field. Such a
deviation can be viewed as an additional random force HF
that acts in the opposite direction to 0 ( )F x . In the limit of
Fig. 5. Regions of positive and negative values of the increment
coefficient 1( , )D A h for = 10Γ . Solid (dashed) lines indicate
where the Wigner distribution function for the oscillation ampli-
tude A has a local maximum (minimum) and hence where the
stationary state 1[ ( , ) = 0]D A h is stable (unstable) with respect
to small perturbations.
784 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
large tunneling rate, /HΓ µ , and small vibration ampli-
tude a spin flip occurs with a probability 2( / ) /( )DH∝ µ ωΓ
during one oscillation period and is instantly accompanied
by the tunneling of the dot electron into the drain electrode,
thereby triggering the force HF . The duration of this force
is determined by the time 1/ ( ( ))St x tδ Γ it takes for the
spin of the dot to be “restored” by another electron tunnel-
ing from the source electrode.
The spin-flip induced random force 0 ( )HF F x= − is al-
ways directed towards the drain electrode. Hence, its effect
depends on the dot’s direction of motion: as the dot moves
away from the source electrode it will be accelerated,
while as it moves towards the source it will be decelerated.
Since a spin flip may occur at any point on the trajectory
one needs to average over different spin-flip positions in
order to calculate the net work done on the dot. The result,
which depends on the competition between the effect of
spin flips that occur at the same position but with the dot
moving in opposite directions, is nonzero because tδ is
different in the two cases. As the dot moves away from the
source electrode the tunneling rate to this electrode will
decrease while as the dot moves towards the source it will
increase. This means that the duration of spin-flip induced
acceleration will prevail over the one for deceleration. As a
result, in weak magnetic fields, the dot will accelerate with
time and one can expect a spintro-mechanical shuttle in-
stability in this limit.
The situation is qualitatively different in the opposite
limit of strong magnetic fields, where /HΓ µ and the
spin rotation frequency therefore greatly exceeds the tun-
neling rates. In this case the quick precession of the elec-
tron spin in the dot averages the exchange force to zero if
one neglects the small effects of electron tunneling to and
from the dot. If one takes corrections due to tunneling into
account (having in mind that the source electrode only
supplies spin-up electrons) one comes to the conclusion
that the average spin on the dot will be directed upwards.
This results in a net spintro-mechanical force in the direc-
tion opposite to that of the net force occurring in a weak
magnetic field limit. As a result, in strong magnetic fields
one expects on the average a deceleration of the dot.
Therefore, there will be no shuttle instability for such
magnetic fields.
As we have discussed above spin-flip assisted electron
tunneling from source to dot to drain in our device results
in a magnetic exchange force that attracts the dot to the
source electrode. It is interesting to note that this is contra-
ry to the effect of the Coulomb force in the same device.
Indeed, since the Coulomb force depends on the electric
charge of the dot it repels the dot from the source elec-
trode. Hence, while the dot is empty as the result of a spin-
flip assisted tunneling event from dot to drain, an “extra”
attractive Coulomb force QF is active. An analysis fully
analogous with our previous analysis of the “extra” repul-
sive magnetic exchange force HF leads to the conclusion
that the effect of the Coulomb force will be just the oppo-
site to that of the exchange force. This means that in the
Coulomb blockade regime in the limit of weak magnetic
field there is no shuttle instability, while in strong magnet-
ic fields electron shuttling occurs. As was shown the de-
tailed analysis confirms these predictions.
3.5. Mechanically assisted magnetic coupling between
nanomagnets
The mechanical force caused by the exchange interac-
tion represents only one effect of the coupling of magnetic
and mechanical degrees of freedom in magnetic nano-
electromechanical device. A complementary effect is the
of mechanical transportation of magnetization, which we
are going to discuss in this subsection.
In the magnetic shuttle device presented in Fig. 6, a fer-
romagnetic dot with total magnetic moment m is able to
move between two magnetic leads, which have total mag-
netization ,L RM . Such a device was suggested in Ref. 64
in order to consider the magnetic coupling between the
leads (which in their turn can be small magnets or
nanomagnets) produced by a ferromagnetic shuttle. It is
worth to point out that the phenomenon we are going to
discuss here has nothing to do with transferring electric
charge in the device and it is valid also for a device made
of nonconducting material. The main effect, which will be
in the focus of our attention, is the exchange interaction
between the ferromagnetic shuttle (dot) and the magnetic
leads. This interaction decays exponentially when the dot
moves away from a lead and hence it is only important
when the dot is close to one of the leads. During the peri-
odic back-and-forth motion of the dot this happens during
short time intervals near the turning points of the mechani-
cal motion. An exchange interaction between the magneti-
zations of the dot and a lead results in a rotation of these
Fig. 6. Single-domain magnetic grains with magnetic moments
LM and RM are coupled via a magnetic cluster with magnetic
moment m, the latter being separated from the grains by insulat-
ing layers. The gate electrodes induce an ac electric field, concen-
trated in the insulating regions. This field, by controlling the
heights of the tunnel barriers, affects the exchange magnetic cou-
pling between different components of the system. Reprinted
with permission from [64], L.Y. Gorelik et al., Phys. Rev. Lett.
91, 088301 (2003). 2003, American Physical Society.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 785
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
two magnetization vectors in such a way that the vector
sum is conserved. This is why the result of this rotation can
be viewed as a transfer of some magnetization ∆m from
one ferromagnet to the other. As a result, the magnetization
of the dot experiences some rotation around a certain axis.
The total angle φ of the rotation accumulated during the
time when the dot is magnetically coupled to the lead is an
essential parameter which depends on the mechanical and
magnetic characteristics of the device. The continuation of
the mechanical motion breaks the magnetic coupling of the
dot with the first lead but later, as the dot approaches the
other magnetic lead an exchange coupling is established
with this second lead with the result that magnetization
which is “loaded” on the dot from the first lead is “trans-
ferred” to this second lead. This is how the transfer of
magnetization from one magnetic lead to another is in-
duced mechanically. The transfer creates an effective cou-
pling between the magnetizations of the two leads. Such a
nonequilibrium coupling can be efficiently tuned by con-
trolling the mechanics of the shuttle device. It is particular-
ly interesting that the sign of the resulting magnetic inter-
action is determined by the sign of cos ( /2)φ . Therefore,
the mechanically mediated magnetic interaction can be
changed from ferromagnetic to antiferromagnetic by
changing the amplitude and the frequency of mechanical
vibrations [64].
4. Resonance spin-scattering effects. Spin shuttle as
a “mobile quantum impurity”
Many-particle effects add additional dimension to the
shuttling phenomena. These effects accompany electronic
tunneling between the gate electrodes and the moving
nanoisland. The common source of many-particle effects is
the so-called “orthogonality catastrophe” related to multi-
ple creation of electron–hole pairs both with parallel and
antiparallel spins [65,66] as a response of electronic gas in
the leads to single electron tunneling. The second-order
cotunneling processes under strong Coulomb blockade
result in effective indirect exchange between the shuttle
and the leads. This exchange is the source of strong scatter-
ing and the many-particle reconstruction of the electron
ensemble in the leads known as the Kondo effect. Various
manifestations of the Kondo effect in shuttling are re-
viewed in this section.
The Kondo effect in electron tunneling close to the uni-
tarity limit manifests itself as a sharp zero bias anomaly in
the low-temperature tunneling conductance. Many-particle
interactions renormalize the electron spectrum enabling
“Abrikosov–Suhl resonances” both for odd [67] and even
[68,69] electron occupations. In the latter case the reso-
nance is caused by the singlet-triplet crossover in the
ground state (see [70] for a review). In the simplest case
of odd occupancy a cartoon of a quantum well and a sche-
matic density of states (DoS) is shown in Fig. 7. For sim-
plicity we consider a case when the dot is occupied by one
electron (as in a SET transistor). The corresponding elec-
tronic level in the dot is located at an energy dE− , deep
beyond the Fermi level of the leads ( Fε ). The dot is in the
Coulomb blockade regime, and the corresponding charging
energy is denoted as CE . The Abrikosov–Suhl resonance
[71–73] at Fε arises due to multiple spin-flip scattering, so
that the narrow peak in the DoS is related mainly to the
spin degrees of freedom (see Fig. 7, upper right panel). The
width of this resonance is defined by the unique energy
scale, the Kondo temperature KT , which determines all
thermodynamic and transport properties of the SET device
through a one-parametric scaling [73]. The Breit–Wigner
(BW) width Γ of the dot level associated with the tunnel-
ing of dot electrons to the continuum of levels in the leads,
is assumed to be smaller than the charging energy CE ,
providing a condition for nearly integer valency regime.
Building on an analogy with the shuttling experiments
of Refs. 74 and 75, let us consider a device where an iso-
lated nanomachined island oscillates between two elec-
trodes. The applied voltage is assumed low enough so that
the field emission of many electrons, which was the main
mechanism of tunneling in those experiments, can be ne-
glected. We emphasize that the characteristic de Broglie
wave length associated with the dot should be much short-
er than typical displacements allowing thus for a classical
treatment of the mechanical motion of the nanoparticle.
The condition 0 B Kk Tω , necessary to eliminate deco-
herence effects, requires for, e.g., planar quantum dots with
the Kondo temperature 100KT mK, the condition
0 1ω GHz for oscillation frequencies to hold; this fre-
quency range is experimentally feasible [74,75]. The shut-
tling island is then to be considered as a “mobile quantum
impurity”, and transport experiments will detect the influ-
ence of mechanical motion on the differential conductance.
If the dot is small enough, then the Coulomb blockade
guarantees the single electron tunneling or cotunneling
regime, which is necessary for the realization of the Kondo
effect [70,76].
Fig. 7. Nanomechanical resonator with spin as a “mobile quan-
tum impurity”.
786 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
The above configuration is illustrated in the lower panel
of Fig. 7: the shuttle of nanoscale size is mounted at the
tight string. Its harmonic oscillations are induced by exter-
nal elastic force. Unlike the conventional resonance case
the resonance level belongs not to the moving shuttle but
develops as a many-body peak at the Fermi level of the
leads. When the shuttle moves between source (S) and
drain (D) (see the lower panel of Fig. 7), both the energy
dE and the width Γ acquire a time dependence. This time
dependence results in a coupling between mechanical,
electronic and spin degrees of freedom. If a source-drain
voltage sdV is small enough ( sd B KeV k T ) the charge
degree of freedom of the shuttle is frozen out while spin
flips play a very important role in cotunneling processes.
Namely, the Abrikosov–Suhl resonance is viewed as a
time-dependent Kondo cloud built up from conduction
electrons in the leads dynamically screening moving spin
localized at the shuttle. Since the electrons in the cloud
contain information about the same impurity, they are mu-
tually correlated. Thus, NEM providing a coupling be-
tween mechanical and electronic degrees of freedom intro-
duces a powerful tool for manipulation and control of the
Kondo cloud induced by the spin scattering and gives a
very promising and efficient mechanism for electrome-
chanical transduction on the nanometer length scale.
Cotunneling is accompanied by a change of spin projec-
tion in the process of charging/discharging of the shuttle
and therefore is closely related to the spin/charge pumping
problem [77].
A generic Hamiltonian for describing the resonance
spin-scattering effects is given by the same Anderson
model as above,
† † 2
0 , ,,
,
= [ ]k k d i Cik
k i
H a a E eEx d d E nσ α σ α σσσ α
α σ
ε + − +∑ ∑ ,
†( )
tunnel ,
,
= ( )[ h.c.] ,i
ik
ik
H T x a dα σσ α
σ α
+∑ (36)
where E is the electric field between the leads. The tunnel-
ing matrix element depends exponentially on the ratio of
the time-dependent displacement ( )x t and the electronic
tunneling length λ , see Eq. (15). The time-dependent
Kondo Hamiltonian for slowly moving shattle can be ob-
tained by applying a time-dependent Schrieffer–Wolff
transformation [78,79]:
†
,,
,
1= ( )[ ]
4K kk
k k
H t a aαα σσ σσ σ α′ ′ ′ ′ ′ ′σ α
ασ α σ′ ′ ′
+ δ∑ S , (37)
where
, 0( ) = ( ) ( )/( ( ))dt t t E tα α α α′ ′Γ Γ πρ
and †1
2= d dσ σσ σ′ ′S , 2
0( ) = 2 | ( ( )) |t T x tα αΓ πρ are level
widths due to tunneling to the left and right leads.
As long as the nanoparticle is not subject to an external
time-dependent electric field, the Kondo temperature is
given by [ ]0
0 0= exp ( )/(8 )B K Ck T D E− π Γ (for simplicity
we assumed that 0(0) = (0) =L RΓ Γ Γ ; 0D plays the role of
effective bandwidth). As the nanoparticle moves adiabati-
cally, 0 0ω Γ , the decoherence effects are small pro-
vided 0
0 B Kk Tω .
Let us first assume a temperature regime KT T (weak
coupling). In this case we can build a perturbation theory
controlled by the small parameter 0 0( ) ln [ /( )] < 1Bt D k Tρ
assuming time as an external parameter. The series of per-
turbation theory can be summed up by means of a renor-
malization group procedure [73,79]. As a result, the Kondo
temperature becomes oscillating in time:
0
( ) = ( )exp .
8 cosh (2 ( )/ )
C
B K
E
k T t D t
x t
π
− Γ λ
(38)
Neglecting the weak time-dependence of the effective
bandwidth 0( )D t D≈ , we arrive at the following expres-
sion for the time-averaged Kondo temperature:
20
2
0
( ( )/ )sinh= exp .
( ( )/ )4 sinh1 2
C
K K
E x tT T
x t
π λ
〈 〉 λΓ +
(39)
Here 〈...〉 denotes averaging over the period of the mecha-
nical oscillation. The expression (39) acquires an especial-
ly transparent form when the amplitude of the mechanical
vibrations A is small: A λ . In this case the Kondo tem-
perature can be written as 0= exp ( 2 )K KT T W〈 〉 − , with the
Debye–Waller-like exponent 2 2
0= ( ) /(8 )CW E x t−π 〈 〉 Γ λ ,
giving rise to the enhancement of the static Kondo tempe-
rature.
The zero bias anomaly (ZBA) in the tunneling conduct-
ance is given by
2
0 2 2
4 ( ) ( )3 1( ) = ,
8 [ ( ) ( )] [ln ( / ( ))]
L R
L R K
t t
G T G
t t T T t
Γ Γπ
Γ + Γ
(40)
where 2
0 = /G e h is a unitary conductance. Although the
central position of the island is most favorable for the BW
resonance ( =L RΓ Γ ), it corresponds to the minimal width
of the Abrikosov–Suhl resonance. The turning points cor-
respond to the maximum of the Kondo temperature given
by the Eq. (38) while the system is away from the BW re-
sonance. These two competing effects lead to the effective
enhancement of G at high temperatures (see Fig. 8).
Summarizing, it was shown in Ref. 80 that Kondo shut-
tling in a NEM-SET device increases the Kondo tempera-
ture due to the asymmetry of coupling at the turning points
compared to at the central position of the island. As a re-
sult, the enhancement of the differential conductance in the
weak coupling regime can be interpreted as a pre-cursor of
strong electron–electron correlations appearing due to for-
mation of the Kondo cloud.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 787
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
Next we turn to the strong coupling regime, KT T .
We consider this regime for an oscillating cantilever with a
nanotip at its end (Fig. 9). Then the motion of a shuttle in
y direction is described by the Newton equation which we
rewrite in a form
20
0
0
1=y y y F
Q m
ω
+ + ω , (41)
where 0 = /k mω is the oscillator frequency of free canti-
lever, 0Q is the quality factor. F is the Lorentz force acting
on moving cantilever in perpendicular magnetic field
= = (0, ,0) .L F⋅ ×F I B (42)
Here L is the length of the cantilever, I is the current
through the system.
In this configuration the Kondo cloud induced by spin
scattering is formed both in the immovable part of the set-
up (drain electrode) and in the oscillating cantilever. The
current I subject to a constant source-drain bias sdV can be
separated in two parts: a dc current associated with a time-
dependent dc conductance and an ac current related to the
periodic motion of the shuttle. While the dc current is
mostly responsible for the frequency shift, the ac current
gives an access to the dynamics of the Kondo cloud and
provides information about the kinetics of its formation. In
order to evaluate both contributions to the total current we
rotate the electronic states in the leads in such a way that
only one combination of the wave functions is coupled
to the quantum impurity. The cotunneling Hamiltonian
may be rationalized by means of the Glazman–Raikh rota-
tion, parametrized by the angle tϑ defined by the relation
tan = | ( ) / ( ) |t R Lt tϑ Γ Γ .
Both the ac and dc contributions to the current can be
calculated by using Nozière’s Fermi-liquid theory (see [81]
for details). The ac contribution, associated with the time
dependence of the Friedel phase σδ [82], is given by
0
2
0 0
tanh (2[ ( ) ]/ )( )( ) =
8 ( ) (2[ ( ) ]/ )cosh
C sd
ac
B K
eE eV y t yy tI t
k T t y t y
− λ
λ Γ − λ
(43)
( 0exp (4 / ) = (0)/ (0)R Ly λ Γ Γ ). The Eq. (43) acquires a sim-
ple form if we assume that the size of Kondo cloud
( ( )) = /( ( ( )))K F B KR y t v k T y t where Fv is a Fermi veloci-
ty. According to Nozières [81], the Friedel phase σδ can be
Taylor-expanded in the vicinity of its resonance value
0 = /2σδ π as
( ( )) ( ) ( ( ))
( ) =
2
sd K B K
F F
eV R y t g B R y t
t
v vσ
µ σπ
δ + +
(44)
and, therefore, ( )/ ( )/Kd dt y dR y dy↓↑δ + δ ∝ . As a result,
0
( )( )( ) = 2 .K
ac sd
F
dR yy tI t G V
v dy
(45)
Thus, the ac current generated in the device due to the me-
chanical motion of the shuttle contains information about
spatial variation of the Kondo cloud.
The “ohmic” dc contribution is fully defined by the adi-
abatic time dependence of the Glazman–Raikh angle
2 2
0( ) = sin 2 sindc sd tI t G V σ
σ
ϑ δ∑ . (46)
As a result, the ac contribution to the total current can be
considered as a first nonadiabatic correction:
ad
tot ad (0)
0
= ( ( ))
16
C
B K
dI E
I I y t y
dy k T
π
−
Γ
, (47)
where 2
ad 0 0= 2 cosh (2[ ( ) ]/ )sdI G V y t y− − λ and (0)
KT is
the Kondo temperature at the equilibrium position. The
small correction to the adiabatic current in (47) may be
Fig. 8. Differential conductance G of a Kondo shuttle for which
0 / 0.4CEΓ = . The solid line denotes G for a shuttle with L RΓ = Γ ,
A=λ , the dashed line shows G for a static nanoisland with
=L RΓ Γ , A=0, the dotted line gives G for LΓ / RΓ =0.5, A=0.
The inset shows the temporal oscillations (here 0Ω ≡ ω ) of KT
for small A=0.05 λ (dotted line) and large A= 2.5 λ (solid line)
shuttling amplitudes. Reprinted with permission from [80], M.N.
Kiselev et al., Phys. Rev. B 74, 233403 (2006). 2006, Ameri-
can Physical Society.
Fig. 9. Shuttling quantum dot mounted on a moving metallic
pendulum. Magnetic field B is applied along z axis.
788 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7
Electronic spin working mechanically
considered as a first term in the expansion over the small
nonadiabatic parameter 0 1ω τ , where τ is the retardation
time associated with the inertia of the Kondo cloud. Using
such an interpretation one gets
(0)
0= / (16 )C B KE k Tτ π Γ .
Equation (47) allows one to obtain information about
the dynamics of the Kondo clouds from an analysis of an
experimental investigation of the mechanical vibrations.
The retardation time associated with the dynamics of the
Kondo cloud is parametrically large compared with the
time of formation of the Kondo cloud = / ( )K B Kk Tτ and
can be measured owing to a small deviation from adia-
baticity. Also we would like to emphasize a supersen-
sitivity of the quality factor to a change of the equilibrium
position of the shuttle characterized by the parameter u
(see Fig. 10). The influence of strong coupling between
mechanical and electronic degrees of freedom on the me-
chanical quality factor has been considered in Ref. 82. It
has been shown that both suppression 0>Q Q and en-
hancement 0<Q Q of the dissipation of nanomechanical
vibrations (depending on external parameters and the equi-
librium position of the shuttle) can be stimulated by Kondo
tunneling. The latter case demonstrates the potential for a
Kondo induced electromechanical instability.
In order to describe these instability, one should discuss
the contribution of “Kondo force” KF to the right hand
side part (42) of Eq. (41). This force consists of two com-
ponents [83]:
ret
22
0 0
=
( )cosh
K
KF
y y
α + α
−
− ω λ
, (48)
where
0
( )
=
8
C B K
K
E k T tπ
α
Γ λ
, (49)
[1 tanh ( )]/20ret 0 bias 0 ret = 2 tanh ( ) e y yyG V BL y y −β + −α − τ .
Here 0= /4CEβ π Γ is the coupling strength of electronic
states. The first term stems from the Kondo cloud adiabati-
cally following the change of ( )KT t induced by the moving
shuttle in the source electrode and metallic cantilever. The
second term describes the temporal retardation related to dy-
namics of Kondo cloud with the characteristic time retτ =
min
0 /(2 )B Kk T= ω β . The time-dependent Kondo tempera-
ture in the strong coupling limit at min
KT T is given by
min
0( ) = exp [1 tanh ( ( ) )]) .
2B K B Kk T t k T y t yβ + −
(50)
The min
B Kk T plays the role of the cutoff energy for Kondo
problem.
The instability is controlled by the bias biasV entering
retα . Figure 11 illustrates two regimes of Kondo shuttling.
Namely, at small bias the Kondo force controlled by exter-
nal fields further damps the oscillator, and we obtain an
efficient mechanism of cooling the nanoshuttle. On the
other hand, at biasV above some treshold value, the contri-
bution of the Kondo force enhances the oscillations, and
we arrive at the nonlinear steady state regime of selfsus-
tained oscillations.
Summarizing, we emphasize that the Kondo phenome-
non in single electron tunneling gives a very promising and
Fig. 10. (Color online) Time dependence of the current 0I for
different values of asymmetry parameter 0= /u x λ . Here red,
blue and black curves correspond to = 0.5; 1.0; 1.5u . For all
three curves shuttle oscillates with amplitude max =x λ ,
min 3
0 /( ) = 10B Kk T −ω , min min
bias| | /( ) = /( ) = 0.1B K B B KeV k T g B k Tµ
with (0) = 2 KKT , 4/ = 10L −λ . Reprinted with permission from
[82], M.N. Kiselev et al., Phys. Rev. Lett. 110, 066804 (2013).
2013, American Physical Society.
Fig. 11. (Color online) (a) Amplitude dynamics at different va-
lues of the dimensionless force α (see details in the text). Insets:
time trace of the oscillation at two different fixed point indicated
by arrow. (b) Saturation amplitude as a function of dimensionless
force. Different colors denote initial conditions near (black dots)
and far (red dots) from the equilibrium position 0y . Insets: ampli-
tude envelope as a function of dimensionless time calculated by
using Eq. (49). The parameter α varies from = 0α (black)
to = 0.1α (magenta). The equations are solved for the following
set of parameters: = 8β , 5= 10−γ , 0 = 0.5y and min
0 /( )B Kk Tω =
310−= .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 7 789
R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, S.I. Kulinich, A.V. Parafilo, K. Kikoin, and M. Jonson
efficient mechanism for electromechanical transduction on
a nanometer length scale. Measuring the nanomechanical
response on Kondo-transport in a nanomechanical single-
electron device enables one to study the kinetics of the
formation of Kondo-screening and offers a new approach
for studying nonequilibrium Kondo phenomena. The Kon-
do effect provides a possibility for superhigh tunability of
the mechanical dissipation as well as supersensitive detec-
tion of mechanical displacement.
5. Conclusions
During the last several years there has been significant
activity in the study of nanoelectromechanical shuttle
structures. In this review we concentrate on description of
the influence of spin-related effects on the functionality of
shuttle devices. In particular, we emphasize the importance
of electronic spin in shuttle devices made of magnetic ma-
terials. Spin-dependent exchange forces can be responsible
for a qualitatively new nanomechanical performance open-
ing a new field of study that can be called spintro-
mechanics. Electronic many-body effects, appearing be-
yond the weak tunneling approach, result in single electron
shuttling assisted by Kondo-resonance electronic states.
The possibility to achieve a high sensitivity to coordinate
displacement in electromechanical transduction along with
the possibility to study the kinetics of the formation of
many-body Kondo states has also been demonstrated.
There are still a number of unexplored shuttling re-
gimes and systems, which one could focus on in the near-
est future. In addition to magnetic shuttle devices one
could explore hybrid structures where the source/drain and
gate electrodes are hybrids of magnetic and superconduct-
ing materials. Then one could expect spintro-mechanical
actions of a supercurrent flow as well as superconducting
proximity effects in the spin dynamics in magnetic NEM
devices. An additional direction is the study of shuttle op-
eration under microwave radiation. In this respect micro-
wave assisted spintro-mechanics is of special interest due
to the possibility of microwave radiation to resonantly flip
electronic spins. As in ballistic point contacts such flips
can be confined to particular locations by the choice of
microwave frequency, allowing for external tuning of the
spintro-mechanical dynamics of the shuttle.
Acknowledgments
Financial support from the Swedish VR, and the Korean
WCU program funded by MEST/NFR (R31-2008-000-
10057-0) is gratefully acknowledged. This research was
supported in part by the Project of Knowledge Innovation
Program (PKIP) of Chinese Academy of Sciences, Grant
No. KJCX2.YW.W10. I.V.K. and A.V.P. acknowledge fi-
nancial support from the National Academy of Sciences of
Ukraine (grant No. 4/13-N). I.V.K. thanks the Department
of Physics at the University of Gothenburg for hospitality.
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1. Introduction
2. Shuttling of single electrons
2.1. Shuttle instability in the quantum regime of Coulomb blockade
3. Electro- and spintro-mechanics of magnetic shuttle devices
3.1. Spin-controlled shuttling of electric charge
3.2. Spintro-mechanics of magnetic shuttle devices
3.3. Spintronics of shuttles
3.4. Electron shuttle based on electron spin
3.5. Mechanically assisted magnetic coupling between nanomagnets
4. Resonance spin-scattering effects. Spin shuttle as a “mobile quantum impurity”
5. Conclusions
Acknowledgments
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/ENU (Use these settings to create Adobe PDF documents that are to be checked or must conform to PDF/A-1b, an ISO standard for the long-term preservation \(archival\) of electronic documents. For more information on creating PDF/A compliant PDF documents, please refer to the Acrobat User Guide. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
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>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|