NANOBUBBLE
Recently, a cylindrical nanotube with a controllable radius can be fabricated by a
method known as the self-rolling strained semiconductor layers.29 After this advance,
the study of the nanotube system has increasingly attracted researchers’ attention. For
instance, the experimental measurement of magnetotransport in a tubular 2DEG system
has been carried out.30,31 Also, a spherical nanobubble with a controllable curvature can
be formed on a very elastic graphene film.32 The electronic properties of the graphene
nanobubble are strongly modified by the strain and the surface curvature.13 Nevertheless,
very few theoretical studies consider the curvature-induced terms in the Hamiltonian,
and thus the pseudo-potential, pseudo-kinetic, and pseudo-momentum terms derived in
Sec.IIwere hitherto neglected.33,34 We give the exact derivations of 2D nanotubular and
nanobubble systems below, and we also discuss the associated curvature-induced physical
phenomena.
For a [111]-grown quantum well in Cartesian coordinates, the x, y, and z axes
corre-spond to [100], [010], and [001] crystallographic directions, respectively, and the Rashba
SOC25,26 is expressed as
HLRS=α
¯h (σxpy− σypx) +α
¯h (σypz− σzpy) +α
¯h (σzpx− σxpz) ,
(3.1)
where α is the Rashba coupling strength whose unit is meV·nm.
For a [100]-grown quantum well in Cartesian coordinates, the x, y, and z axes
corre-spond to [100], [010], and [001] crystallographic directions, respectively, and the
Dres-selhaus SOC27,28 is expressed as
HCDS=β
¯h3σxpx
py 2− pz 2 + β
¯h3σypy
pz 2− px 2
+ β
¯h3σzpz
px 2− py 2 ,
(3.2)
where β is the Dresselhaus coupling strength whose unit is meV·nm3.
In such conditions, the components of the Rashba tensor Si j in a Cartesian coordinate
system are
and the components of the Dresselhaus tensor Sii j j in a Cartesian coordinate system are
Sxxyy= Syyzz= Szzxx= β
¯h3 and Szzyy= Syyxx= Sxxzz= −β
¯h3.
(3.4)
For a 2D nanotubular system with SOC based on Eqs. (12) and (13), we will express
the Hamiltonian of the nanotubular system in cylindrical coordinates (ρ, φ , z). Using the
tensor transformations given in AppendixFand Eq. (10), we obtain a Hamiltonian with
linear Rashba SOC in a cylindrical coordinate system:
H˜KLR= − ¯h2
The first two terms in Eq. (16) are the kinetic terms and the geometric effect due to the
contribution of the kinetic term for cylindrical geometry. The third and fourth terms are
the Rashba SOC along the unconfined φ - and z-directions, respectively. It is interesting
to point out that the linear derivatives with respect to φ and z become anisotropic in
cylindrical symmetry. The effective strength of the Rashba SOC along z-direction remains
as it is on a flat surface; the effective strength of the Rashba SOC along φ -direction
increases inversely with the radius. As for the remaining term, the linear momentum in the
confined ρ-direction contributes a geometric potential with an inverse linear dependence
on the radius (1/ρ).
In order to compare our results with the previously studied SOC in a half-ring
chan-nel,19we let z → 0, and Eq. (16) reduces to a Hamiltonian with Rashba SOC in a 1D toric
ring of the nanotubular system. If we assume that the momentum along z-direction tends
to zero in the nanotubular system, the simplified Hamiltonian must remain Hermitian.
Hence, its form must be
H˜KLR(z → 0) = − ¯h2 2m
1 ρ2
∂2
∂ φ2− ¯h2 2m
1 2ρ
2
− iα [cos φ σx+ sin φ σy
− (sin φ + cos φ ) σz]1 ρ
∂
∂ φ +1
2iα [sin φ σx
− cos φ σy+ (cos φ − sin φ ) σz] 1 ρ
.
(3.6)
Similarly, from the tensor transformations given in AppendixFand Eq. (11), a
Hamil-tonian with cubic Dresselhaus SOC in a cylindrical coordinate system will be
The first two terms in Eq. (18) are the kinetic terms and the geometric effect due to the
contribution of the kinetic term for cylindrical geometry. The cubic momenta in the third
term are related to the Dresselhaus SOC in both unconfined φ - and z-directions. 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 the unconfined φ - or z-direction in the fourth term. Also, the
square momentum in the confined 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 - or z-direction in the last term.
For a 2D nanobubble system with SOC based on Eqs. (12) and (13), we will express
the Hamiltonian of the nanobubble system in spherical coordinates (ρ, θ , φ ). Similar to
the previous case, using the tensor transformations given in AppendixGand Eq. (10), we
obtain a Hamiltonian with linear Rashba SOC in a spherical coordinate system:
The first term in Eq. (19) only includes the kinetic terms, and there is no geometric
effect in the kinetic term for spherical geometry. The second and third terms are the
Rashba SOC along the unconfined θ - and φ -directions, respectively. We also point out
that the linear derivatives with respect to θ and φ are anisotropic in spherical symmetry.
The effective strength of the Rashba SOC along θ - or φ -direction increases inversely with
the radius; the effective strength of the Rashba SOC along φ -direction is also a function of
θ . As for the remaining term, the linear momentum in the confined ρ -direction contributes a geometric potential with an inverse linear dependence on the radius (2/ρ).
For a comparison of our results and the SOC in a half-ring channel,19 we let θ →
π /2, and Eq. (19) reduces to a Hamiltonian with Rashba SOC in a 1D toric ring of the
nanobubble system. Assume further that the momentum along θ -direction tends to zero in
the nanobubble system, and thus the influence of the principal curvature along θ -direction
becomes insignificant. Hence, in order to maintain the Hermitian property, the simplified
Hamiltonian must be
Then, from the tensor transformations given in AppendixGand Eq. (11), a
Hamilto-nian with cubic Dresselhaus SOC in a spherical coordinate system will be
H˜KCD= − ¯h2
The first term in Eq. (21) only includes the kinetic terms, and the geometric effect
in the kinetic term does not arise here for spherical geometry. The cubic momenta in the
second term are related to the Dresselhaus SOC in both unconfined θ - and φ -directions.
The linear momentum in the confined ρ-direction contributes a geometric potential with
an inverse linear dependence on the radius (2/ρ), and the geometric potential will be
coupled with the square momentum in the unconfined θ - or φ -direction in the third term.
Also, the square momentum in the confined ρ-direction contributes a geometric potential
with an inverse quadratic dependence on the radius (8/ρ2), and the geometric potential
will be coupled with the linear momentum in the unconfined θ - or φ -direction in the last
term.
We use the nearly-free electron approximation to numerically calculate the electronic
band structures and their associated spinors. In these calculations we take the Rashba
cou-pling strength to be α = 60 meV·nm, the Dresselhaus coucou-pling strength β = 60 meV·nm,
the radius of the curvature ρ = 9 nm, the circumference of the toric rings a = 2πρ, and
the effective electron mass m = 0.04me, where meis the electron rest mass.
The solid line in Fig. 3(a) shows the band structure for the kinetic energy and its
geo-metric potential, and for comparison the dashed line in Fig. 3(a) only refers to the kinetic
energy. For a curved surface with a constant curvature, the geometric potential causes a
constant shift in the kinetic energy. The band structures of the three different toric rings
with SOC are shown in Figs. 3(b), 3(c) and 3(d). Fig. 3(b) shows the band structure given
19
Fig. 3. Electronic band structures of the different toric rings for (a) Kinetic energy and geometric potential, (b) Rashba or Dresselhaus SOC on a flat surface, (c) Rashba SOC on a cylindrical surface, (d) Rashba SOC on a spherical surface, (e) Rashba SOC with band mixing on a cylindrical surface, and (f) Rashba SOC with band mixing on a spherical surface. The model parameters are α = 60 meV·nm, β = 60 meV·nm, ρ = 9 nm, a = 2πρ, and m = 0.04me.
shows that two more states appear from the symmetry breaking of the up- and down-spin
under the SOC. It is interesting to mention that the two dispersions of the Rashba and
Dresselhaus spin-splittings are the same if their coupling strengths are equal. However,
the associated spinors are dramatically different, and they are shown in Figs. 4(a) and
4(b). With the equal coupling strengths, both Rashba and Dresselhaus SOCs provide an
equal k-dependent spin-splitting while the spin precessional phases are different.21,22
When we compare the band structure of the half-ring channel on a flat surface (Fig.
3(b)) with the band structure of the toric ring on a cylindrical surface (Fig. 3(c) from
Fig. 4. The various spinors on the different toric rings for (a) Rashba SOC on a flat surface, (b) Dresselhaus SOC on a flat surface, (c) Rashba SOC on a cylindrical surface, and (d) Rashba SOC on a spherical surface. The model parameters are α = 60 meV·nm, β = 60 meV·nm, ρ = 9 nm, a = 2π ρ , and m = 0.04me.
Eq. (17)), it seems that the band structure for the toric ring is shifted, and a larger
k-dependent spin-splitting is observed. This can be understood from the modification of the
geometric potential and of the k-dependent Rashba SOC. For a comparison between Eqs.
(17) and (20), the difference of the two Hamiltonians on the different toric rings is only
a constant potential change. Therefore, the band structure of the toric ring on a spherical
surface (Fig. 3(d) from Eq. (20)) is just shifted by the geometric potential. The various
spinors on the cylindrical and spherical surfaces are shown in Figs. 4(c) and 4(d) from
Eqs. (17) and (20), respectively, and their difference is also quite significant. Hence, it
is still necessary to take into account the out-of-plane spin component on a cylindrical or
spherical surface. The nonzero principal curvature on a curved surface is significant for
the motion of electrons, and thus spin precession on a curved surface is different from that
on a flat surface.
Furthermore, we consider the Hamiltonian of a corrugated single-layer graphene
com-bined with the atomic SOC of carbon,16 the effective SOC for π band states is written as
HSOC=
pseudo-spin Pauli vector, respectively, ζ and ζ0are the material parameters, ν = ν
*
are the geometric parameters, and τz = 1 or −1 denotes the Dirac point. Here,
the angle ω is defined counterclockwise from the y-axis to one carbon-carbon bond. For
ω = 0, the Hamiltonian in Eq. (22) can be separated into two terms, and then we obtain
HSOC= ζ ν [σx⊗ τzsy− σy⊗ sx] + ζ0τzI⊗*µ ·
*s, (3.12)
where I is a unit matrix, and the symbol ⊗ denotes the direct product. The first term in
Eq. (23) is the off-diagonal part of the Hamiltonian, and such form is the Rashba-type
SOC; it corresponds to the first term of the linear Rashba SOC on a curved surface in Eq.
(7). The second term in Eq. (23) is the diagonal part of the Hamiltonian, and it is the
geometric potential induced by the surface curvature; it corresponds to the second term
of the linear Rashba SOC on a curved surface in Eq. (7). The forms of the two
Hamilto-nians in Eqs. (7) and (23) are the same, but with a modification of the Rashba coupling
strength for different materials. Hence, in order to construct the linear Rashba-type SOCs
of most curved materials, we need to modify the Rashba coupling strength α by inserting
the material parameters ξ and ξ0 for different materials,10,16 i.e., αξ and αξ0. In the
in-stance of a corrugated single-layer graphene with π band states, the material Rashba-type
coupling strength is αξ = aα (εp− εs) Vppπ +Vppσ /12Vspσ2≈ 0.15 meV·nm,10,16and the
curvature-induced Rashba-type coupling strength is αξ0= aαVppπ /2 Vppπ −Vppσ ≈ 0.21
meV·nm,10,16 where a is the lattice constant, α is regarded as the atomic SOC strength of
p orbitals, εs and εpare the atomic energies for s and p orbitals, respectively, and Vspσ, Vppσ
and Vppπ represent the coupling strengths for σ and π bands between nearest-neighbor s
and p orbitals or between nearest-neighbor p orbitals. Similarly, for constructing the cubic
Dresselhaus-type SOCs of most curved materials, we need to modify the Dresselhaus
cou-pling strength β by inserting the material parameters ξ and ξ0for different materials, i.e.,
β ξ and β ξ0. Nevertheless, for a curved surface, the linear Rashba SOC only induces the extra potential term, but the cubic Dresselhaus SOC can induce the extra
pseudo-kinetic and pseudo-momentum terms. Therefore, for the cubic Dresselhaus-type SOC,
the second term in Eq. (23) must be modified according to our Eq. (9).
For spin transport in the ballistic regime, we use the nearly-free electron
approxi-plane-wave bases with spinors. Here only the band mixings from an external magnetic
field are discussed, and the other possible band mixing between s and p orbitals is not
analyzed. However, from the second-order k · p perturbation theory, we can obtain the
effective mass of the n-th band, and also obtain the modification of the Rashba coupling
strength, i.e., the material parameters ξ and ξ0. Hence, the band mixing between
multi-bands can be introduced by the material parameters ξ and ξ0. For a corrugated
single-layer graphene with π band states, the influence of the band mixing can be studied using
the second-order k · p perturbation theory with the atomic SOC and the local curvature
as two weak perturbations.10,16 The curvature effect breaks the isotropy of the lattice,
and leads to an effective anisotropic coupling between σ and π bands in the momentum
space.10 Moreover, it was reported that an external magnetic field also plays an important
role in controlling the band mixing between σ and π bands.10 Here we apply an external
magnetic field to a cylindrical or spherical surface, and the resultant spin-splitting with
band mixing due to the external magnetic field appears as shown in Figs. 3(e) and 3(f).
In the absence of an external magnetic field, the spinors will be position-dependent, and
no band gap will arise, as is apparent in Figs. 3(b), 3(c) and 3(d). However, when an
ex-ternal magnetic field is uniformly applied to the entire cylindrical and spherical surfaces,
and the direction of the uniform field is not parallel to the spin directions, the external
magnetic field asymmetrically shifts the energies of different spinors with opposite
mo-and 3(f). Hence, the bmo-and-crossing points are obviously broken, mo-and local bmo-and gaps are
also observed.