Single-magnon tunneling through a ferromagnetic nanochain
Magnon transmission between ferromagnetic contacts coupled by a linear ferromagnetic chain is studied at the condition when the chain exhibits itself as a tunnel magnon transmitter. It is shown that dependently on magnon energy at the chain, a distant intercontact magnon transmission occurs either i...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
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| Цитувати: | Single-magnon tunneling through a ferromagnetic nanochain / E.G. Petrov, V. Ostrovsky // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 958–963. — Бібліогр.: 24 назв. — англ. |
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Petrov, E.G. Ostrovsky, V. 2017-05-23T15:18:26Z 2017-05-23T15:18:26Z 2010 Single-magnon tunneling through a ferromagnetic nanochain / E.G. Petrov, V. Ostrovsky // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 958–963. — Бібліогр.: 24 назв. — англ. 0132-6414 PACS: 05.60.Gg, 85.75.–d https://nasplib.isofts.kiev.ua/handle/123456789/117441 Magnon transmission between ferromagnetic contacts coupled by a linear ferromagnetic chain is studied at the condition when the chain exhibits itself as a tunnel magnon transmitter. It is shown that dependently on magnon energy at the chain, a distant intercontact magnon transmission occurs either in resonant or off-resonant tunneling regime. In the first case, a transmission function depends weakly on the number of chain sites whereas at off-resonant regime the same function manifests an exponential drop with the chain length. Change of direction of external magnetic field in one of ferromagnetic contacts blocks a tunnel transmission of magnon. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 80-летию со дня рождения В.Г. Барьяхтара Single-magnon tunneling through a ferromagnetic nanochain Article published earlier |
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Single-magnon tunneling through a ferromagnetic nanochain |
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Single-magnon tunneling through a ferromagnetic nanochain Petrov, E.G. Ostrovsky, V. К 80-летию со дня рождения В.Г. Барьяхтара |
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Single-magnon tunneling through a ferromagnetic nanochain |
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Single-magnon tunneling through a ferromagnetic nanochain |
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Single-magnon tunneling through a ferromagnetic nanochain |
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Single-magnon tunneling through a ferromagnetic nanochain |
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single-magnon tunneling through a ferromagnetic nanochain |
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Petrov, E.G. Ostrovsky, V. |
| author_facet |
Petrov, E.G. Ostrovsky, V. |
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К 80-летию со дня рождения В.Г. Барьяхтара |
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К 80-летию со дня рождения В.Г. Барьяхтара |
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2010 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Magnon transmission between ferromagnetic contacts coupled by a linear ferromagnetic chain is studied at the condition when the chain exhibits itself as a tunnel magnon transmitter. It is shown that dependently on magnon energy at the chain, a distant intercontact magnon transmission occurs either in resonant or off-resonant tunneling regime. In the first case, a transmission function depends weakly on the number of chain sites whereas at off-resonant regime the same function manifests an exponential drop with the chain length. Change of direction of external magnetic field in one of ferromagnetic contacts blocks a tunnel transmission of magnon.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117441 |
| citation_txt |
Single-magnon tunneling through a ferromagnetic nanochain / E.G. Petrov, V. Ostrovsky // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 958–963. — Бібліогр.: 24 назв. — англ. |
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2025-11-26T00:08:25Z |
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2025-11-26T00:08:25Z |
| _version_ |
1850592153239027712 |
| fulltext |
© E.G. Petrov and V. Ostrovsky, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9, p. 958–963
Single-magnon tunneling through a ferromagnetic
nanochain
E.G. Petrov
N.N. Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine
14-b Metrologichna Str., Kiev 03680, Ukraine
E-mail: epetrov@bitp.kiev.ua
V. Ostrovsky
Department of Physics, Polytechnic Institute of NYU, Brooklyn, New York 11201, USA
Received December 29, 2009
Magnon transmission between ferromagnetic contacts coupled by a linear ferromagnetic chain is studied at
the condition when the chain exhibits itself as a tunnel magnon transmitter. It is shown that dependently on mag-
non energy at the chain, a distant intercontact magnon transmission occurs either in resonant or off-resonant
tunneling regime. In the first case, a transmission function depends weakly on the number of chain sites whereas
at off-resonant regime the same function manifests an exponential drop with the chain length. Change of direc-
tion of external magnetic field in one of ferromagnetic contacts blocks a tunnel transmission of magnon.
PACS: 05.60.Gg Quantum transport;
85.75.–d Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated
magnetic fields.
Keywords: magnon, nanochain, transmission, tunneling.
1. Introduction
Modern electronics including spintronics, operates with
the structures that have an effective size of the order of
several tens nanometers [1–6]. Further minimization of
element base for electronics is associated with molecular
architecture where single molecules or their combinations
have to demonstrate the properties of wires, diodes, tran-
sistors, storage cells, etc. [7–12]. In specific cases when
single molecules contain paramagnetic ions, these ions can
polarize an electron current through a molecule and even
block the current [13–16]. The work of electronic devices
is based on switching on/off the microcurrents and thus
the physics of information transmission is associated with
the transfer of electrons or holes. Such transfer is accom-
panied by a rather large energy dissipation. It is obviously
that much more less power is required if the information
is transmitted by uncharged carriers. In present communi-
cation, a principally new mechanism of information trans-
mission is proposed. It is associated with a distant transfer
of spin excitation (magnon) from one magnetic contact
to another magnetic contact via magnetically ordered na-
nochain.
2. Model and theory
We consider the simplest magnetic device that consists
of ferrodielectric contacts A and B connected by a ferro-
magnetic nanochain (AFB-device, Fig. 1). The chain in-
volves a regular interior part and edge groups a and b
coupled to respective contacts. Let Bμ be the Bohr magne-
Fig. 1. Magnon transferring device. Exchange couplings in the
device are characterized by parameters AJ and BJ (between
sites related to ferromagnetic contacts), AaJ and BbJ (between
edge sites of ferromagnetic chain and surface sites belonging the
adjacent contacts), 1aJ and bNJ (between edge sites and end
sites of regular part of chain), and J (between interior sites of
chain). ( ) ,A BS ( )a bS and S are the spins belonging the contact
sites, the edge sites of chain and the interior sites of chain, respec-
tively.
Contact
A
SA A, J
Contact
B
SB B, J
Regular part of chain
a 1 2 N–1 N b
JAa
Ja1 J J
JBb
JbN
s s ss sbsa
Single-magnon tunneling through a ferromagnetic nanochain
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 959
ton and let ˆ
mS , mD , mg and mH be, respectively, the
spin operator, the parameter of single-ion anisotropy, the
g-factor and the magnetic field related to device site m.
Denoting via (> 0)mnJ the parameter that characterizes an
exchange interaction between the nearest neighboring sites
m and n , we represent the device's magnetic Hamiltonian
in conventional form [17]
2ˆ ˆ= ( )z
B m m m m m
m m
H g D Sμ − −∑ ∑H S
1 ˆ ˆ .
2 mn m n
mn
J− ∑ S S (1)
Ground state | 0〉 of device corresponds to minimal spin
projection =m mM S− for each site m ( mS is the spin
value at site m ) so that | 0 = | =m m mm S M S〉 − 〉∏ where
| =m m mS M S− 〉 denotes a ground spin state for the mth
site Ground magnetic energy of device appears as (magnet-
ic fields are directed along axis z)
0 = 0 | | 0 = B j m m
m
E H g H S〈 〉 −μ −∑
2 .m m mn m n
m mn
D S J S S− −∑ ∑ (2)
In what follows, the lowest spin excitations (magnons) are
considered only. This excitation appears at site n if the
spin projection varies from =n nM S− to = ( 1).n nM S− −
Respective device state becomes | =| =n nn S M〉
= ( 1) | =n m m mm nS S M S≠− − 〉 − 〉∏ . In line with theory of
magnetic excitons [18–20] we introduce the operators of
creation and annihilation of spin excitation as =| 0 |nb n+ 〉〈
and =| 0 |nb n〉〈 . Expanding Hamiltonian (1) with respect
to these excitations one derives the following form of Ha-
miltonian of spin excitations
=s n n n nm n m
n nm
H E b b V b b+ +Δ +∑ ∑ (3)
where 0= | |nE n H n EΔ 〈 〉 − and = | |nmV n H m〈 〉 are,
respectively, the energy of local spin excitation in device
and the matrix element characterizing the hopping of spin
excitation between sites n and m . Restriction by a qua-
dratic form over nb+ and nb supposes small number of
spin excitations in device. This is satisfied at condition
1n nb b+〈 〉 where symbol ...〈 〉 denotes the thermodynam-
ic average. Bearing in mind the fact that matrix element
from the product ˆ ˆ
m nS S has a form
ˆ ˆ, | | , =n n m m m n m m n nS M S M S M S M′ ′〈 〉S S
, ,
1 ( )( )
2m n M M M M m m m mm m n n
M M S M S M′ ′ ′= δ δ + + − ×
, 1 , 1( )( )n n n n M M M Mm m n n
S M S M ′ ′− +′× − + δ δ +
1 ( )( )
2 m m m mS M S M ′+ − + ×
, 1 , 1( )( ) ,n n n n M M M Mm m n n
S M S M ′ ′+ −′× + − δ δ (4)
one derives
= (2 1)n B n n n n mn m
m n
E g H D S J S
≠
Δ μ + − + ∑ (5)
and
= | | = .nm nm n mV n H m J S S〈 〉 − (6)
We rewrite now a Hamiltonian of spin excitations with
taken into account the fact that contacts A and B are regular
structures and thus spin excitations in these structures are
magnons. Let vector n indicates the position of site n be-
longing to the rth contact ( , ).r A B= Using the transfor-
mation ( )= ( )r
n n rb T b∑ k
k
k we achieve the following di-
agonal form of respective Hamiltonian,
= ( )r r r rH E b b+∑ k k
k
k (7)
where ( )
cont( ) = ( )r
r rE E z− β γk k is the energy of magnon
with wave vector k [21]. Position of the center of magnon
band is determined by expression ( )
cont =r
B r rE g Hμ +
(2 1) .r r rD S z+ − + β In a simple case of cubic crystal where
the number of nearest neighbors z is equal to 6, one derives
=r r rJ Sβ and ( ) = (1/ 3)(cos cos cos )x y zak ak akγ + +k
where rS is the site spin in the rth contact, rJ is the ex-
change parameter for the nearest neighbors, and a is the
cell constant. For a regular part of chain, we utilize an exact
transformation
=1
=
N
n nb U bμ μ
μ
∑ where 1/2= ( 1)nU N −
μ + ×
sin( / 1)n N× π μ + Such transformation diagonalizes a
Hamiltonian of interior part of nanochain yielding
reg
=1
=
N
H E b b+μ μ μ
μ
∑ (8)
where
(0)
reg= 2 cos
1
E E
Nμ
πμ
− β
+
(9)
is the energy of spin excitation in regular chain with
(0)
reg = (2 1) 2BE gH D Sμ + − + β being the center of discrete
magnon band. Here, g, D and J are the g-factor, para-
meter of single-ion anisotropy and exchange parameter,
respectively, while .JSβ ≡
After above transformations, Hamiltonian of spin exci-
tations appears in the form
0 tr= .SH H V+ (10)
The first term,
0 reg= ,A B a bH H H H H H+ + + + (11)
E.G. Petrov and V. Ostrovsky
960 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9
includes Hamiltonians related to the contacts A and B, the
interior part of chain, and the edge chain sites a and b .
The latter Hamiltonians read as
= , ( = , ),l l l lH E b b l a b+ (12)
where
1= (2 1) ,a B a a a a Aa aE g H D Sμ + − +β +β (13)
and
= (2 1)b B b b b b Bb bNE g H D Sμ + − +β +β (14)
are the energies of spin excitation at the edge sites. In Eqs.
(13) and (14), we have introduced the following notations
(see also Fig. 2) Aa A Aa Az J Sβ ≡ , Bb B Bb Bz J Sβ ≡ and
1 1a aJ Sβ ≡ , bN bNJ Sβ ≡ with Az and Bz being the
number of identical contact sites coupled to the adjacent
edge site. Quantities AaJ and BbJ are the parameters that
characterize an exchange coupling of edge sites to respec-
tive contacts whereas 1aJ and bNJ are the parameters
related to an exchange coupling of edge sites a and b to
the 1st and the Nth sites of interior part of chain, respec-
tively. Operator
tr = Aa ac Bb bcV V V V V+ + + (15)
describes the transfer of spin excitation between edge site
( )a b and adjacent contact A(B) as well as between the
same edge site ( )a b and site 1( N ) of regular chain (terms
( )Aa BbV V and ( )ac bcV V , respectively). The terms read
( = , )rl Aa Bb
*
, ,= [ ],rl l r l r l r r l
k
V b b b b+ +β +β∑ k k k k (16)
and ( = , )l a b
*
=1
= [ ].
N
lc l l l lV b b b b+ +
μ μ μ μ
μ
β +β∑ (17)
In Eqs. (16) and (17), the coupling parameters are defined
through the relations ( )
, = ( )r
l r rl nl
Tβ βk k and =l lm mUμ μβ β
with =rl rl r lJ S Sβ − and =lm lm lJ S Sβ − . Symbol ln
indicates the position of surface contact site coupled to the
chain edge site = ( , )l a b coupled to respective contact
(= , )r A B while index (= 1, )m N numbers the end site of
interior part of chain. (Fig. 2 shows a relative position of
magnon energies in the AFB-device along with the coupl-
ings responsible for magnon hoppings).
Our aim is to derive expression for a distant flow of mag-
nons from one magnetic contact to another one. To this end,
we suppose that interaction between nanochain and precise
macroscopic contact does not distinctly perturb the contact's
magnon energy ( )rE k so that magnon vector k can be refer
to a good quantum number. Quantum mechanics shows [22]
that in a dynamic system, the probability ,Pβ α of a transition
from the state α to the state β per unit time is given by ex-
pression 2
,
ˆ= (2 / ) | | | | ( )P T E Eβ α α βπ 〈β α〉 δ − where
int int int
ˆ = ( )T H H G E H+ is the operator for a transition on
the energy shell =E Eα . Quantity 1( ) = ( 0 )G E E H i + −− +
is the Green's operator with 0 int=H H H+ being the Ha-
miltonian of entire dynamic system. In the case of distant
magnon transmission under consideration, the states α
and β are associated with magnon wave vectors k and q.
Therefore, a probability to transfer a separate magnon
from the contact A to the contact B is given by expres-
sion 2
, ,= (2 / ) | | ( ( ) ( ))A B B A A BP T E Eπ δ −k q q k k q where
, tr tr= | ( ) |B AT B V G E V A〈 〉q k q k , = ( )AE E k , and ( ) =G E
1= ( 0 )sE H i + −− + . Since the device is an open quantum
system, an integral transmission probability A BP → appears
as the sum of probabilities ,A BP k q each weighted with
magnon distribution function ( ( ))A AW E k . Thus,
,= ( ( )) .A B A A A BP W E P→ ∑ k q
kq
k (18)
Analogous form is valid for reverse probability B AP → (in
Eq. (18), one has only to substitute ( ( ))A AW E k for the
( ( ))B BW E q ).
Let transform now Eq. (18) to more convenient form.
Bearing in mind that exchange interaction couples only the
nearest neighbors one derives *
, , ,= ( )B A b B ba a AT G Eβ βq k q k
where 1( ) = | ( 0 ) |ba sG E b E H i a+ −〈 − + 〉 is the quantity
that establishes a coupling between spin states of spatially
separated edge sites a and b . Its form is similar those
used in theory of elastic electron transmission through or-
ganic molecules [23,24]. Following the method derived
in Ref. 24 we reduce exact Hamiltonian sH to the
(eff )(eff )
chain
= ,
=s r
r A B
H H H+∑ where contact Hamiltonians
conserve their form (7) whereas the effective chain Hamil-
tonian reads
(eff )
chain = .H b b+λ λ λ
λ
∑ E (19)
Fig. 2. Relative position of magnon energies in AFB-device.
Magnon energy at contact A(B) is presented in the effective mass
model. Couplings of edge chain site ( )a b to the contact A(B) and
to the end site 1( )N of interior part of chain are characterized by
quantities , ,( )a A b Bβ βk q and 1( )a bNβ β , respectively.
Contact
A Contact
B
Magnon tunneling
Econt
(A)
E ( )A k
E (0)A
EN
Ereg
(0)
E1
Ea
Eb
Econt
(B)
E q(B)
E (0)B
�a�
�a, Ak
�b�
�b, Bq
Single-magnon tunneling through a ferromagnetic nanochain
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 961
Here, λE is the magnon energy in proper state λ of Hamil-
tonian (19), = j j
j
b bλ λΘ∑ and *= j j
j
b b+ +
λ λΘ∑ are new
operators of annihilation and creation of magnon in the chain,
and = |j jλΘ 〈 λ〉 are the elements of matrix that trans-
forms chain states (= , , )j a b μ formed at 1 = = 0a bNV V
and = = 0Aa BbV V , into proper states λ formed with taken
into consideration off-diagonal interaction (15). Substitution
of Hamiltonian sH for the (eff )
sH yields
( ) = .b a
baG E
E
λ λ
λλ
Θ Θ
−∑ E
(20)
Note an important fact that proper chain energy λE con-
tains an image addition caused by interactions of edge
chain sites with macroscopic contacts (operators aAV and
bBV in Eq. (15)). As an example, we consider the case
where exchange interaction of edge sites a and b with
adjacent contacts and interior part of chain does not exceed
exchange interactions within the contacts as well as within
the interior part of chain. At such conditions, a mixture of
extended states | ,|A B〉 〉k q and |μ〉 with localized states
| a〉 and | b〉 is not large. This allows one to reduce exact
form (20) to the expression
1( )
( ( ))( ( ))
a bN
ba
a b
G E
E E E E
β β
×
− −E E
1
=1 ( )
N NU U
E E
μ μ
μμ
×
−∑ E
(21)
where jE is the proper chain energy for magnon states
= , ,j a b μ . Proper energy is derived from relation
( ) = ( )j j jE E E+ ΣE with
2
,
= ,
| |
( ) =
( ) 0
j rk
j r A B r
E
E E i +
β
− +
∑ ∑ ∑
k k
(22)
being the magnon self-energy. Self-energy characterizes
the influence of macroscopic contacts on the chain through
exchange couplings ,j rβ k . Real part of self-energy deter-
mines a small alteration of energies and can be omitted. It
is not the case for image part which plays a fundamental
role in magnon transmission. Thus, in Eq. (21), magnon
chain energies appear as
( ) ( ) / 2j j jE E i E− ΓE (23)
where quantities jE are defined through expressions (9),
(13), and (14) while image additions read ( = ( )l a b if
= ( )r A B )
2
,( ) = 2 | | ( ( )),l l r rE E EΓ π β δ −∑ k
k
k (24)
and ( = 1,...Nμ )
2
,
= ,
( ) = 2 | | ( ( )).r r
r A B
E E Eμ μΓ π β δ −∑ ∑ k
k
k (25)
Couplings ,a Aβ k and ,b Bβ k have been written above
whereas *
, tr tr ,= | ( ) | = / ( )r l r l lV G E V r Eμ μβ 〈μ 〉 β β −k kk E ,
( = ( )l a b if = ( )r A B ).
Quantity =A B A B B AQ P P→ → →− characterizes a nor-
malized net flow of magnons. With introduction of width
parameters (24) and (25), this flow can be represented in
form
1= ( , )
2A BQ dE T N E
+∞
→
−∞
×
π ∫
( ( ) ( ))A BW E W E× − (26)
where
2( , ) = ( ) | ( ) | ( )b ba aT N E E G E EΓ Γ (27)
is the transmission function that specifies a dynamics of
direct contact-contact transfer of single magnon depen-
dently on both the number of chain sites and the character
of exchange couplings within the AFB-device.
3. Results and discussion
Magnon flow (26) depends strongly on precise form of
distribution functions ( )AW E and ( )BW E as well as
transmission function (27). To specify distribution function
one has to know a regime of magnon formation in the con-
tacts. This problem requires a separate consideration. In
this communication, we discuss only the properties of
transmission function. To this end, let rewrite the ( , )T N E
in more detail form
1
2 2
( )
( , ) =
[( ) ( ) / 4]
a a
a a
E
T N E
E E E
Γ β
×
− +Γ
2
1
2 2 2 2
=1
( )( )
[( ) ( ) / 4] ( ) ( ) / 4
N Nb bN
b b
U U E EE
E E E E E E
μ μ μ
μ μ μ
⎡⎛ ⎞−Γ β ⎢⎜ ⎟× +⎢⎜ ⎟− + Γ − +Γ⎝ ⎠⎢⎣
∑
2
1
2 2
=1
( ( ) / 2)
.
( ) ( ) / 4
N NU U E
E E E
μ μ μ
μ μ μ
⎤⎛ ⎞Γ ⎥⎜ ⎟+ ⎥⎜ ⎟− + Γ⎝ ⎠ ⎥⎦
∑ (28)
Bearing in mind that, generally, the magnons are generated
at 0k ≈ , the widths can be calculated in the effective mass
approximation with taken into account the fact that
(0)rE E≥ . Calculations yield
2 (0)1( ) = ,
2
rl r
l
r r
E E
E
β −
Γ
π β β
( = ( ), = ( ))r A B l a b (29)
E.G. Petrov and V. Ostrovsky
962 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9
Fig. 3. Behavior of transmission function at different length of
ferromagnetic chain. Peaks correspond to such transmission ener-
gies that coincide with magnon energies within the ferromagnetic
chain of 2N + sites (regime of resonant tunneling). The curves
are calculated in using Eq. (28) with = = = =A B Aa Bbβ β β β
1= = = 10a bNβ β cm–1, = 15β cm–1, (0)
reg = 80E cm–1, (0) =AE
(0) = 80BE= cm–1, = = 40a bE E cm–1.
N = 3
N = 5
N = 7
T
(N
,
E
)
E, cm–1
0 50 100 150 200
10
–1
10
–5
10
–9
10
–13
10
–1
10
–5
10
–9
10
–13
10
–1
10
–5
10
–9
10
–13
T
(N
,
E
)
T
(N
,
E
)
Fig. 4. Off-resonant regime of magnon tunneling. At fixed trans-
mission energy, the transmission function drops exponentially
when the number of chain sites increases. The curves are calculated
in using Eq. (28) with the same parameters as those for Fig. 3.
4 6 8 10 12
N
10
0
10
–4
10
–8
10
–12
10
–16
10
–20
E = 10 cm
–1
E = 30 cm
–1
E = 40 cm
–1
T
(N
,E
)
and
2[ / ( 1)]sin( ) =
1
NE
Nμ
πμ +
Γ ×
+
2 2
1
2 2 2 2
( ) ( )
.
( ) ( ) / 4 ( ) ( ) / 4
a a bN b
a a b b
E E
E E E E E Eμ μ
⎡ ⎤β Γ β Γ⎢ ⎥× +
⎢ ⎥− + Γ − + Γ⎣ ⎦
(30)
Figure 3 manifests a typical dependence of transmission
function on magnon energy = ( ) = ( ).A BE E Ek q The
peaks appear at elastic resonant transmission regime when
magnon energy at the contacts coincides exactly with mag-
non energies at the ferromagnetic chain. The coincidence
occurs at condition = ,jE E ( = , , )j a b μ . Broadening the
peaks is completely determined by quantities (29) and (30).
It is necessary to note an important fact that independently
on the number of chain sites, the peak's heights differ insig-
nificantly from each other. It is not the case at off-resonant
transmission regime. As it follows from Fig. 4, at such re-
gime, a transmission function exhibits an exponential drop
showing, thus, the tunneling like behavior. Respective ana-
lytical expression for transmission function follows from Eq.
(28) if one sets 2 2( ) ( ) / 4.E E Eμ μ− Γ Then at
(0)
reg| | 2E E− ≥ β one derives
( , ) = ( )T N E D E ×
2 2 2
1
2 2 2 2
( ) ( )( / )
[( ) ( ) / 4][( ) ( ) / 4]
a b a bN
a a b b
E E
E E E E E E
Γ Γ β β β
×
− + Γ − + Γ
(31)
where we have introduced a specific superexchange de-
crease factor
sinh ( )( ) =
sinh[ ( )( 1)]
ED E
E N
ζ
ζ +
(32)
with
2(0) (0)
reg reg| |
( ) = ln 1
2 2
E E E E
E
⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟ζ + −⎢ ⎥⎜ ⎟β β⎢ ⎥⎝ ⎠⎣ ⎦
(33)
being the decrease parameter. When exp ( ) 1Eζ the de-
crease factor reduces to simple form ( ) exp[ ( ) ]D E E N−ζ
reflecting thus an exponential drop of transmission func-
tion. The drop strongly depends on transmission gap
(0)
reg| |E E− .
It is seen from Fig. 4 that the less is the gap, the slower
drop of transmission function. For instance, if generation
of magnons in contact A occurs near the bottom of magnon
band so that magnon distribution function ( )AW E has a
maximum value at 0k , then the main contribution in
integral of expression (26) give the energies of the order
(0)AE E and thus, exp[ ( (0)) ]A B AQ E N→ −ζ∼ .
4. Conclusion
In this communication, we propose a physical mechan-
ism for a coherent distant transmission of spin excitation
(magnon) from one magnetic contact to another magnetic
contact via a linear ferromagnetic chain embedded between
the contacts, Fig. 1. Coupling of structure units in such
Single-magnon tunneling through a ferromagnetic nanochain
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 963
device is performed through the Heisenberg's site-site ex-
change interaction which is also responsible for a magnon
hopping within the device. It is assumed that a magnon
transit-time is mach less the magnon life-time and, thus,
one can describe a magnon transmission as a stationary
transfer process. Zero temperature case is considered only
so that a temperature excitation of magnons is ignored. It is
assumed that generation of magnons in ferromagnetic con-
tacts is caused by an external source. But, the concentra-
tion of magnons is too small that the use of single-magnon
model is quite enough to describe a magnon transmission
between the contacts. We show that a ferromagnetic chain
is able to form a distant (superexchange) coupling ,B AT q k
between single-magnon states | A 〉k and | B 〉q related to
different contacts. The character of superexchange coupl-
ing depends strongly on value of magnon energies ( )rE k
and jE at the rth contact and at the chain, respectively
(see Fig. 2). When magnon energy = ( ) = ( )A BE E Ek q
coincides with the ,jE transmission function ( , )T N E
demonstrates the presence of resonant peaks well seen at
the Fig. 3. Appearance of the peaks indicates an effective
coherent contact-contact magnon transfer independently of
the number of chain sites. Another situation occurs if the
contact's magnon energy differs from the chain's magnon
energy. In this case, a transmission function drops expo-
nentially with increase of chain length, Fig. 4. Such expo-
nential drop corresponds to a tunnel mechanism of trans-
mission and, thus, one can say about magnon tunneling.
Magnon transmission in the AFB-device under considera-
tion is strongly controlled by an external magnetic field.
For instance, let one change a direction of magnetic field
applied to one of the contacts (or to the chain). After such
change, a direction of spins in the contact and the chain
becomes opposite. As a result, the hopping matrix element
(4) vanishes, and magnon hopping from the contact to the
chain is blocked.
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