Spin-orbit coupling effect on the persistent currents in mesoscopic ring
  with an Anderson impurity
Spin-orbit coupling effect on the persistent currents in mesoscopic
ring with an Anderson impurity
Guo-Hui Ding and Bing Dong
Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China
(Dated: November 4, 2018)
Abstract
Based on the finite U slave boson method, we have investigated the effect of Rashba spin-
orbit(SO) coupling on the persistent charge and spin currents in mesoscopic ring with an Anderson
impurity. It is shown that the Kondo effect will decrease the magnitude of the persistent charge
and spin currents in this side-coupled Anderson impurity case. In the presence of SO coupling,
the persistent currents change drastically and oscillate with the strength of SO coupling. The SO
coupling will suppress the Kondo effect and restore the abrupt jumps of the persistent currents. It
is also found that a persistent spin current circulating the ring can exist even without the charge
current in this system.
PACS numbers: 73.23.Ra, 71.70.Ej, 72.25.-b
http://arxiv.org/abs/0704.0319v1
I. INTRODUCTION
Recently the spin-orbit(SO) interaction in semiconductor mesoscopic system has attracted
a lot of interest[1]. Due to the coupling of electron orbital motion with the spin degree of
freedom, it is possible to manipulate and control the electron spin in SO coupling system by
applying an external electrical field or a gate voltage, and it is believed that the SO effect
will play an important role in the future spintronic application. Actually, various interesting
effects resulting from SO coupling have already been predicted, such as the Datta-Das spin
field-effect transistor based on Rashba SO interaction[2] and the intrinsic spin Hall effect[3].
In this paper we shall focus our attention on the persistent charge current and spin cur-
rent in mesoscopic semiconductor ring with SO interaction. The existence of a persistent
charge current in a mesoscopic ring threaded by a magnetic flux has been predicted decades
ago[4], and has been extensively studied in theory[5, 6, 7, 8, 9] and also observed in various
experiments[10, 11, 12]. The reason that a persistent charge current exists may be inter-
preted as that the magnetic flux enclosed by the ring introduces an asymmetry between
electrons with clockwise and anticlockwise momentum, thus leads to a thermodynamic state
with a charge current without dissipation. For a mesoscopic ring with a texture like inho-
mogeneous magnetic field, D. Loss et al.[13] predicted that besides the charge current there
are also a persistent spin current. The origin of the persistent spin current can be related
to the Berry phase acquired when the electron spin precesses during its orbital motion. The
persistent spin current has also been studied in semiconductor system with Rashba SO cou-
pling term[14, 15, 16]. Recently it is shown that a semiconductor ring with SO coupling can
sustain a persistent spin current even in the absence of external magnetic flux[17].
For the system of a mesoscopic ring with a magnetic impurity, the persistent charge
current has been investigated in the context of a mesoscopic ring coupled with a quantum
dot[18, 19, 20, 21, 22, 23, 24], where the quantum dot acts as an impurity level and will
introduce charge or spin fluctuations to the electrons in the ring. The Kondo effect arising
from a localized electron spin interacting with a band of electrons will be essential in the
charge transport in the ring. But to our knowledge in these systems the SO effect hasn’t
been considered. It might be expected that the interplay between the Kondo effect and
the SO coupling in the ring can give new features in the persistent currents. In this paper
we shall address this problem and investigate the SO effect on persistent charge and spin
currents in the ring system with an Anderson impurity. The Anderson impurity can act as
a magnetic impurity when the impurity level is in single electron occupied state and as well
as a barrier potential in empty occupied regime.
The outline of this paper is as follows. In section II we introduce the model Hamiltonian
of the system and also the method of calculation by finite-U slave boson approach[25, 26,
27, 28]. In section III the results of persistent charge current and spin current are presented
and discussed. In Section IV we give the summary.
II. MESOSCOPIC RING WITH AN ANDERSON IMPURITY
The electrons in a closed ring with SO coupling of Rashba term can be described by
following Hamiltonian in the polar coordinates[14, 29]
Hring = ∆(−i
[(σx cosϕ+ σy sinϕ)(−i
) + h.c.] , (1)
where ∆ = h̄2/(2mea
2), a is the radius of the ring. αR will characterize the strength of
Rashba SO interaction. Φ is the external magnetic flux enclosed by the ring, and Φ0 =
2πh̄c/e is the flux quantum.
We can write the above Hamiltonian in terms of creation and annihilation operators of
electrons in the momentum space,
Hring =
mσcmσ + 1/2
[tm(c
m+1↓cm↑ + c
m−1↑cm↓) + h.c.] , (2)
where ǫm = ∆(m+φ)
2, tm = αR(m+φ),(m = 0,±1, · · · ,±M) with φ = Φ/Φ0. One can see
that the SO interaction causes the m mode electrons coupled with m + 1 and m − 1 mode
electrons and spin-flip process. We consider the system with a side-coupled impurity which
can be described by the Anderson impurity model,
σdσ + Und↑nd↓ . (3)
The tunneling between the impurity level and the ring are given by
Hd−ring = tD
(d†σcmσ + h.c) . (4)
Then the total Hamiltonian for the system should be
H = Hring +Hd +Hd−ring . (5)
In order to treat the strong on-site Coulomb interaction in the impurity level. we adopt
the finite-U slave boson approach[25, 26]. A set of auxiliary bosons e, pσ, d is introduced for
the impurity level, which act as projection operators onto the empty, singly occupied(with
spin up and spin down), and doubly occupied electron states on the impurity, respectively.
Then the fermion operators dσ are replaced by dσ → fσzσ, with zσ = e
†pσ + p
σ̄d. In order
to eliminate un-physical states, the following constraint conditions are imposed :
σpσ +
e†e+ d†d = 1, and f †σfσ = p
σpσ + d
†d(σ =↑, ↓). Therefore, the Hamiltonian can be rewritten
as the following effective Hamiltonian in terms of the auxiliary boson e, pσ, d and the pesudo-
fermion operators fσ:
Heff =
mσcmσ + 1/2
[tm(c
m+1↓cm↑ + c
m−1↑cm↓) + h.c.]
σfσ + Ud
σcmσ + h.c.) + λ
p†σpσ + e
†e+ d†d− 1)
λ(2)σ (f
σfσ − p
σpσ − d
†d) , (6)
where the constraints are incorporated by the Lagrange multipliers λ(1) and λ(2)σ . The first
constraint can be interpreted as a completeness relation of the Hilbert space on the impurity
level, and the second one equates the two ways of counting the fermion occupancy for a given
spin. In the framework of the finite-U slave boson mean field theory[25, 26], the slave boson
operators e, pσ, d and the parameter zσ are replaced by real c numbers. Thus the effective
Hamiltonian is given as
HMFeff =
mσcmσ + 1/2
[tm(c
m+1↓cm↑ + c
m−1↑cm↓) + h.c.]
ǫ̃dσf
σfσ +
(t̃Dσf
σcmσ + h.c.) + Eg , (7)
where t̃Dσ = tDzσ represents the renormalized tunnel coupling between the impurity and
the mesoscopic ring. zσ can be regarded as the wave function renormalization factor. ǫ̃dσ =
σ is the renormalized impurity level and Eg = λ
2+d2−1)−
d2) + Ud2 is an energy constant.
In this mean field approximation the Hamiltonian is essentially that of a non-interacting
system, hence the single particle energy levels can be calculated by numerical diagonalization
of the Hamiltonian matrix. Then the ground state of this system |ψ0 > can be constructed
by adding electrons to the lowest unoccupied energy levels consecutively . By minimizing
the ground state energy with respect to the variational parameters a set of self-consistent
equations can be obtained as in Ref.[27,28], and they can be applied to determine the
variational parameters in the effective Hamiltonian.
III. THE PERSISTENT CHARGE CURRENT AND SPIN CURRENT
In this section we will present the results of our calculation of the persistent charge current
and spin current circulating the mesoscopic ring. Since there is still some controversial in the
literature for the definition of the spin current operator in the ring system with SO coupling
term[30]. We give both the formula of charge and spin currents used in this paper explicitly.
It is easy to obtain that the ϕ component of electron velocity operator in this SO coupled
ring is
[2∆(−i
+ φ) + αR(σx cosϕ+ σy sinϕ)] . (8)
Thereby the charge current operator is define as Î = −evϕ, and in terms of creation and
annihilation operator it can be written as
Î = −
c†mσcmσ(m+ φ) + αR
m+1↓cm↑ + c
m−1↑cm↓)] . (9)
At zero temperature, the persistent charge current is given by the expectation value of the
above charge current operator in the ground state, I = 1
< ψ0|Î|ψ0 >, and it can also be
calculated from the expression
I = −c
< ψ0|
|ψ0 > , (10)
where Egs is the ground state energy.
In Fig.1 the persistent charge current vs. the enclosed magnetic flux is plotted for a
set of values for the SO coupling strength. Here we have taken the model parameters
∆ = 0.01, tD = 0.3, U = 2.0 and the total number of electrons N is around 100. In this
case one can obtain the Fermi energy of the system EF = 6.25 and the level spacing δ = 0.5
around the Fermi surface. We consider the energy level of the Anderson impurity is well
below the Fermi energy( with ǫd − EF = −1.0), therefore the Anderson impurity is in the
Kondo regime. One can see in Fig.1 that the characteristic features of persistent charge
current depends on the parity of the total number of electrons(N), and can be distinguished
by two cases with N odd and N even. This is attributed to the different occupation patterns
of the highest occupied single particle energy level in the mean field effective Hamiltonian.
The persistent charge current for the system with N +2 electrons is different from that with
N electrons by a π phase shift IN+2(φ) = IN(φ+ π). In case (I) where the electron number
is odd(N = 4n− 1 and N = 4n + 1), one electron is almost localized on the impurity level
and forming a singlet with electron cloud in the conducting ring. This phenomena leads to
the well known Kondo effect. Fig.1 shows that the Kondo effect decreases the magnitude
of the persistent charge current, and also makes its curve shape resemble sinusoidal. In the
presence of finite SO coupling(αR < ∆), the spin-up and spin-down electrons are coupled and
it causes the splitting of the twofold degenerated energy levels in the effective Hamiltonian.
It turns out that the Kondo effect is suppressed and the abrupt jumps of the persistent
charge current with similarity to that of ideal ring case appears. It is explained in Ref.[14]
that the jumps of the persistent charge current in the case of odd number of electrons are
due to a crossing of levels with opposite spin. In case (II) where N is even (N = 4n and
N = 4n+2), The Kondo effect is manifested that the magnitude of persistent charge current
is significantly suppressed compared with ideal ring case and the rounding of the jumps of
persistent charge current due to the level crossing. In the presence of finite SO coupling, the
persistent charge current decreases with increasing the SO coupling strength when αR < ∆.
Fig.2 displays the persistent charge current as a function of the SO coupling strength
αR at different enclosed magnetic flux. The persistent charge current exhibits oscillations
with increasing the value of αR for both the systems with even or odd number of electrons.
Therefore by tuning the SO coupling strength, the magnetic response of this system can
change from paramagnetic to diamagnetic and vice versa. It indicates that SO coupling
can play a important role in electron transport in this mesoscopic ring. The curve of the
persistent charge current for odd number of electrons shows discontinuity in its derivation,
this can be attributed the level crossing in the energy spectrum by changing αR. It is also
noted that the position of this discontinuity for odd N also corresponds to the peak or valley
in even N case.
Since the electron has the spin degree of freedom as well as the charge, the electron
motion in the ring may give rise to a spin current besides the charge current. Now we turn
to study the persistent spin current in the ground state. The spin current operator is defined
by Ĵv = (v
ϕσv + σvv
ϕ)/2, which can be written explicitly as
Ĵv =
{2∆(−i
+ φ)σv +
[(σx cosϕ+ σy sinϕ)σv + h.c.]} , (11)
Therefore the three component of spin current operator in terms of creation and annihi-
lation operators are given by
Ĵz =
m↑cm↑ − c
m↓cm↓)(m+ φ)] , (12)
Ĵx =
m↑cm↓ + c
m↓cm↑)(m+ φ) +
m+1σ + c
m−1σ)cmσ] , (13)
Ĵy =
[−2i∆
m↑cm↓ − c
m↓cm↑)(m+ φ)− i
m+1σ − c
m−1σ)cmσ] , (14)
The expectation value of the spin current Jv =
< ψ0|Ĵv|ψ0 >.
In our calculation we find that only the z component of the spin current is nonzero in the
ground state. Fig.3 shows the persistent spin current Jz vs. magnetic flux at different SO
coupling strength. The persistent spin current is a periodic function of the magnetic flux
φ, which has the even parity symmetry Jz(−φ) = Jz(φ) and also an additional symmetry
Jz(φ) = Jz(π−φ). It is noted that the persistent spin current has quite different dependence
behaviors on magnetic flux compared with the persistent charge current in Fig.1. In the
presence of finite SO coupling, the persistent spin current is nonzero both for the systems
with odd N and even N at zero magnetic flux, it indicates that a persistent spin current can
be induced solely by SO interaction without accompany a charge current. This phenomena
is also shown in Ref.[17] where a SO coupling/normal hybrid ring was considered.
In Fig.4 the persistent spin current Jz as a function of SO coupling strength is plotted.
In the absence of SO coupling αR = 0, the persistent spin current is exactly zero for both
even and odd number electron system. In the presence of SO coupling, The persistent spin
current becomes nonzero and shows oscillations with increasing αR. It can change from
positive to negative values or vice versa by tuning the SO coupling strength. The sign of the
persistent spin current also shows dependence on the enclosed magnetic flux. For the system
with odd N , there is abrupt jumps in the curve of persistent spin current at certain value of
αR, the reason for the jump is the same as that in the charge current, and is due to the level
crossing in the energy spectrum. It is noted that the position of the jump coincides with
that of the persistent charge current. This kind of characteristic feature of the persistent
currents might provide a useful way to detect the SO coupling effects in semiconductor ring
system.
IV. CONCLUSIONS
In summary, we have investigated the Rashba SO coupling effect on the persistent charge
current and spin current in a mesoscopic ring with an Anderson impurity. The Anderson
impurity leads to the Kondo effect and decreases the amplitude of the persistent charge and
spin current in the ring. In the semiconducting ring with SO interaction, the persistent
charge current changes significantly by tuning the SO coupling strength, e.g. from the
paramagnetic to diamagnetic current. Besides the persistent charge current, there also
exists a persistent spin current, which also oscillates with the SO coupling strength. It is
shown that at zero magnetic flux a persistent spin current can exist even without the charge
current. Since the persistent spin current can generate an electric field[31], one might expect
that experiments on semiconductor ring with Rashba SO coupling can detect the persistent
spin current.
Acknowledgments
This project is supported by the National Natural Science Foundation of China, the
Shanghai Pujiang Program, and Program for New Century Excellent Talents in University
(NCET).
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http://arxiv.org/abs/cond-mat/0605748
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
(d)(b)
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 1: The persistent charge current vs. magnetic flux for a set of values for the spin-orbit
coupling strength(αR/∆ = 0.0(solid line),0.5(dashed line), 0.7(dotted line),1.0(dash-dotted line)).
The total number of electrons N = 99 (a), 100(b), 101(c), 102(d). We take the other parameters
∆ = 0.01, td = 0.3, ǫd − EF = −1.0, U = 2.0 in the calculation. The persistent charge current is
measured in units of I0 = eN∆.
0 1 2 3 4
-0.15
-0.10
-0.05
0 1 2 3 4
-0.10
-0.05
(c)(a)
0 1 2 3 4
-0.10
-0.05
0 1 2 3 4
-0.10
-0.05
FIG. 2: The persistent charge current as a function of the spin-orbit coupling strength. The
magnetic flux (Φ/Φ0 = 0.125(solid line),0.25(dashed line), 0.375(dotted line)).
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
0.0 0.2 0.4 0.6 0.8 1.0
-0.10
-0.05
0.0 0.2 0.4 0.6 0.8 1.0
-0.05
0.0 0.2 0.4 0.6 0.8 1.0
-0.05
FIG. 3: FIG.3: The persistent spin current Jz vs. magnetic flux for a set of values for the spin-
orbit coupling strength( with αR/∆ = 0.5(solid line),0.7(dashed line), 1.0(dotted line)). The panel
(a), (b), (c) and (d) corresponds the system with total number of electrons N = 99, 100, 101, 102,
respectively. The persistent spin current is measured in units of J0 = N∆, and we have taken the
other parameter values the same as that in Fig.1.
0 1 2 3 4
-0.10
-0.05
0 1 2 3 4
-0.10
-0.05
0 1 2 3 4
-0.10
-0.05
0 1 2 3 4
-0.10
-0.05
FIG. 4: FIG.4: The persistent spin current Jz as a function of the spin-orbit coupling strength.
The magnetic flux takes the value (Φ/Φ0 = 0.0(solid line),0.125(dashed line), 0.25(dotted line),
0.5(dash-dotted line)).
	introduction
	Mesoscopic ring with an Anderson impurity
	the persistent charge current and spin current 
	conclusions
	Acknowledgments
	References
