The cubic Rashba SOC is due to a structure inversion asymmetry (SIA) of the
interfa-cial potential, and the cubic Dresselhaus SOC is due to a bulk inversion asymmetry (BIA)
of the crystal potential. The spin-splitting of the heavy-hole states is mediated by a
cou-pling to the light-hole states, so that the spin-splitting is proportional to k3.35–37 For heavy
holes in the quantum well grown along the [001]-direction, the x, y, and z axes correspond
to [100], [010], and [001] crystallographic directions, respectively, and we have the total
angular momentum with spin projection ±3/2 along the growth direction for heavy holes.
The cubic Rashba SOC in a Cartesian coordinate system is represented as
HCRS= i α
2¯h3 σ+p3−− σ−p3+ , (4.1)
and the cubic Dresselhaus SOC in a Cartesian coordinate system is represented as
HCDS= β
2¯h3(σ+p−p+p−+ σ−p+p−p+) , (4.2)
where the notations p± = px± ipy and σ± = σx± iσy denote the ladder operators for
the momentum components and the Pauli matrices, respectively, ¯h is the reduced Planck
constant, α is the cubic Rashba coupling strength whose unit is meV·nm3, and β is the
cubic Dresselhaus coupling strength whose unit is meV·nm3.
To study the cubic Rashba or Dresselhaus SOC in a curved two-dimensional hole
system, we need to analyze the curved system in curvilinear coordinates q1, q2, q3. In
general, the position in curvilinear coordinates is
*
R q1, q2, q3 =*r q1, q2 +q3N qˆ 1, q2,
where *r q1, q2 and ˆN q1, q2 are the parametric equation and the unit normal vector
of the curved system. The confined potential of the curved system is similar to a delta
function potential well, and its form is expressed as Eq. (2).
For the Rashba or Dresselhaus SOC with cubic momenta, the general form in
curvi-linear coordinates is written as
HCS=Schi jσcphpipj
where Schi jare the components of the cubic Rashba or Dresselhaus tensor, σcare the Pauli
matrices, and the index values are 1, 2, and 3. We note that the metric components are
defined as gi j = ∂ For obtaining the cubic Rashba or Dresselhaus Hamiltonian in curvilinear coordinates,
we can use the tensor transformation rules:
pi= ∂ qi
where ¯qand q represent the old and new coordinates, respectively, and the index values
are 1, 2, and 3.
We apply the eigenfunction transformation to the cubic Rashba or Dresselhaus SOC.
The derivation is quite nontrivial for its nonlinear dependence on the momentum, and it is
shown in AppendixHfor brevity and clarity of the main text. Hence, from the effective
eigen equation in Eq. (H2), the effective Hamiltonian of the cubic Rashba or Dresselhaus
SOC in curvilinear coordinates can be written as
H˜CS=i¯h3Sclmnσcglu ∂
∂ qu
gmv ∂
∂ qv
gnw ∂
∂ qw
−3
2i¯h3Sclm3σcTr(αmn) glv ∂
∂ qv
gmw ∂
∂ qw
+3
4i¯h3Scl33σc h
3Tr (αmn)2− 4Det (αmn)i glu ∂
∂ qu +3
8i¯h3Sc333σc h
12Tr (αmn) Det (αmn) − 5Tr (αmn)3i ,
(4.5)
where the indices l, m and n are 1 and 2, and other index values are 1, 2, and 3. We
note that αmn are the Weingarten curvature matrix elements of the curved surface,24 and
the associated eigenvalues are usually the orthogonal principal curvatures. Tr (αmn) is the
trace of the Weingarten curvature matrix, and Det (αmn) is the determinant of the
Wein-garten curvature matrix. The first term in Eq. (28) is the k3-dependence of the Rashba or
Dresselhaus SOC, and others are the curvature-induced geometric effects from the cubic
Rashba or Dresselhaus SOC. Those curvature-induced terms correspond to the
pseudo-kinetic, pseudo-momentum, and pseudo-potential terms on a curved surface.
From Eqs. (4) and (28), we obtain a general form of the Hamiltonian which includes
the effect of the confinement of the kinetic term and of the cubic Rashba or Dresselhaus
SOC in curvilinear coordinates:
is the reduced metric tensor, and m is the effective heavy-hole mass in solids. The first
term is the ordinary kinetic term and its geometric potential induced by the surface
cur-vature. The other terms in Eq. (29) are just from the cubic Rashba or Dresselhaus
Hamil-tonian on a curved surface.
We emphasize that the momentum along the confined normal direction of the
ultra-thin film will induce a geometric effect in the Hamiltonian. Once the momentum in the
confined normal direction is coupled with the spin, the geometric effect on the SOC will
appear due to the surface curvature. For the cubic Rashba or Dresselhaus SOC on a curved
surface, the effective Hamiltonian becomes very complicated because the momentum in
the confined normal direction is also coupled with the momentum in the unconfined
di-rection. Hence, on a curved surface, the geometric potential with a linear dependence
of the curvature will be coupled with the k2-dependent term in the unconfined direction.
The k2-dependent spin-splitting will appear, and it is similar to that induced by an
ex-tra pseudo-kinetic term. Similarly, the geometric potential with a quadratic dependence
of the curvature will also be coupled with the k-dependent term in the unconfined
di-rection. The k-dependent spin-splitting will also arise, and it is similar to that induced
by an extra momentum term. The strengths of the kinetic and
pseudo-momentum terms depend on the surface curvature and the cubic Rashba or Dresselhaus
coupling strength. Obviously, the k-cubic term in the unconfined directions can lead to a
k3-dependent spin-splitting. The k-cubic term in the confined normal direction contributes
a geometric potential with a cubic dependence of the curvature. Its influence is similar to
an extra pseudo-potential term, and its strength depends on the surface curvature and the
cubic Rashba or Dresselhaus coupling strength. Therefore, there are four different origins
of effective magnetic fields. The first effective magnetic field is due to the k3-dependence
of the Rashba or Dresselhaus SOC in the unconfined directions, and the other effective
magnetic fields originate, respectively, from the extra pseudo-kinetic, pseudo-momentum,
and pseudo-potential terms.