Magnetically controlled single-electron shuttle
A theory of single-electron shuttling in an external magnetic field in nanoelectromechanical system with magnetic leads is presented. We consider partially spin-polarized electrons in the leads and electron transport in both the Coulomb blockade regime and in the limit of large bias voltages when th...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2015
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/122023 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Magnetically controlled single-electron shuttle / O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2015. — Т. 41, № 1. — С. 90-95. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-122023 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1220232017-06-27T03:02:56Z Magnetically controlled single-electron shuttle Ilinskaya, O.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань A theory of single-electron shuttling in an external magnetic field in nanoelectromechanical system with magnetic leads is presented. We consider partially spin-polarized electrons in the leads and electron transport in both the Coulomb blockade regime and in the limit of large bias voltages when the Coulomb blockade is lifted. The influence of the degree of spin polarization on shuttle instability is considered. It is shown that there is certain degree of spin polarization above which the magnetic field ceases to control electron transport. In the Coulomb blockade regime the dependence of the threshold magnetic field, which separates the “shuttle” and vibron regimes, on the degree of polarization is evaluated. The possibility of re-entrant transitions to the shuttle phase is discussed. 2015 Article Magnetically controlled single-electron shuttle / O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2015. — Т. 41, № 1. — С. 90-95. — Бібліогр.: 11 назв. — англ. 0132-6414 PACS: 81.07.Oj, 72.25.–b, 73.23.Hk http://dspace.nbuv.gov.ua/handle/123456789/122023 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань |
| spellingShingle |
Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань Ilinskaya, O.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. Magnetically controlled single-electron shuttle Физика низких температур |
| description |
A theory of single-electron shuttling in an external magnetic field in nanoelectromechanical system with magnetic leads is presented. We consider partially spin-polarized electrons in the leads and electron transport in both the Coulomb blockade regime and in the limit of large bias voltages when the Coulomb blockade is lifted. The influence of the degree of spin polarization on shuttle instability is considered. It is shown that there is certain degree of spin polarization above which the magnetic field ceases to control electron transport. In the Coulomb blockade regime the dependence of the threshold magnetic field, which separates the “shuttle” and vibron regimes, on the degree of polarization is evaluated. The possibility of re-entrant transitions to the shuttle phase is discussed. |
| format |
Article |
| author |
Ilinskaya, O.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. |
| author_facet |
Ilinskaya, O.A. Kulinich, S.I. Krive, I.V. Shekhter, R.I. Jonson, M. |
| author_sort |
Ilinskaya, O.A. |
| title |
Magnetically controlled single-electron shuttle |
| title_short |
Magnetically controlled single-electron shuttle |
| title_full |
Magnetically controlled single-electron shuttle |
| title_fullStr |
Magnetically controlled single-electron shuttle |
| title_full_unstemmed |
Magnetically controlled single-electron shuttle |
| title_sort |
magnetically controlled single-electron shuttle |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2015 |
| topic_facet |
Актуальные проблемы магнитного резонанса и его приложений: Анатоль Абрагам, Евгений Завойский, Казань |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/122023 |
| citation_txt |
Magnetically controlled single-electron shuttle / O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, M. Jonson // Физика низких температур. — 2015. — Т. 41, № 1. — С. 90-95. — Бібліогр.: 11 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT ilinskayaoa magneticallycontrolledsingleelectronshuttle AT kulinichsi magneticallycontrolledsingleelectronshuttle AT kriveiv magneticallycontrolledsingleelectronshuttle AT shekhterri magneticallycontrolledsingleelectronshuttle AT jonsonm magneticallycontrolledsingleelectronshuttle |
| first_indexed |
2025-07-08T20:59:47Z |
| last_indexed |
2025-07-08T20:59:47Z |
| _version_ |
1837113947543568384 |
| fulltext |
© O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1, pp. 90–95
Magnetically controlled single-electron shuttle
O.A. Ilinskaya
1
, S.I. Kulinich
1
, I.V. Krive
1,2
, R.I. Shekhter
3
, and M. Jonson
3,4
1
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine,
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: kulinich@ilt.kharkov.ua
2
Physical Department, V.N. Karazin National University, Kharkov 61077, Ukraine
3
Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden
4
SUPA, Institute of Photonics and Quantum Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK
Received November 4, 2014, published online November 24, 2014
A theory of single-electron shuttling in an external magnetic field in nanoelectromechanical system with
magnetic leads is presented. We consider partially spin-polarized electrons in the leads and electron transport in
both the Coulomb blockade regime and in the limit of large bias voltages when the Coulomb blockade is lifted.
The influence of the degree of spin polarization on shuttle instability is considered. It is shown that there is cer-
tain degree of spin polarization above which the magnetic field ceases to control electron transport. In the Cou-
lomb blockade regime the dependence of the threshold magnetic field, which separates the “shuttle” and vibron
regimes, on the degree of polarization is evaluated. The possibility of re-entrant transitions to the shuttle phase is
discussed.
PACS: 81.07.Oj Nanoelectromechanical systems (NEMS);
72.25.–b Spin-polarized transport;
73.23.Hk Coulomb blockade; single-electron tunneling.
Keywords: nanoelectromechanical systems, Coulomb blockade, spin-polarized transport.
1. Introduction
Electron transport through a quantum dot (QD) in me-
chanically “soft” systems can be realized as shuttling of
electrons [1] (see also the reviews [2–4]). The shuttle re-
gime of charge transport is characterized by a strong en-
hancement of the electrical current at a certain bias volt-
age, which determines the threshold of shuttle instability.
In an ideal nonmagnetic system the threshold voltage in the
weak tunneling limit under certain conditions is deter-
mined only by the frequency of QD vibrations [5]. In real-
istic systems, when dissipation and defects in the mechani-
cal subsystem are present, the threshold voltage depends
on the friction coefficient and the characteristics of the
pinning potential. In this case the shuttle electrical current
abruptly appears when the QD is de-pinned by external
sources (microwave electromagnetic or acoustic fields) and
a small bias voltage drives the system to the regime of self-
sustained mechanical vibrations (see, e.g., the experiment
[6]). Although electron shuttling in the Coulomb blockade
regime has not been observed yet, the experimental realiza-
tion of a single electron shuttle is expected in the nearest
future.
It has been predicted [7,8] that in magnetic nano-
electromechanical systems the tunnelling of spin-polarized
electrons could be sensitive to an external magnetic field.
In particular, in an idealized situation, when electrons in
the leads are 100% spin-polarized and the source and drain
leads have opposite polarization, the electrical current is
blocked in the absence of an external magnetic field (“spin
blockade” [9]). It was shown [7] that even a small magnet-
ic field can trigger a shuttle instability in magnetic nano-
electromechanical systems. In principle, this mechanism
allows one to realize magnetically controlled electron
transport in single-electron transistors. In Ref. 7 the calcu-
lations were performed in the limit of high voltages
V when the Coulomb blockade is lifted. In this case
the shuttle instability, in the absence of dissipation, appears
in arbitrarily small magnetic fields. The magnetic field,
however, strongly influences the increment ( ) > 0r h of the
exponentional growth of classical shuttle coordinate
exp( )cx rt [7].
Magnetically controlled single-electron shuttle
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1 91
The purpose of the present paper is to generalize the
model of Ref. 7 to a more realistic situation when electrons
in the leads are not fully spin-polarized (Fig. 1). We also
consider both the Coulomb blockade ( ,eV U U is the
charging energy) and V regimes of electron
transport. We show that in the Coulomb blockade regime
there is a threshold magnetic field which separates the vi-
bronic (small oscillations around the equilibrium position
of QD), th< ,H H and shuttling, th> ,H H regimes of
quantum dot vibrations.
The dependence of the threshold magnetic field on the
degree of electron polarization is the main result of our
paper. It is shown that the threshold magnetic field de-
creases with the decrease of the degree of polarization, and
at certain value of polarization (numerically 64%) the
threshold field vanishes.
The paper is organized as follows. In Sec. 2 we general-
ize the model of Ref. 7 to the case of partially spin-
polarized electrons in the leads. In Sec. 3 an analytic solu-
tion for the increment of exponentional growth of shuttle
coordinate is obtained. We discuss the shuttle instability in
the absence of a Coulomb blockade and in the Coulomb
blockade regime in Sec. 4. In the Conclusion section we
summarize the main results of our paper.
2. Hamiltonian and equations of motion
The Hamiltonian ˆ ˆ ˆ ˆ ˆ= l d tv of our sys-
tem consists of four terms. The first term, ˆ ,l describes
noninteracting electrons in the leads,
†
, , , ,, ,
, ,
ˆ =l k kk
k
a a . (1)
Here the operator
†
, ,, ,
( )kk
a a creates (destroys) an
electron with momentum k and the spin projection
= ( , ) in the lead , ,= ( , ) = ( 1,1); kS D is the
electron energy.
The second term is the quantum dot Hamiltonian, ˆ ,d
which reads
†
0
ˆ = ( )d e x a a
† † † †
( ) .
2
g H
a a a a Ua a a a (2)
It describes the single electron state in the dot and its cou-
pling to an electric field and a magnetic field H ( is
the Bohr magneton, g is the gyromagnetic ratio). In
Eq. (2) † ( )a a is the creation (annihilation) operator for
electron on the dot. The intra-dot electron correlations are
characterized by the Coulomb energy U.
Vibrations of the dot are described by the harmonic os-
cillator Hamiltonian
2 2 2ˆ ˆˆ = ,
2 2
p m x
m
v (3)
where x̂ is the dot displacement operator, p̂ is the cano-
nical conjugated momentum ˆ ˆ([ , ] = ),x p i m is the mass
and is the vibrational frequency of the dot. The last term
in our Hamiltonian represents spin-conserving tunnelling
of electrons between dot and leads,
†
, , ,
, ,
ˆ ˆ= ( ) H.c.t k
k
T x a a (4)
Here , ,ˆ ˆ( ) = exp( / )T x T x is the position-dependent
tunnelling amplitude, being the tunnelling length. The
electrons in each lead are held at a constant electrochemi-
cal potential , = | | /2S D e V (relative to the Fermi lev-
el), where > 0V is the bias voltage. The electron density
of states =j in the leads is assumed to be independent
of energy.
To solve the problem, one needs to know the evolution
of a reduced density matrix operator , which describes
the vibrational degree of freedom coupled to a single elect-
ronic dot state. The electronic state is spanned by the
four basis vectors | 0 ,
†
| = | 0 ,a
†
| = | 0 ,a and
† †
| 2 = | 0 .a a We first consider the Coulomb blockade
regime, < ,eV U where the tunnelling of a second electron
onto the dot is blocked by the Coulomb interaction
2( 0).
It is convenient to introduce dimensionless variables for
time, ,t t dot displacement 0/x x x (where
0 = /x m is the zero-point oscillation amplitude), mo-
mentum 0 /px p and various characteristic energies,
1, / ,g H h 0 / ,eEx d ( )/ ( )x x
2
,( ( ) = 2 | ( ) | exp(2 / )x T x x are partial level
widths).
Fig. 1. Sketch of the nanomagnetic device studied: a movable
quantum dot with a single spin-degenerate electron level is cou-
pled to two partially spin-polarized leads. and are the tunne-
ling rates for the two spin projections (we assume that ). The
potential difference S – D = |e|V between the leads is due to a
bias voltage V. An external magnetic field H induces flips bet-
ween the spin-up and spin-down states on the dot.
O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson
92 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1
Following Ref. 7 one gets the equations of motion for
the reduced density matrix operators 0 0 | | 0 ,
| | , | | and | | . These
equations are
0
0 0
1ˆ= [ , ] { ( ) ( ), }
2
S Si dx x x
t
v
( ) ( ) ( ) ( ),D D D Dx x x x (5)
†ˆ= [ , ] ( )
2
ih
i
t
v
0
1
( ) ( ) { ( ), },
2
S S Dx x x (6)
†ˆ= [ , ] ( )
2
ih
i
t
v
0
1
( ) ( ) { ( ), },
2
S S Dx x x (7)
ˆ= [ , ] ( )
2
ih
i
t
v
( ( ) ( )) ( ( ) ( )) .
2 2
S D S Dx x x x (8)
It is easy to check that for the case of fully spin-
polarized electrons in the leads, Eqs. (5)–(8) are reduced to
the equations given in Ref. 7. In what follows we will re-
strict ourselves to the symmetrical case, ,S D
= .S D
We are interested in the classical motion of the dot. By
using Eqs. (5)–(8) it is easy to get the classical equations of
motion for coordinate and momentum:
ˆ= Tr ( ) = ,c
c
x
x p
t t
(9)
0ˆ= Tr ( ) = Tr .c
c
p
p x d
t t
(10)
Therefore one needs to know the equations of motion for
the zeroth moments, = Tri iR (the index i runs over all
the sub-indices in Eqs. (5)–(8)).
The dynamics of the zeroth moments is coupled to the
dynamics of the first moments, which in turn are coupled
to higher moments. We will decouple at the level of the
first moments by using the rule ˆTr{ } Tr ,i c ix x where
cx is the classical shuttle coordinate. In addition to re-
stricting our study to the vibrational dynamics near the
ground state we will assume the parameters , 1/d to be
small and linearize all equations with respect to the classi-
cal displacement .cx
It is convenient to introduce the following linear com-
binations of ,iR
0 0 1 2= Tr , = 1 Tr , = Tr ( ),R R R i
3 = Tr ( ).R (11)
Using the approximations described the equations of mo-
tion for the zeroth moments take the form
0
0
2
= 2
R x
R
t
1
2 2
( ) 1 1 ,
x x
R (12)
1
0 1
2 2
= 1 1
R x x
R R
t
2
2
1 ,
2
h x
R (13)
2
0 1 2
2
= (1 2 ) 1 .
2
R x
h R R R
t
(14)
(Note that the equation for 3R is decoupled from the other
equations and not relevant in what follows.)
3. Analytical solution
For small vibrations an analytical solution can be found
by perturbation theory in terms of the small parameters
= { , 1/ }.d We solve these equations by perturbation
expansions,
(0) (1)
( ) = ( ) ...,i i iR t R R t (15)
where
( )n
iR is of nth order in . It is evident from
Eqs. (12)–(14) that the functions
(0)
iR do not depend on
time. Hence,
2 2 2
(0) (0)
0 1
2
= , = ,
h h
R R
(0)
2
2( )
= ,
h
R (16)
where
2 2 2= 3 .h
It is convenient to define the vector-function | =R
(1) (1) (1)
0 1 2( , , ) .TR R R Then to first order in perturbation
theory one has
| 2ˆ= | ( ) | ,cA x t
t
R
R e (17)
Magnetically controlled single-electron shuttle
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1 93
where
(2 ) 0
ˆ /2
2 ( )/2
A h
h h
, (18)
and the vector | e is defined as follows:
2
2 2
2 2
2( )( )
1
| = 2 ( )
( )
h
h
h
e . (19)
Consequently, the eigenfrequencies of the shuttle vibra-
tions can be found from the equation
2
ˆ ( )
02
( ) 2
( ) = ( ) | e | ,
t
A t tc
c c
x t d
x t dt x t
t
e e (20)
where 0| = (1,0,0) .T
e
We are interested in the sign of the imaginary part of the
correction to the shuttle eigenfrequency, exp( ),cx i t
=1 ( / ) ,d due to coupling with the leads (the incre-
ment of exponentional growth is = Im > 0).r i It follows
from Eq. (20) that this correction takes the form
1
0
1
= , (21)
where
2 2 2 2 2
1 = ( )[2( ) 2 ( 2 ) ( )]h h h
2 2 2 2 2 2[3 ( ) 2 2 ( )]i h , (22)
and
2 2 2
0 = (5 3 )
2
h
2 2 2(1 2 3 2 ).i h (23)
Therefore, the condition for being in the shuttle domain is
that the inequality
6 4 2
4 2 0 > 0,h C h C h C (24)
is fulfilled. Here the coefficients iC take the form
2 2
4 = 2( 1),
2
C (25)
4 4 2 2
2 2 2 2
2
3
= ( ) 5 1,
4 2
C
(26)
2 2 2 2
4 4 2 2
0
5
= ( ) ( ) .
4 4 2
C
(27)
In the next section, based on this inequality, we will
discuss the specific features of the shuttle domain.
4. Analysis of the solution
4.1. Shuttle dynamics in the Coulomb blockade regime
The inequality (24) defines the shuttle instability do-
main. In Fig. 2 we plot the extent of this domain in the
( ,h)-plane for several values of . The case of fully spin-
polarized leads ( = 0) was considered in detail in Ref. 10:
When 0< (4/3) (now we return to dimensional varia-
bles), only the “shuttle phase” is stable (for arbitrary values
of h). If 0 0(4/3) < < 2 , there is certain interval in
h when the “vibronic phase” is stable. When 0> 2 ,
the transformation from the vibronic to the shuttle region
occurs at the threshold magnetic field, thh (for
1).
Increasing from zero, the shuttle domain of electron
transport expands while the vibronic domain becomes nar-
rower and vanishes completely for a definite value
( / )m = 0.22 (see Fig. 3). The corresponding critical de-
gree of spin polarization, defined as
1 /
= ,
1 /
(28)
is therefore 64%.m If the spin polarization is lower
than this value the magnetic field ceases to cause any tran-
sition between the vibronic and shuttle phases.
The threshold magnetic field thh is plotted as a func-
tion of in Fig. 3. This function has a vertical tangent at
Fig. 2. Schematic dependence of magnetic field h on the tunnel-
ing rate (solid lines) at the border between the shuttle- and vi-
bronic domains for (a) = 0 (full spin polarization), (b) = b and
(c) = c, where c > b. In each case the vibronic domain is be-
low (above) the upper (lower) branch of the border line. With an
increase of the shuttle domain expands and eventually the vi-
bronic domain vanishes.
O.A. Ilinskaya, S.I. Kulinich, I.V. Krive, R.I. Shekhter, and M. Jonson
94 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1
the point M, which defines ( / ) .m The point M divides
the plotted curve into an upper and a lower branch. Part of
the lower branch is dashed to indicate that for a given dis-
sipation rate there can be no shuttle instability for small
enough magnetic fields. This is because the rate r (incre-
ment) at which the amplitude of the dot oscillations would
increase in the absence of dissipation is small 2( ).r h
Therefore any amount of friction will prevent the instabil-
ity to develop for low enough magnetic fields.
Note that thh tends to zero and thh as 0. In
the limits / 1 and 1 it follows that the “large-h”
branch is given by the relation th / (1/ 2)(1 3 /2 ).h
We see that at 1 threshold magnetic field normalized
on depends only on one variable / . It is precisely this
dependence that is shown in Fig. 3. The maximum value of
ratio ( / ) 0.22m gives the minimum level of spin polar-
ization, = 0.64m (see Eq. (28)), below which magnetic
field can not induce transition between vibronic and shuttle
“phases”. It is interesting to note that there is finite interval
of polarizations when the increase of magnetic field from
small to large values is accompanied by re-entrant transition
to the shuttling phase (the vibronic phase is “inside” the
curve in Fig. 3).
4.2. Shuttle instability in the regime V
If >eV U the Coulomb blockade is lifted and there is a
finite probability for electrons to occupy both interaction-
split energy levels. For fully spin-polarized electrons (i.e.,
in the case when the leads are half-metals) the shuttle dy-
namics was considered in Ref. 7, where it was shown that
in the absence of dissipation in the mechanical subsystem a
shuttle instability takes place for arbitrary values of an ex-
ternal magnetic field. The only condition for the realization
of electron shuttling is to direct the external magnetic field
not parallel to the direction of magnetization in the leads.
The strength of the magnetic field, however, strongly in-
fluences the increment of exponentional growth of shuttle
coordinate.
In this subsection we derive the rate of the development
of shuttle instability for partially spin-polarized electrons in
the leads and analyze the conditions under which a shuttle
instability occurs in the presence of weak dissipation. In the
absence of a Coulomb blockade the equations of motion for
the matrix elements of the density operator take a form simi-
lar to the system of Eqs. (5)–(8). The only distinction is the
presence of an additional equation for the matrix element of
the doubly occupied state 2 ˆ= 2 | | 2 ,
2
2 2
1ˆ= [ , ] { , }
2
D Di dx
t
v
S S S S (29)
and an additional term
( )
2S in the equations of motion
for and . The analysis of the new system of equa-
tions is completely analogous to the procedure described in
Secs. 2 and 3 and results in the analytical expression for
the increment
2
2 2 2
( ) = 1 ,
1
d
r h
h
(30)
where = . We see that r is never negative since
2 2 2 .h In the limit = 0 (fully spin-polarized
electrons) Eq. (30) is reduced to the corresponding formula
derived in Ref. 7 (notice that in Eq. (67) in [7] there is a
misprint: factor 2 1
0( 1) is missing). In the opposite lim-
it of unpolarized electrons ( = ) we get
2
2
= .
1
d
r (31)
We see that for unpolarized electrons the magnetic field
has no influence at all (as it should be). The rate of in-
crease of shuttle amplitude, Eq. (31), for 1 is reduced
to the increment of shuttle instability for unpolarized elec-
trons derived in [5] (see also [11]) in perturbation theory
with as the small parameter. One of the main conclu-
sions of Ref. 7 was the assertion that for weak dissipation
(phenomenologically introduced as friction in the equation
of motion for coordinate, f is the friction coefficient) a
weak magnetic field can trigger a shuttling instability when
( ) > .fr h The corresponding critical magnetic field
scales as
3 1/2( / )c fh d for 1 and we assume that
0f (otherwise the shuttle instability can not be con-
trolled by the magnetic field).
Fig. 3. The threshold magnetic field hth plotted as a function of
the minority spin tunnelling rate (normalized to the majority
spin tunneling rate ). Part of the “lower” branch is shown as a
dashed curve because for small magnetic fields the rate of insta-
bility is small and dissipation prevents the development of insta-
bility. Point M defines the maximum value of / above which
magnetic field does not cause the transition between shuttle and
vibronic regime of electron transport.
Magnetically controlled single-electron shuttle
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 1 95
For partially polarized electrons an external magnetic
field influences ( )r h when 2 2 2h (see
Eq. (30)). For a realistic situation ( 1) a magnetic con-
trol of electron shuttling can be achieved by external fields
obeying the inequality / 1,h i.e., only for a de-
gree of polarization close to 100%.
5. Conclusions
Single-electron shuttling is a nonequilibrium phenome-
non, which under certain conditions determines the elec-
tron transport in some nanoelectromechanical devices. The
possibility to control this shuttle current by external fields
is an important problem in nanoelectromechanics.
It is especially interesting to consider magnetically con-
trolled single-electron shuttling — something that can be
realized in a magnetic shuttle system. A theory of spin-
controlled electron shuttling was formulated in Ref. 7 for
the case of 100% spin-polarized electrons in the source-
and drain leads. In this ideal case the magnetic control of
the electrical current is most effective. Here we have ana-
lyzed how the onset of shuttling can be controlled mag-
netically in the more realistic case of partially spin-
polarized leads. Two different regimes of electron tran-
sport have been studied: (i) the Coulomb blockade regime
and (ii) the regime of large bias voltages when the Cou-
lomb blockade is lifted.
In the Coulomb blockade regime we obtained a univer-
sal curve, which for the realistic case that 0 deter-
mines how the threshold magnetic field (separating the
vibron and shuttle domains) depends on the degree of po-
larization. Using this result a numerical value for the min-
imum degree of spin polarization ( 64%) was found. For
partially polarized leads we also predicted the existence of
re-entrant transitions to the shuttle phase as the magnetic
field is increased. In the absence of a Coulomb blockade,
we showed that magnetic control of electron shuttling can
be realized only for almost 100%-polarized leads (so-cal-
led half metals).
Acknowledgments
We thank L. Gorelik for fruitful discussions. Financial
support from the Swedish VR is gratefully acknowledged.
SK thanks the Department of Physics at the University of
Gothenburg and the Department of Physics and Astronomy
at Seoul National University for their hospitality.
1. L.Y. Gorelik, A. Isacsson, M.V. Voinova, B. Kasemo, R.I.
Shekhter, and M. Jonson, Phys. Rev. Lett. 80, 4526 (1998).
2. R.I. Shekhter, Y. Galperin, L.Y. Gorelik, A. Isacsson, and
M. Jonson, J. Phys.: Condens. Matter 15, R441 (2003).
3. R.I. Shekhter, L.Y. Gorelik, M. Jonson, Y.M. Galperin, and
V.M. Vinokur, J. Comput. Theor. Nanosci. 4, 860 (2007).
4. R.I. Shekhter, L.Y. Gorelik, I.V. Krive, M.N. Kiselev, A.V.
Parafilo, and M. Jonson, Nonoelectromech. Systems 1, 1
(2013).
5. D. Fedorets, L.Y. Gorelik, R.I. Shekhter, and M. Jonson,
Europhys. Lett. 58, 99 (2002).
6. D.R. Koenig and E.M. Weig, Appl. Phys. Lett. 101, 213111
(2012).
7. L.Y. Gorelik, D. Fedorets, R.I. Shekhter, and M. Jonson,
New J. Phys. 7, 242 (2005).
8. D. Fedorets, L.Y. Gorelik, R.I. Shekhter, and M. Jonson,
Phys. Rev. Lett. 95, 057203 (2005).
9. L.Y. Gorelik, S.I. Kulinich, R.I. Shekhter, M. Jonson, and
V.M. Vinokur, Phys. Rev. Lett. 95, 116806 (2005).
10. S.I. Kulinich, L.Y. Gorelik, A.N. Kalinenko, I.V. Krive, R.I.
Shekhter, Y.W. Park, and M. Jonson, Phys. Rev. Lett. 112,
117206 (2014).
11. D. Fedorets, Phys. Rev. B 68, 033106 (2003).
|