The persistent spin current (PSC) was proposed by many researchers in an equilibrium
system.38–40 Rashba first found out a background PSC in an infinite system with Rashba
SOC, and he pointed out that the definition of the conventional spin current need to be
redefined. The PSC is protected by the time reversal symmetry, and it can not be detected
by the method of the conventional charge current or of the spin accumulation. Some recent
results indicated that the PSC on a nanoring with SOC can be detected.39,40 Here we
analytically study the curvature-induced quantized energy levels on a nanoring and their
associated spin precessions. The quantized energy levels of the PSC and the transitions
between different energy levels provide a feasible way to detect the existence of the PSC.
To express the cubic Rashba Hamiltonian of the nanoring system in polar coordinates
(ρ, φ ), we will take a 1D nanoring system with cubic Rashba SOC based on Eq. (24).
Using the tensor transformations shown in Appendix I and Eq. (29), we can derive a
Hamiltonian with cubic Rashba SOC in a polar coordinate system:
H˜KCR= − ¯h2
where the Pauli matrices in a polar coordinate system are
σρ = ∂ ρ
In Eq. (30), the kinetic term in the unconfined φ -direction and the kinetic-induced
geometric effect are obtained in the first two terms. The cubic momentum in the third
term is related to the Rashba SOC in the unconfined φ -direction. The geometric potential
with an inverse linear dependence on the radius (1/ρ) will be coupled with the square
momentum in the unconfined φ -direction, and the coupling can lead to an extra
pseudo-dependence on the radius (3/ρ2) will be coupled with the linear momentum in the
uncon-fined φ -direction, and the coupling can lead to an extra pseudo-momentum term in the
fifth term. Finally, the geometric potential with an inverse cubic dependence on the radius
(−5/ρ3) give rise to an extra pseudo-potential term in the last term.
To express the cubic Dresselhaus Hamiltonian of the nanoring system in polar
coor-dinates (ρ, φ ), we will take a 1D nanoring system with cubic Dresselhaus SOC based on
Eq. (25). Similar to the previous case, using the tensor transformations shown in
Ap-pendix I and Eq. (29), we can derive a Hamiltonian with cubic Dresselhaus SOC in a
polar coordinate system:
In Eq. (32), the first two terms are the kinetic term in the unconfined φ -direction and
the kinetic-induced geometric effect. The cubic momentum in the third term is related to
the Dresselhaus SOC in the unconfined φ -direction. The linear momentum in the confined
ρ -direction contributes a geometric potential with an inverse linear dependence on the radius (1/ρ), and the geometric potential will be coupled with the square momentum in
ρ -direction contributes a geometric potential with an inverse quadratic dependence on the radius (3/ρ2), and the geometric potential will be coupled with the linear momentum in
the unconfined φ -direction in the fifth term. Finally, the cubic momentum in the confined
ρ -direction contributes a geometric potential with an inverse cubic dependence on the radius (−5/ρ3) in the last term.
We calculate straightforward the eigenenergies and eigenstates for the cubic Rashba
Hamiltonian of the nanoring system in Eq. (30). The eigenstates ψCR,m±
jfor each quantum number mjsatisfies the eigen equation
The exact eigenenergies for a Hamiltonian with cubic Rashba SOC will be
ECR,m±
and the corresponding eigenstates can be written as
Here I is the moment of inertia, ρ is the radius of the curvature, and mj refers to the
projection of the total angular momentum along a specified axis.
The cubic Rashba SOC is determined by the off-diagonal part of the Hamiltonian
matrix, and the effective cubic Rashba field is in the plane of the nanoring. The angular
kinetic energy is determined by the diagonal part of the Hamiltonian matrix, and the
orbital angular momentum causes the cubic Rashba spinors to tilt out of the ring plane.
In principle, the spin and orbital angular momenta affect the corresponding tilt angle,
and then the tilt angle is dependent of the quantum number mj. Furthermore, for the
cubic Rashba case, the eigenenergies and eigenstates in Eqs. (33) and (34) depend on
the integer quantum number mj≥ 0 and the signs ±. Because of the cubic Rashba
spin-splitting which corresponds to the signs ±, there are two quantum states for each quantum
number mj.
As for the cubic Dresselhaus Hamiltonian of the nanoring system in Eq. (32). The
eigenstates ψCD,m±
j for each quantum number mjsatisfies the eigen equation
The exact eigenenergies for a Hamiltonian with cubic Dresselhaus SOC will be
ECD,m±
and the corresponding eigenstates can be written as
ψCD,m±
∆CD,m± j = ± ECD,m±
j−2I¯h2h
mj+ 32
−14i
−β
ρ3
m3j+12m2j−34mj−158 .
The off-diagonal part of the Hamiltonian matrix determines the cubic Dresselhaus
SOC, and there exists the effective cubic Dresselhaus field in the plane of the nanoring.
The diagonal part of the Hamiltonian matrix determines the angular kinetic energy, and
the cubic Dresselhaus spinors are tilted out of the ring plane as a result of the orbital
angular momentum. We note that the corresponding tilt angle is determined by the spin
and orbital angular momenta, and thus the tilt angle is dependent of the quantum number
mj. Moreover, for the cubic Dresselhaus case, the eigenenergies and eigenstates in Eqs.
(35) and (36) depend on the integer quantum number mj≤ 0 and the signs ±. The cubic
Dresselhaus spin-splitting which corresponds to the signs ± can lead to two quantum
states for each quantum number mj. Hence, due to opposite quantum numbers mj, the
cubic Rashba and Dresselhaus spinors are travelling in opposite directions.
For a comparison of our result and the linear SOCs in a ring channel,41,42 we take
account of the linear Rashba and Dresselhaus SOCs in a Cartesian coordinate system:25,27
HLR= α
¯h(σxpy− σypx) and HLD= β
¯h(σypy− σxpx) .
(5.8)
Similarly, by applying the eigenfunction transformation to the linear SOCs based on
Eq. (37), a Hamiltonian with linear Rashba SOC in a polar coordinate system19 is
and a Hamiltonian with linear Dresselhaus SOC in a polar coordinate system19 is
H˜KLD= − ¯h2
Considering the linear Rashba Hamiltonian of the nanoring system in Eq. (38), we
can easily compute the eigenenergies and eigenstates. The eigenstates ψLR,m±
j for each quantum number mjsatisfies the eigen equation
The exact eigenenergies for a Hamiltonian with linear Rashba SOC are
and the eigenstates corresponding to the eigenenergies are given as
ψLR,m+ j(φ ) = exp imjφ ·
with a tilt angle given by
tan θLR= 2mρα
¯h2 .
The effective linear Rashba field lies in the plane of the nanoring, but a out-plane tilt
angle is used to describe the linear Rashba spinors in the presence of the orbital angular
momentum. Furthermore, in the case of a Hamiltonian with linear Rashba SOC, the
tilt angle becomes independent of the quantum number mj in the absence of an external
magnetic field.42 The eigenenergies and eigenstates in Eqs. (40) and (41) rely on the
integer quantum number m ≥ 0 and the signs ±. For each quantum number m , two
quantum states will arise due to the linear Rashba spin-splitting.
As regards the linear Dresselhaus Hamiltonian of the nanoring system in Eq. (39).
The eigenstates ψLD,m± j for each quantum number mjsatisfies the eigen equation
The exact eigenenergies for a Hamiltonian with linear Dresselhaus SOC are
ELD,m±
and the eigenstates corresponding to the eigenenergies are given as
ψLD,m+ j(φ ) = exp imjφ ·
with a tilt angle given by
tan θLD= 2mρβ
¯h2 .
The effective linear Dresselhaus field lies in the plane of the nanoring, but the linear
Dresselhaus spinors are characterized by a out-plane tilt angle due to the presence of the
orbital angular momentum. Moreover, in the case of a Hamiltonian with linear
Dressel-haus SOC, the tilt angle becomes independent of the quantum number mj in the absence
of an external magnetic field. The eigenenergies and eigenstates in Eqs. (42) and (43) rely
on the integer quantum number mj≤ 0 and the signs ±. For each quantum number mj,
there exist two quantum states as a result of the linear Dresselhaus spin-splitting. Similar
to the cubic SOCs, the linear Rashba and Dresselhaus spinors are travelling in opposite
directions because of opposite quantum numbers mj. Nevertheless, the eigenenergies of
the Rashba and Dresselhaus SOCs are the same if their coupling strengths are equal.
From the eigenenergies and eigenstates on the Rashba or Dresselhaus nanoring, the
results are just similar to the Rashba or Dresselhaus spinors on an infinite straight line
when the ring radius tends to infinity. However, for a nanoring with a rather small
ra-dius, the curvature-induced geometric effect becomes dominant. The nanoring confines
the Rashba or Dresselhaus spinors to the curved structure, and the curvature induces the
atomic-like bound states of the PSC on a nanoring.43 Hence, the quantized energy levels
in the atomic-like bound states strongly depend on the curvature of the nanoring for the
Fig. 5. The various spinors on the Rashba nanoring for (a) the ground state, (b) the first excited state, (c) the second excited state, (d) the third excited state, (e) the fourth excited state, and (f) the fifth excited state. The model parameters are α = 60 meV·nm, ρ = 9 nm, a = 2πρ, and m = 0.28me.
Fig. 6. The various spinors on the Rashba nanoring for (a) the sixth excited state, (b) the seventh excited state, (c) the eighth excited state, (d) the ninth excited state, (e) the tenth excited state. The model parameters are α = 60 meV·nm, ρ = 9 nm, a = 2πρ, and m= 0.28me.
existence of the PSC on a nanoring.
Furthermore, we take the Rashba coupling strength to be α = 60 meV·nm, the radius
of the curvature ρ = 9 nm, the circumference of the toric rings a = 2πρ, and the effective
heavy-hole mass m = 0.28me, where me is the electron rest mass. We then use these
parameters and apply the nearly-free electron approximation to numerically study the
complicated spin precession in the quantized energy levels. The various spinors for the
different quantized energy levels are shown in Figs.5and6.
For the quantized energy levels on the Rashba nanoring, we find that the spin
preces-sion can be either clockwise or counterclockwise in opposite spinors. We note that the
spinor and anti-spinor have opposite signs ±, and we can then define the chirality
associ-ated with the signs. When the sign is a plus, the precession direction is always clockwise,
and the spinor normal to the nanoring will point towards the ring center. When the sign is
a minus, the precession direction is always counterclockwise, and the spinor normal to the
nanoring will point away from the ring center. Therefore, the positive chirality associates
with a plus sign, and the negative chirality associates with a minus sign. The chirality
associated with the spinor and anti-spinor satisfies the time reversal symmetry. Moreover,
the vertical intersection number of the spin precession and the nanoring is defined to be
the winding number. We also find that the winding number is dependent of the quantum
number mj, and always equals to 2mj+ 1. There is no even winding number because of
For the quantized energy levels on the Dresselhaus nanoring, the physical phenomenon
is similar to that on the Rashba nanoring. The time reversal symmetry can lead to the
chi-rality associated with the spinor and anti-spinor, and the space inversion asymmetry can
lead to the absence of even winding number. For the ground state of the spin precession on
the Rashba or Dresselhaus nanoring, it is a small modification of the Goldstein mode on
a flat surface, and the winding number of the ground state is suppressed by the Goldstein
mode.
For comparison, the topological insulator is a new state of the quantum matter with
strong SOC and have the insulating bulk and the metallic surface with an odd number of
Dirac cones. The Rashba and Dresselhaus SOCs are just two kinds of SOCs. In general,
the SOC break the space inversion symmetry, but preserve the time reversal symmetry.
The SOC and the time reversal symmetry combine to form the topologically nontrivial
state described by a topological invariant, like the Chern number or the winding number.