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Intersubband-induced spin-orbit interaction in quantum wells
Rafael S. Calsaverini,1 Esmerindo Bernardes,1, ∗ J. Carlos Egues,1, † and Daniel Loss2
1Instituto de Fı´sica de Sa˜o Carlos, Universidade de Sa˜o Paulo, 13560-970 Sa˜o Carlos, SP, Brazil
2Department of Physics, University of Basel, CH-4056 Basel, Switzerland
(Dated: October 20, 2008)
Recently, we have found an additional spin-orbit (SO) interaction in quantum wells with two subbands
[Bernardes et al., Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is non-zero even in symmet-
ric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its
strength is comparable to that of the ordinary Rashba. Starting from the 8 × 8 Kane model, here we present
a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the
self-consistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric
modulation-doped wells with two subbands. We consider gated structures with either a constant areal electron
density or a constant chemical potential. In the parameter range studied, both models give similar results. By
considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells,
we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly,
we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly
across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric
configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them
essentially constant as function of the applied bias.
1. INTRODUCTION
The coupling between spatial and spin degrees of freedom
in semiconductors provides an interesting possibility for co-
herently manipulating the electron spin via its orbital (charge)
motion. For instance, the proposal of Datta and Das1 for a spin
field-effect transistor highlights the use of the spin-orbit (SO)
interaction of Rashba,2,3,4 which is electrically tunable,5,6 to
control – via spin rotation – the flow of electrons between fer-
romagnetic source and drain.
In addition to the Rashba SO coupling present in het-
erostructures with structural inversion asymmetry in the con-
fining potential, there is the Dresselhaus SO interaction7
present in both bulk and confined structures with inversion
asymmetry in the underlying crystal lattice. These spin or-
bit interactions have played an important role in the exciting
field of semiconductor spintronics as they underlie a num-
ber of interesting physical phenomena and potential spin-
tronic applications.8,9,10 For instance, the effective zitterbe-
wegung of spin-polarized wave packets injected into SO cou-
pled two-dimensional (2D) electron gases is a very interest-
ing possibility.11,12 The interplay of the Rashba and Dres-
selhaus interactions can give rise to conserved spin-rotation
symmetries13,14 relevant for devising robust SO-based devices
operating in the nonballistic regime.13
Recently, a new type of SO interaction arising in quantum
confined systems with two subbands has been found15. Un-
like the usual Rashba SO, this new SO term is nonzero even
in wells with full structural inversion symmetry (and hence
it does not produce spin splitting). This essentially follows
from the distinct parities of the confined states (even and odd),
which can couple via the derivative of a symmetric poten-
tial. This intersubband-induced SO coupling is quadratic in
the crystal momentum, unlike the Rashba and the (linearized)
Dresselhaus terms in wells.16 As shown in Ref. 15, this SO
coupling can give rise to an unusual zitterbewegung (both in
position and in spin17) and a nonzero spin Hall conductivity.
Here we complement and extend the work of Ref. 15: (i)
We present a more thorough derivation of the intersubband-
induced SO interaction, starting from the 8 × 8 Kane
model18,19,20 within the k · p approach. We also slightly gen-
eralize the derivation for confined systems with more than
two subbands and structurally asymmetric potentials in which
the usual Rashba-type SO interaction is present. (ii) We per-
form a detailed investigation of the relevant SO couplings via
a self-consistent scheme where we solve both Poisson and
Schro¨dinger equations numerically (Numerov method) within
the Hartree approximation. We consider realistic modulation-
doped single and double quantum wells with applied external
biases, which can change the spatial symmetry of the wells,
and having either a constant areal electron density or a con-
stant chemical potential.
Our simulations focus on wells with two subbands.
For nonzero applied biases we calculate not only the
intersubband-induced SO coupling η but also the Rashba-type
couplings α0, α1 for the first and second subbands, respec-
tively. For completeness, we also calculate the linearized
Dresselhaus SO couplings for each subband.21 For both the
constant density and constant chemical-potential models con-
sidered, we find sizable values of the intersubband SO cou-
pling η as compared to the usual Rashba and Dresselhaus
couplings. Interestingly, for double wells near the symmetric
(zero-bias) configuration we find that η has a resonant behav-
ior, changing its magnitude by a factor of 10. On the other
hand, the Rashba couplings for the first and second subbands
abruptly change signs around the zero-bias voltage. The Dres-
selhaus couplings do not show any noticeable behavior around
this point, being essentially constant as a function of the ap-
plied bias.
We note that the SO coupling constants η, α0, and α1 con-
tain contributions from the potential well (and barrier) offsets,
the electronic Hartree potential, and the external gate plus the
modulation doping potentials. For the single wells investi-
gated here, the external gate (+ modulation doping) is the

2dominant contribution to α0 and α1, while η is mostly de-
termined by the Hartree and structural offset contributions.
On the other hand, for the double wells studied the electronic
Hartree potential is the dominant contribution to η, α0, and
α1. Interestingly, the Hartree potential in this case is highly
influenced by the external gate, particularly around the sym-
metric (zero-bias) configuration, as the electrons can localize
in either well for small (positive or negative) changes in the
gate potential. This renders η, α0, and α1 more amenable to
gate modulations in double wells as compared to single wells.
Next we outline our work.
In Sec. 2 we review the k · p approach and the Kane
model. In Sec. 3 we present a detailed derivation of our effec-
tive Hamiltonian for electrons in heterostructures with many
confined states within the Kane model. In this section we
also show the relevant expressions for the new intersubband-
induced SO coupling η and those for the Rashba α (and Dres-
selhaus β) SO couplings as well. In Sec. 4 we describe the
quantum wells investigated and (briefly) the standard self-
consistent calculation performed. We present our results and
discussions in Sec. 5. In this section we focus specifically on
realistic single and double-well systems. Section 6 summa-
rizes our work. In Appendices A and B we show details of
our self-consistent scheme to solve the relevant Schro¨dinger
and Poisson equations.
2. k · p APPROACH AND KANE MODEL
Here we briefly review the k·p approach and use it to obtain
the 8 × 8 Kane model relevant for our derivation of the new
intersubband-induced SO coupling.19,20
2.1. Basics of the k · p method
The single-particle Hamiltonian for an electron with mo-
mentum p in a periodic potential19,22 V(r) with SO is
H =
p2
2m0
+ V(r) + ~
4m20c2
σ × ∇V(r) · p, (1)
where m0 is the bare electron mass and σ is a vector oper-
ator defined in terms of the Pauli matrices. With the help of
Bloch’s theorem ψnk(r) = exp(ik · r) unk(r) [unk(r) has the pe-
riodicity of the underlying Bravais lattice] we can rewrite the
Schro¨dinger’s equation Hψnk = εnkψnk, where n indexes the
distinct solutions for each k vector, in the form
[H(k = 0) + W(k)] unk(r) =
(
εnk −
~2k2
2m0
)
unk(r), (2)
with
H(k = 0) = − ~
2
2m0
∇2 + V(r) + ~
4m20c2
σ × ∇V(r) · p, (3)
W(k) = ~
m0
k ·
(
p + ~
4m0c2
σ × ∇V(r)
)
. (4)
As usual, to solve Eq. (2), we expand unk in terms of the
eigenstates ul0 at k = 0 [i.e., W(k = 0) = 0] obtained from
H(k = 0) ul0(r) = εl0 ul0(r), (5)
where l = 1, 2, . . .N (in principle, N → ∞) indexes the dis-
crete set of levels at k = 0 [note that Eq. (5) contains the SO
interaction, even though W(0) = 0]. Substituting
unk(r) =
N
∑
l=1
anl(k) ul0(r), (6)
into Eq. (2) and projecting the resulting expression onto the
ul′0(r) eigenstate, we find19
N
∑
l=1
[
(εl0 − εnk +
~2k2
2m0
)δll′
+
〈l′| ~
m0
k · p+ ~
2
4m20c2
k · σ × ∇V(r)|l〉
]
anl(k) = 0. (7)
Here we use the notation 〈r|l〉 = ul0(r) and define
〈l′|A|l〉 =
∫
d3r u∗l′0 A ul0, (8)
with A denoting a Hermitian operator.
2.2. 8 × 8 Kane model - bulk case
As usual, in order to solve Eq. (7) we have to truncate the
basis set by considering a finite number N of zone-center ba-
sis functions ul0(r). In addition, since the k = 0 Hamiltonian
[Eq. (3)] contains a SO term, it is convenient to choose linear
combination of basis functions which are eigenstates of the to-
tal angular momentum J = L+S, and its z component Jz; here
L and S denote the orbital and spin angular momenta, respec-
tively. In II-VI and III-V (both zincblend) compounds the rel-
evant conduction and valence bands arise from the “bonding”
p-type and “anti-bonding” s-type states, respectively. Follow-
ing the notation of Refs. 19 and 23, we summarize in Table I
the set of eight zone-center wave functions we consider here
(the kets |JJz〉 are also shown), which are the eigenstates of the
zone-center Schro¨dinger’s equation (5) for the periodic part of
the Bloch function. Note that we use the standard state vector
notation |S 〉, |X〉, |Y〉, and |Z〉 to denote the symmetry of the
corresponding “atomic orbitals” (tight-binding view).
Using the ordered basis states u1, . . . , u8 in Table I we can
easily write out the matrix Hamiltonian [Eq. (7)]19,20,24

3H8×8 =

























































~2k2
2m0 0 −
1√
2
Pk+
√
2
3 Pkz
1√
6
Pk− 0 − 1√3 Pkz −
1√
3
Pk−
0 ~2k22m0 0 −
1√
6
Pk+
√
2
3 Pkz
1√
2
Pk− − 1√3 Pk+
1√
3 Pkz
− 1√
2
Pk− 0 ~
2k2
2m0 − Eg 0 0 0 0 0
√
2
3 Pkz −
1√
6
Pk− 0 ~
2k2
2m0 − Eg 0 0 0 0
1√
6
Pk+
√
2
3 Pkz 0 0
~2k2
2m0 − Eg 0 0 0
0 1√
2
Pk+ 0 0 0 ~
2k2
2m0 − Eg 0 0
− 1√3 Pkz −
1√
3 Pk− 0 0 0 0
~2k2
2m0 − Eg − ∆g 0
− 1√3 Pk+
1√
3 Pkz 0 0 0 0 0
~2k2
2m0 − Eg − ∆g.

























































(9)
where P is the usual Kane matrix element18
P = −i ~
m0
〈
S |px|X
〉
= ~
√
EP
2m0
, (10)
expressed in terms of the parameter EP (Ref. 25) and k± = kx±
iky. We have also used that
〈
S |px|X
〉
=
〈
S |py|Y
〉
=
〈
S |pz|Z
〉
.
Equation (9)) is the 8×8 Kane Hamiltonian18 describing the s-
type conduction and p-type valence bands around the Γ point
in zincblend compounds. Note that the diagonal elements in
Hamiltonian (9) correspond to the eigenenergies εl0 of Eq. (5):
ε10 = ε20 = 0 (“conduction-band states,” defined as the zero
of energy), ε30 = ε40 = ε50 = ε60 = −Eg (“heavy” and “light”
hole bands), and ε70 = ε80 = −Eg −∆g (“split-off” hole band).
Here,
∆g =
3~2
4m20c2
〈
X|∂V∂y
∂
∂x −
∂V
∂x
∂
∂y |Y
〉 (11)
is the “atomic” SO parameter defining the split-off gap; see
Fig. 1(a), which schematically shows the conduction and va-
lence bands of a zincblend structure. The circles indicate the
k = 0 eigenenergies.
The Kane model treats exactly the conduction-valence band
couplings within the truncated set of eight band-edge wave
functions. It is important to emphasize that we have neglected
contributions from the k-dependent SO term in Eq. (7), when
constructing the Kane Hamiltonian (9).18 The SO interaction
is accounted for only within the zone center Schro¨dinger’s
equation (5) (parameter ∆g above). The diagonalization of
the Kane Hamiltonian gives the dispersions εn,k around the Γ
point. It is known that the Kane model presented here is not
accurate for valence bands19 (e.g., wrong sign of the heavy
hole masses). However, it provides a simplified and accurate
description for the conduction electrons, which is the focus of
our work. Next we discuss the Kane model in the context of
heterostructures.
2.3. Kane model for quantum wells
Following Refs. 19 and 20 we can straightforwardly gen-
eralize the bulk Kane model of Sec. 2.2 to heterostructures.
Essentially, we have to introduce position-dependent (growth
direction) band gaps which represent the different compounds
comprising the heterostructure, e.g., Fig. 1(b). In this case, the
form of the resulting Kane Hamiltonian is similar to that of
bulk but with z-dependent diagonal matrix elements and with
kz → −id/dz. More specifically, defining E6 = H11 = H22,
E8 = H33 = H44 = H55 = H66, and E7 = H77 = H88, we have
for the double quantum well of Fig. 1(b)
E6 =
~2k2
2m0
+ VH(z) + h6(z), (12)
E8 =
~2k2
2m0
+ VH(z) − h8(z) − Eg, (13)
E7 =
~2k2
2m0
+ VH(z) − h7(z) − Eg − ∆g, (14)
with k2 = k2q + k2z , Eg and ∆g being the fundamental and split-
off band gaps in the well region, respectively, and
hi(z) = δi hw(z) + δb i hb(z), i = 6, 7, 8, (15)
where hw(z) is a dimensionless profile function describing the
shape of a square well of width Lw [and unit depth, Fig. 1(b)];
similarly, hb(z) describes the shape of the central square bar-
rier, Fig. 1(b). The parameters δ6, δ7, δ8, δb6, δb7, and δb8 de-
note the relevant band offsets between the well and the lateral
and central barriers for conduction and valence bands. Defin-
ing the zero of energy at the bottom of the conduction well
[see Fig. 1(b)], we have
δ8 = Ew − Eg − δ6, δ7 = δ8 + ∆w − ∆g, (16)
δb8 = Eb − Eg − δb6, δb7 = δb8 + ∆b − ∆g. (17)
The corresponding expressions for a single well can readily
be obtained from the above by setting the δbi’s to zero (i.e., no
central barrier).
Finally, note that we have added a “Hartree” potential VH(z)
in the diagonal elements; see Eqs. (12)–(14). The Hartree po-
tential VH(z) here contains contributions from the electron-
electron interaction (mean field) relevant in quantum wells
containing many electrons, the external gate potentials, and

4the modulation-doped potential (i.e., ionized impurities out-
side the well region). In Appendix A, we describe in detail
these distinct contributions to VH and how they are calculated
in our system. As we will see next, both VH(z) and the struc-
tural confining potentials contribute to the effective SO cou-
pling for electrons.
Γ7
Γ8
Γ6
∆g
Eg
(a) Energy band structure.
δ6 δb6
δ7
δ8
δb7
δb8
Eg
∆g
Ew
∆w
Eb
∆b
Lw
Lb
(b) Band offsets.
Figure 1: (a) Schematic of the band structure of direct gap zincblend
semiconductors near the Γ point (k = 0). The label Γ6 represents
the s states in the conduction band, while Γ7 (split-off holes) and Γ8
(heavy holes and light holes) represent the p-states in the valence
bands. (b) Band offsets for a double quantum well of width Lw with
a central barrier of width Lb. The relationships among the several
offset parameters are given in (16)–(17).
Table I: Truncated set of zone center wave functions ul0 (for simplic-
ity we denote them by ul) used in constructing the matrix Hamilto-
nian (9)
ui Γ |J,mJ
〉
uJ,mJ
u1 Γ6 | 12 ,+
1
2
〉
i|S 〉 ⊗ | + 12
〉
u2 Γ6 | 12 ,−
1
2
〉
i|S 〉 ⊗ | − 12
〉
u3 Γ8 | 32 ,+
3
2
〉 − 1√
2
(
|X〉 + i|Y〉
)
⊗ | + 12
〉
u4 Γ8 | 32 ,+
1
2
〉 − 1√
6
(
|X〉 + i|Y〉
)
⊗ | − 12
〉 +
√
2
3 |Z
〉 ⊗ | + 12
〉
u5 Γ8 | 32 ,−
1
2
〉 + 1√6
(
|X〉 − i|Y〉
)
⊗ | + 12
〉 +
√
2
3 |Z
〉 ⊗ | − 12
〉
u6 Γ8 | 32 ,−
3
2
〉
+ 1√
2
(
|X〉 − i|Y〉
)
⊗ | − 12
〉
u7 Γ7 | 12 ,+
1
2
〉 − 1√
3
(
|X〉 + i|Y〉
)
⊗ | − 12
〉 − 1√
3
|Z〉 ⊗ | + 12
〉
u8 Γ7 | 12 ,−
1
2
〉 − 1√3
(
|X〉 − i|Y〉
)
⊗ | + 12
〉 + 1√3 |Z
〉 ⊗ | − 12
〉
3. EFFECTIVE SPIN-ORBIT HAMILTONIAN FOR
ELECTRONS
3.1. Folding down
Since we are interested in SO effects for the conduction
electrons, here we derive an effective Hamiltonian for them.
To this end, let us rewrite our 8 × 8 Hamiltonian [Eq. (9)] in
the block form,
H8×8 =









Hc Hcv
H†cv Hv









, (18)
where Hc is a 2 × 2 diagonal matrix in the sector Γ6 (conduc-
tion band) with identical diagonal elements E6 [(12)] and Hv
is a 6 × 6 diagonal matrix in the sectors Γ8 and Γ7 (valence
bands) with diagonal elements E8 [(13)] and E7 [(14)], re-
spectively. The 2 × 6 matrix Hcv can be read off directly from
the corresponding 2 × 6 block in Eq. (9).
Using the block form of our Hamiltonian (18)) the eigen-
value problem can be written in the compact form









Hc Hcv
H†cv Hc


















ϕc
ϕv









= E









ϕc
ϕv









, (19)
where ϕc is a two-component spinor (conduction sector) and
ϕv is a six-component spinor (valence sector). Straightforward
manipulations19,20 yield the effective Schro¨dinger’s equation
H(E)ϕ˜c = Eϕ˜c, (20)
with
H(E) = Hc + Hcv(E − Hv)−1H†cv, (21)
and ϕ˜c is a properly renormalized conduction-electron
spinor.26
The matrix elements of H(E) are given by27
H(E)11 = H(E)22 = E6 + P
2
3
(k2q γ1 + kz γ1kz
), (22)
H(E)12 = H(E)†21 =
P2
3 k−
[γ2, kz
]
= −P
2
3 k−kz γ2, (23)
where k2q = k±k∓ = k2x + k2y and
γ1(z) =
( 2
E − E8
+
1
E − E7
)
, (24)
γ2(z) =
( 1
E − E8
− 1
E − E7
)
, (25)
We should emphasize that Eq. (20) is not really an eigenvalue
equation as H(E) depends on E. However, as we show in
Sec. 3.2, we can still obtain a true eigenvalue problem by per-
forming suitable expansions.

53.2. Energy denominator expansions
Since Eg and Eg + ∆ are the largest energy scales in our
system, i.e.,
χ8 =
E −
[
~2k2
2m0
+ VH(z) − h8(z)
]
Eg
≪ 1, (26)
χ7 =
E −
[
~2k2
2m0
+ VH(z) − h7(z)
]
Eg + ∆g
≪ 1, (27)
we can expand the energy denominators in the γi’s [Eqs. (24)
and (25)] in the form
γ1 =
2
Eg
(
1 − χ8 + · · ·
)
+ 1
Eg + ∆g
(
1 − χ7 + · · ·
) (28)
γ2 =
1
Eg
(
1 − χ8 + · · ·
) − 1
Eg + ∆g
(
1 − χ7 + · · ·
). (29)
For the diagonal matrix elements H(E)11 = H(E)22 we keep
only zeroth-order (i.e., energy-independent) terms, while for
the off-diagonal matrix elements H(E)12 = H(E)†21 we keep
in addition the first-order terms as they give the lowest non-
vanishing contribution (because the off-diagonal matrix el-
ements contain derivatives with respect to z). Straightfor-
wardly, we then obtain the energy-independent one electron
Hamiltonian
H(E) = HQW1 + η(z)









0 −ik−
ik+ 0









, (30)
where
HQW =
~2k2q
2m∗
+ ~
2
2m∗
∂2
∂z2
+ Vsc(z), (31)
[the subscript “sc” emphasizes that the potential is to be de-
termined self-consistently – see Appendices A and B] and15
1
m∗
= 1
m0
+ 2P
2
3~2
( 2
Eg
+ 1
Eg + ∆g
)
, (32)
Vsc(z) = VH(z) + δ6hw(z) + δb6hb(z), (33)
η(z) = ηw dhw(z)dz + ηb
dhb(z)
dz − ηH
dVH(z)
dz , (34)
with
ηH =
P2
3
[ 1
E2g
− 1(Eg + ∆g)2
]
, (35)
ηw =
P2
3
[ δ8
E2g
− δ7(Eg + ∆g)2
]
, (36)
ηb =
P2
3
[δb8
E2g
− δb7(Eg + ∆g)2
]
. (37)
3.3. Projection into the quantum well subbands
Here we define a quasi-two-dimensional (2D) model start-
ing from the three-dimensional (3D) Hamiltonian (30). The
idea is essentially to obtain a 2D effective model similar to the
well-known Rashba model, but now for the case of wells with
many subbands. To this end we (i) first project [(30) into the
spin-degenerate eigenstates of HQW [Eq. (31)] (note that HQW
does not contain the SO interaction):29 |kqv〉σz = |kqv〉|σz〉,
〈r|kqv〉 = exp(ikq·rq)ϕv(z), v = 0, 1, 2, . . ., andσz = ± (or ↑, ↓),
which correspond to the subband energies Ekqv =
~2k2q
2m∗ + Ev,
with Ev being the quantized levels of the well, and then (ii)
consider a reduced set of subbands (e.g., two) by truncating
the basis set used. In this section we simply assume that we
know the eigensolutions of HQW ; later on we actually calcu-
late them within a self-consistent procedure, from which we
can explicitly determine the relevant SO coupling constants in
our problem.
The matrix elements of H(ε) in the {|kqv〉σz } basis are
〈kqv|
〈±|H(E)|kqv′〉|±〉 = (~
2k2q
2m∗
+ Ev
) δvv′ , (38)
〈
kqv|
〈±|H(E)|kqv′〉|∓〉 = ∓i ηvv′ k∓, (39)
with the generalized SO couplings
ηvv′ = ΓHvv′ + Γwvv′ + Γbvv′ , (40)
where
ΓHvv′ = −ηH
〈
v|dVH(z)dz |v
′〉, (41)
Γwvv′ = +ηw
〈
v|dhw(z)dz |v
′〉, (42)
Γbvv′ = +ηb
〈
v|dhb(z)dz |v
′〉. (43)
The coefficients ΓHvv′ , Γwvv′ , and Γbvv′ denote the contributions
from the Hartree potential, the quantum-well edges, and the
central barrier edges, respectively. It is convenient to split the
Hartree contribution into two terms, i.e., VH(z) = Ve(z)+Vg(z),
where Ve(z) is the purely electronic Hartree potential and
Vg(z) denotes the contributions from the external gate po-
tential and the modulation doping potential. Hence Γ(H)vv′ =
−ηH
〈
v| dVe(z)dz |v′
〉 − ηH
〈
v| dVg(z)dz |v′
〉
. This separation will be use-
ful when discussing our results.
We emphasize that the diagonal (in v, v′) parameters ηvv
correspond to the Rashba coupling in the vth subband, i.e.,
αv = ηvv. The off-diagonal terms ηvv′ arise due to the inter-
subband coupling. Interestingly, these new SO terms can be
non-zero even in structurally symmetric wells, since they arise
from quantum-well states of distinctive parities.
For completeness we present here the linearized Dressel-
haus couplings30 in the vth subband
βv = βD
〈
v|k2z |v
〉, (44)
where the constant βD is the bulk Dresselhaus SO parameter.21

6We can easily rewrite the above expression in the more con-
venient form
βv = βD
2m∗
~2
[Ev −
〈
v|V(z)|v〉] . (45)
In Sec. 5 we shall use the above form to discuss how the Dres-
selhaus couplings vary as a function of the system parameters.
3.4. Two-subband case
To illustrate the procedure of Sec. 3.3, let us explicitly work
out here the case of a quantum well with only two subbands
v = 0, 1. In Sec. 5 we shall investigate in detail the SO
couplings for single and double quantum wells with two sub-
bands.
3.4.1. 4x4 Hamiltonian
With the basis ordering {|kq0〉↑, |kq0〉↓, |kq1〉↑, |kq1〉↓},
Eqs. (38) and (39) yield the effective Hamiltonian
H =























Ekq0 −iα0 k− 0 −iη k−
iα0 k+ Ekq0 iη k+ 0
0 −iη k− Ekq1 −iα1 k−
iη k+ 0 iα1 k+ Ekq1























, (46)
where the Rashba couplings are given by αv = ηvv, v = 0, 1,
and the intersubband SO coupling31 by η = η01 [see Eqs. (40)
and (43)] and
Ekqv = Ev +
(~kq)2
2m∗
, v = 0, 1. (47)
3.4.2. Eigensolutions
The energy eigenvalues Eσλ of Eq. (46) are straightforward
to obtain:
Ekq,λ1,λ2 = Ekq+ + λ2α+ kq + λ1
√
(ηkq)2 + (Ekq− + λ2α−kq)2,
(48)
where λ2 = ± are spin quantum numbers and λ1 = ± are the
subband (or pseudo spin) indices, and
Ekq± =
1
2
(Ekq1 ± Ekq0
), α± =
1
2
(α1 ± α0). (49)
The corresponding (normalized) eigenvectors are
|λ1, λ2
〉 =
√
1 + λ1
ǫλ2 (0)
ǫλ2 (η)




































− iλ1ηkq e−iθ
ǫλ2 (η) + λ1ǫλ2 (0)
λ1λ2ηkq
ǫλ2 (η) + λ1ǫλ2 (0)
−iλ2e−iθ
1




































ei kq ·rq
4π , (50)
where
ǫ±(η) =
√
(ηkq)2 + (Ekq− ± α−kq)2, e±iθ =
k±
kq
. (51)
3.4.3. SO-induced effective mass renormalization
Expanding the energy dispersions [Eq. (48)] around kq = 0,
we obtain to second order
Ekq→0,λ1,λ2 ≈ E+ + λ1 E− + λ2 (α+ + λ1 α−)kq +
~2k2q
2m∗λ1
, (52)
where m∗λ1 are the effective masses
m∗± =
m∗
1 ±
2Eso
∆E
, (53)
where Eso = 12 m∗η2/~2 and ∆E = 2E−. Note that the mass
renormalization is solely due to the intersubband-induced
SO coupling η. For the realistic wells we investigate here
2Eso/∆E << 1 for single wells but can reach ∼ 0.1 for double
wells (Secs. 4 and 5).
3.4.4. Determining the SO couplings
As mentioned previously, we determine the SO orbit cou-
plings (here specifically α0, α1, and η) from the self-consistent
eigensolutions of the quantum well without spin orbit,29 via
Eqs. (40))-(43)). In Sec. 4 we detail the quantum well sys-
tems investigated and briefly outline the self-consistent pro-
cedure used to obtain the eigensolutions (a full description is
provided in Appendices A and B). We then present results for
single and double wells with two subbands; i.e., we calculate
α0, α1, and η and discuss in detail the several distinct contri-
butions to each of these quantities.
4. QUANTUM-WELL SYSTEMS AND
SELF-CONSISTENCY
Figure 2 shows a schematic view of the quantum-well sys-
tem we study: a well of width Lw centered at z = 0 (z: growth
direction), and two adjacent symmetrically doped regions of
widths w in the barriers. We also consider double wells by
inserting an additional (central) barrier of width Lb in the well
region. The doping densities of the left and right regions, ρa
and ρb, respectively, can be used to control the degree of struc-
tural inversion asymmetry of the wells (in Sec. 5, however, we
present results only for ρa = ρb). The external gates Va and
Vb, located at the end points ±L, can also be used to control
the degree of inversion asymmetry and to vary the areal elec-
tron density in the well.
Since our wells have many electrons and are subject to ex-
ternal gates, we have to solve the Schro¨dinger and Poisson

7equations self-consistently (“Hartree approximation”32) in or-
der to determine their potential profile Vsc(z) [see Eq. (33)]
and corresponding eigenfunctions and eigenenergies. In Ap-
pendices A and B we describe in detail our standard self-
consistent procedure.
−L +L
−Ld +Ld
0
w w
ρa ρb
Va Vb
Lw
Quantum Well
dopants
Figure 2: Schematic view of our quantum-well system. The doping
densities ρa, ρb and the external gate voltages Va and Vb can be used
to control the degree of the structural inversion asymmetry.
Before going into the discussion of the SO couplings in
detail, let us first have a look at the outcome of a typical
self-consistent simulation we perform. Figure 3(a) shows
the self-consistent potential Vsc (thick solid line) for a sin-
gle well with two subbands; the corresponding self-consistent
wave functions ψ0(z) and ψ1(z) are also shown. The energies
of the two lowest subband edges (see levels in the well) are
E0 = 309.09 meV and E1 = 406.39 meV (∆E = 97.3 meV).
Here we fix the chemical potential at µ = 413.40 meV with
respect to the V = 0 origin (“constant chemical potential
model”, see below) and set the external gates to Va = 0 and
Vb = 1200 meV. The two subbands are occupied with areal
densities n0 = 18.7422×1011 cm2 and n1 = 1.2578×1011 cm2,
respectively. The electronic Hartree potential Ve (short dashed
line) and the the external gates (plus modulation doping) con-
tribution Vg (long dashed line) are also shown. Figure 3(b)
shows the corresponding “force fields” Fe = −dVe/dz arising
from the confined electrons in the well and Fg = −dVg/dz
coming from the doping regions (±12 nm to ±18 nm) plus
the external gates (Fg and Fe will be useful when discussing
the SO couplings further below). Using the self-consistent
solutions ψv(z), v = 0, 1, we can straightforwardly calcu-
late the relevant SO couplings [via Eqs. (40)–(43)]: η =
−3.81 meV nm, α0 = −5.44 meV nm, α1 = −3.74 meV nm,
β0 = 0.87 meV nm, and β1 = 2.50 meV nm.
5. RESULTS
Here we focus on single and double quantum wells with
only two subbands. More specifically, we calculate three SO
couplings: the intersubband-induced SO coupling η = η01 and
the two Rashba-type couplings α0 = η00 and α1 = η11. We
consider two experimentally relevant cases: the constant areal
density (nT -constant) and the constant chemical potential (µ-
constant) models. In our simulations we always keep Va = 0
as a reference potential and vary Vb; see Fig. 2. For complete-
ness, we also calculate the two Dresselhaus constants β0 and
β1 [see Eq. (44)] within each subband.
-400
-200
0
200
400
600
800
1000
1200
1400
1600
1800
-40 -30 -20 -10 0 10 20 30 40
V
(m
eV
)
z (nm)
ψ0
ψ1
Vsc
Ve
Vg
µ
(a) Self-consistent potential energies.
-20
-10
0
10
20
30
40
50
-40 -30 -20 -10 0 10 20 30 40
(m
eV
/n
m
)
z (nm)
Fe = − dVedz
Fg = − dVgdz
(b) Force fields.
Figure 3: (a): Self-consistent potential energy Vsc (thick solid line)
and the corresponding wave functions ψ0 and ψ1 for the single well
Al0.48In0.52As/Ga0.47In0.53As with external gates Va = 0 e Vb = 1.2 eV
(see Fig. 2). The the electronic Hartree potential Ve (short dashed
line), the external gate plus modulation doping contributions Vg (long
dashed line), and the corresponding force fields Fe = −dVe/dz and
Fg = −dVg/dz are also shown in (b). The two levels in the well (solid
lines) denote the energies of the first and second subband edges,
while the dotted level indicates the chemical potential.
5.1. Single wells
5.1.1. Single-well parameters
We consider a realistic Al0.48In0.52As/Ga0.47In0.53As single
quantum well.33,34 We assume doping densities ρa = ρb =
4 × 1018 cm−3 with widths w = 6 nm (“sample 3” in Ref. 33).
Table II summarizes band parameters, potential offsets,25 well
widths, and other important parameters of our system. The
coefficients ηw and ηH in Tab. II are defined in Eq. (36) and
Eq. (35), respectively. Here, the Dresselhaus parameter βD in
Eq. (44) is assumed to be the same as that of the GaAs (see
Ref. 21).

8Table II: Relevant parameters25 (see Fig. 1) (at 0.3 K) for the sin-
gle quantum well Al0.48 In0.52As/Ga0.47In0.53As system in our simu-
lations. The doping regions have widths w = 6 nm and densities
ρa = ρb = 4 × 1018 cm−3 (see Fig. 2). All energies are in eV and
lengths in nm; the coefficient ηH is in nm2 and ηw in meV nm2. The
Dresselhaus coupling constant21 βD is in meV nm3.
Ew = 1.5296 ∆w = 0.2998 w = 6 EP = 25.3
Eg = 0.8161 ∆g = 0.3296 δ6 = 0.52 m∗/m0 = 0.043
Eb = 0 ∆b = 0 δb6 = 0 ǫr = 14.013
L = 40 Ld = 18 Lb = 0 Lw = 14
ηH = 0.2376 ηw = 0.0533 ηb = 0 βD = 0.0237
5.1.2. SO couplings: single wells
Figure 4 shows the strength of the Rashba (αν, ν = 0, 1;
dashed lines), Dresselhaus (βν, ν = 0, 1; dotted lines) and
intersubband-induced (η, solid line) SO couplings as func-
tions of the gate voltage Vb, for both the nT -constant and
the µ-constant models, Figs. 4(a) and 4(b), respectively. At
Vb = Va = 0 eV, our sample is completely symmetric and,
as expected, the Rashba couplings α0 and α1 are zero. We
note that the Dresselhaus couplings β0 and β1 are practically
constant in both models. This follows from Eq. (45) which
shows that in each subband the Dresselhaus coupling is essen-
tially the difference between the expected value of the self-
consistent potential in the respective subband and the corre-
sponding eigenenergy. The Rashba couplings, on the other
hand, vary considerably with Vb, although showing a similar
trend in both models. Interestingly, they change signs about
Vb = 0 (symmetric configuration), but always with |α0| > |α1|.
Our calculated α0 within the µ-constant model [Fig. 4(b)] is
consistent with the measurements of this quantity by Koga et
al.,33,34 whose samples have a constant chemical potential.
The new intersubband-induced coupling η [see the solid
lines in Figs. 4(a) and 4(b)] is non-zero even in the sym-
metric well configuration (Vb = 0 = Va). It has a strength
comparable to the Rashba and is at least twice as large as the
Dresselhaus. In contrast to the Rashba couplings, the intersub-
band SO η does not change sign with Vb. In fact, for the single
well investigated here η is almost constant with Vb, although
it varies slightly more in the µ-constant model [compare the
solid curves in Figs. 4(a) and 4(b)].
To more easily understand the results above, we analyze the
several contributions to the SO couplings separately. To this
end, we rewrite [see comments following Eq. (40)] ηvv′ for a
single well in the form
ηS Wvv′ = Γevv′ + Γ
g
vv′ + Γ
w
vv′ , (54)
where we have set Γbvv′ = 0 in (40), i.e., no central barrier
contribution, and have split the Hartree contribution into its
purely electronic Γevv′ and the external gate (plus doping po-
tential) Γgvv′ parts. Hence, for two subbands, each of the SO
couplings has three contributions: η = ηS W01 = Γ
e
01 + Γ
g
01 + Γ
w
01,
-6
-4
-2
0
2
4
6
-1.0 -0.5 0 0.5 1.0
η,
α v
,β
v
(m
eV
n
m
)
Vb (eV)
η
α0
α1
β0
β1
(a) nT constant.
-6
-4
-2
0
2
-0.2 0 0.2 0.4 0.6 0.8
η,
α v
,β
v
(m
eV
n
m
)
Vb (eV)
η
α0
α1
β0
β1
(b) µ constant.
Figure 4: Rashba α, Dresselhaus β and intersubband-induced η SO
coupling constants for the Al0.48In0.52As/Ga0.47In0.53As quantum well
as functions of the gate voltage Vb (see Fig. 2). In (a) the total 2D
electron density is kept constant at nT = 20 × 1011 cm−2 and in (b)
the chemical potential is kept constant at µ = 200 meV.
α0 = ηS W00 = Γ
e
00 + Γ
g
00 + Γ
w
00 and α1 = η
S W
11 = Γ
e
11 + Γ
g
11 + Γ
w
11.
Figures 5(a)–5(c) show the above contributions separately for
the nT -constant case (similar results hold for the µ-constant
model, in the parameter range studied).
Figure 5(a) shows that the external gates and doping con-
tributions to η (Γg01 curve) are essentially zero, while the elec-
tronic Hartree contribution (Γe01 curve) and the structural (Γw01
curve) contributions are comparable in magnitudes and both
negative. In contrast, for both α0 and α1 the largest con-
tributions come from the external gates together with dop-
ing regions [see the curve Γg00 in Fig. 5(b) and the curve Γg11
in Fig. 5(c)]; these account for 60% of α0 and 100% of α1.
The electronic Hartree contribution is negligible in α0 [curve
Γe00 in Fig. 5(c)] while the structural part (Γw00) accounts for
about 30% of it. On the other hand, the structural and elec-
tronic Hartree contributions in α1 essentially cancel out (same
magnitude and opposite signs); cf. the Γe11 and Γw11 curves in
Fig. 5(c).

9-4
-3
-2
-1
0
-1.0 -0.5 0 0.5 1.0
η
(m
eV
n
m
)
Vb (eV)
Γe01
Γg01
Γw01
η
(a) Contributions to η.
-4
-3
-2
-1
0
1
2
3
4
-1.0 -0.5 0 0.5 1.0
α 0
(m
eV
n
m
)
Vb (eV)
Γe00
Γg00
Γw00
α0
(b) Contributions to α0 .
-4
-3
-2
-1
0
1
2
3
4
-1.0 -0.5 0 0.5 1.0
α 1
(m
eV
n
m
)
Vb (eV)
Γe11
Γg11
Γw11
α1
(c) Contributions to α1 .
Figure 5: Several distinct contributions to the coupling constants η
(a), α0 (b) and α1 (c) for the single GaInAs quantum well shown
in Fig. 4(a) (nT -constant model) as functions of the external gate
Vb (Va = 0). These contributions arise from: the electron density
(Hartree potential), the external gate (together with donor regions),
and the structural well potential; these are denoted by the superscripts
e, g and w, respectively.
We can understand the above remarks by looking at the self-
consistent potentials and the “force fields” Fe = −dVe(z)/dz
(short dashed curve) and Fg = −dVg(z)/dz (long dashed
curve) – note that Γivv′ ∼
〈
v|Fi|v
〉
, i ∈ {e, g,w}, – in Fig. 3(b).
This figure was obtained for Vb = 1.2 eV, but it does dis-
play the general behavior for all quantities shown. The force
field Fg is essentially constant, except within the donor re-
gions where the wave functions are vanishingly small. Hence,
the matrix element
〈
v|Fg|v
〉 [see Eqs. (40)–(43)] is approxi-
mately linear in the external gate Vb. This explains why the
Rashba couplingsαv are strongly modulated by external gates.
This is even more so for α1, Fig. 5(c), for which the struc-
tural and electronic contributions cancel out. Looking at the
wave functions ψ0 and ψ1 and the force field Fe = −dVe/dz in
Fig. 3(b), we can see that the electronic Hartree contribution
(∼ −〈v|Fe|v〉) is almost zero (though slightly negative) for the
lowest subband and positive for the first subband. The struc-
tural well contributions Γwvv [see Eq. (42)] to αv are similar for
both subbands, though |Γw00| ≥ |Γw11|, because the nonzero bi-
ases (Vb , 0) cause the wave functions to shift toward one
side of the well [e.g., Vb = 1.2 eV in Fig. 3(a)].
On the other hand, the contribution Γg01 ∼ −
〈0|Fg|1
〉
to
the intersubband coupling η is essentially zero since the wave
functions [ψ0 and ψ1 in Fig. 3(a)] are orthogonal and, again,
Fg is constant. Hence η is not as sensitive to the external
gates as the Rashba couplings. Most of the modulation of
η arises from the electronic Hartree and structural contribu-
tions, which both have the same sign and magnitude as shown
in Fig. 5(a).
5.2. Double well
5.2.1. Double-well parameters
Table III shows the band parameters21,25 for the dou-
ble quantum well Al0.4In0.6Sb/InSb with one central barrier
InSb/Al0.12In0.88Sb. Hereafter we refer to this heterostructure
as InSb double well. The meaning of some of these parame-
ters (e.g., band offsets) can be seen in Figs. 1 and 2.
Table III: Relevant parameters21,25 (at 1 K) for the InSb double well
see Figs. 1 and 2). The width of the doping regions is w = 4 nm and
their densities are ρa = ρb = 3 × 1018 cm−3. All energies are in eV
and lengths in nm. The coefficient ηH is measured in nm2 while ηw
and ηb are measured in meV nm2. The Dresselhaus constant βD is
measured in meV nm3. The parameters in the last column are to the
InSb binary compound.
Ew = 0.9922 ∆w = 0.6964 w = 4 EP = 23.3
Eg = 0.2350 ∆g = 0.8100 δ6 = 0.6133 m∗/m0 = 0.0135
Eb = 0.4477 ∆b = 0.7675 δb6 = 0.1723 ǫr = 16.8
L = 100 Ld = 65 Lb = 20 Lw = 50
ηH = 0.2171 ηw = 0.7627 ηb = 5.0873 βD = 0.326
5.2.2. SO couplings: double-well case
Figure 6 shows the Rashba α, Dresselhaus β, and
intersubband-induced η SO couplings as functions of the gate
voltage Vb (here again Va = 0) for both the nT -constant (a)
and µ-constant (b) models. We first discuss the nT -constant
model [Fig. 6(a)]. Here the Rashba couplings (dashed lines)
are most sensitive to the external bias Vb, being essentially

10
the largest of all SO couplings for very asymmetric structures
(i.e., high biases). The Dresselhaus couplings (dotted lines)
are almost identical (β0 ≈ β1) and mostly independent of the
-40
-20
0
20
40
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
η,
α v
,β
v
(m
eV
n
m
)
Vb (eV)
η
α0
α1
β0
β1
(a) nT constant.
-20
-10
0
10
20
30
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
η,
α v
,β
v
(m
eV
n
m
)
Vb (eV)
η
α0
α1
β0
β1
(b) µ constant.
Figure 6: Rashba α, Dresselhaus β and the intersubband-induced η
SO couplings for a InSb double quantum well as functions of the
right gate voltage Vb. In (a) the total electron density is kept constant
at nT = 10 × 1011 cm−2 and in (b) the chemical potential is kept
constant at µ = 100 meV (relative to to initial bottom well).
external gates. The SO coupling η (solid line) is an even func-
tion of the external gate Vb and presents a “resonant behavior”
around the Vb = Va = 0 eV configuration, at which our sample
is symmetric. While the Rashba couplings are both zero at this
symmetric configuration, we note that they are odd functions
of the external gate (with |α0| > |α1|), have opposite signs and
abruptly change magnitudes around Vb = 0 (over a 40 meV
wide region). For the µ-constant model [Fig. 6(b)], a similar
picture as above also holds; note, however, that in contrast to
the nT -constant model, in the µ-constant case the positive and
negative bias configurations are not equivalent as they corre-
spond to the well having different numbers of electrons.
For completeness we show in Fig. 7 the behavior of all cou-
pling constants near the symmetric point Vb = Va = 0 eV for
the double well in Fig. 6(a). Note that the Dresselhaus cou-
plings β0 and β1 present a (double) crossing over a 160 meV
wide region [see Fig. 7(b)]. However, this is a minor effect:
note the change in the scale of the vertical axis. While the
resonant behavior of η is accompanied by an enhancement of
about 10 in its magnitude [see Fig. 6(a)], we see no substantial
change in the magnitudes of the β’s near the zero-bias case [cf.
Figs. 7(a) and 7(b)].
-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20
η,
α v
,β
v
(m
eV
n
m
)
Vb (meV)
η
α0
α1
(a) αv and η near Vb = 0.
4.65
4.75
4.85
4.95
5.05
-100 -50 0 50 100
β v
(m
eV
n
m
)
Vb (meV)
β0
β1
(b) βv near Vb = 0.
Figure 7: Rashba α, intersubband-induced η and Dresselhaus β cou-
plings vs Vb about the symmetric configuration Vb = Va = 0 eV for
the double InSb well in Fig. 6(a).
The relative strengths of the Rashba and Dresselhaus cou-
pling constants to the intersubband-induced SO coupling are
shown in Fig. 8. The Rashba couplings have the largest
strengths (note the pre-factors in front of αv/η in the legends).
In contrast to βv/η, the linear behavior of the Rashba ratios
αv/η near Vb = 0 (see insets) shows that αv and η undergo
similar variations near the symmetric configuration. As ob-
served before, the intersubband-induced coupling η becomes
important near Vb = 0 (Fig. 6).
-7
-5
-3
-1
0
1
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Vb (eV)
40−1 α0/η
10−1 α1/η
β0/η
β1/η
-0.6
0
0.2
-0.02 0 0.02
(a) nT constant.
-8
-6
-4
-2
0
2
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Vb (eV)
40−1 α0/η
10−1 α1/η
β0/η
β1/η
-0.6
0
0.2
-0.02 0 0.02
(b) µ constant.
Figure 8: Ratios αv/η and βv/η for the InSb double well in Fig. 6.
The insets are blowups around Vb = 0, the symmetric configuration.
Figure 9 (similar to Fig. 5 for the single-well case) shows
the several contributions to each of the SO couplings η, α0
and α1 for the double-well case. Here, in addition to the elec-
tronic Hartree, the gate (+ doping regions), and the well con-
tributions, there is an additional structural term arising from
the central barrier (superscript b). A general feature in Figs.
9(a)–9(c) is that the structural contributions (well and central
barrier) almost cancel out because they have opposite signs
(see the curves with superscripts w and b). These terms have
opposite signs because the derivatives dhw(z)/dz (well) and
dhb(z)/dz (barrier), which enter the coupling constants [see
Eqs. (40), (42), and (43)], have opposite slopes. Similarly to
the single-well case [Fig. 5(a)] the contribution of the exter-
nal gates (which includes the doping regions) to the intersub-
band SO coupling η is vanishingly small [see the Γg01 curve in
Fig. 9(a)]. Hence, η is mostly due to the electronic Hartree
contribution [curve Γe01 in Fig. 9(a)]. In addition, the gate con-
tribution to α0 and α1 for the InSb double well is linear in Vb
as for the single-well case. Hence, the Rashba couplings α0
and α1 for the double InSb well are essentially determined by
the electronic (Hartree) contribution and are modulated by the
gate contribution. Summarizing: looking at Fig. 9, we can see

11
-15
-10
-5
0
5
-0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20
η
(m
eV
n
m
)
Vb (eV)
Γe01
Γg01
Γw01
Γb01
η
(a) Contributions to η.
-20
-15
-10
-5
0
5
10
15
20
-0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20
α 0
(m
eV
n
m
)
Vb (eV)
Γe00
Γg00
Γw00
Γb00
α0
(b) Contributions to α0 .
-20
-15
-10
-5
0
5
10
15
20
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20
α 1
(m
eV
n
m
)
Vb (eV)
Γe11
Γg11
Γw11
Γb11
α1
(c) Contributions to α1 .
Figure 9: Different contributions to the SO couplings η, α0 and α1
for the InSb double well in the constant areal density model [see
Fig. 6(a)] as functions of the external gate Vb (Va = 0). In the sub-
figures we show the contributions to the coupling constants coming
from the areal electronic density, indicated by the superscript e, and
from the external gate + donor regions g, and also from the structural
potential, being w for the well and b for the central barrier.
that (i) the structural contributions (well and barrier; dashed
curves) almost cancel out, and (ii) the external gate (dotted
curves) modulates the Rashba couplings αv; therefore, for the
double well investigated here (iii) most of the strength of these
three coupling constants (η, α0, and α1) comes from the elec-
tronic contribution (dot-dashed curves).
It is instructive to investigate in more detail how the reso-
nant behavior in η comes about, as well as the abrupt changes
in the Rashba couplings; see Fig. 6. This can be accomplished
by looking more closely at the self-consistent wave functions
of the InSb double well around the symmetric configuration
(Vb = 0). The top row in Fig. 10 shows the self-consistent
potential profile of the double well and the normalized wave
functions ψ0 (short dashed line) and ψ1 (long dashed line) for
the lowest v = 0 and for the first excited v = 1 subbands at
three distinct gate voltages: Vb = +0.3 eV, Vb = 0 eV and
Vb = −0.3 eV (left, center, and right columns, respectively).
For positive bias ψ0 is mostly localized in the left well and ψ1
in the right well, while for negative biases this configuration is
reversed. The electronic Hartree contribution to the potential
energy Ve and the corresponding force field Fe = −dVe/dz are
shown on the second row, thin and thick lines respectively.
Notice that Fe is practically zero in the central barrier re-
gion (−10 ≤ z ≤ 10 nm) and has opposite signs within the
wells (−25 ≤ z ≤ −10 nm and 10 ≤ z ≤ 25 nm). Hence
the quantities F00e (z) = ψ0(z)Feψ0(z), F11e (z) = ψ1(z)Feψ1(z)
and F01e (z) = ψ0(z)Feψ1(z) have the forms shown on the third
and fourth rows. The integral over z of these quantities de-
fines the electronic Hartree contributions to the spin-orbit cou-
plings α0, α1, and η, i.e., Γe00 ∼
〈0|Fe|0
〉
, Γe11 ∼
〈1|Fe|1
〉
, and
Γe01 ∼
〈0|Fe|1
〉
, respectively. Since the electronic Hartree con-
tributions dominate over the others, see Figs. 9(a)–9(c), the
abrupt changes in the Rashba couplings and the resonant be-
havior of η around Vb = 0 follow straightforwardly.
5.2.3. Density anticrossings and effective masses
Figures 11(a) and 11(b) show anti-crossings of the areal
densities nT for the InSb double well near the symmetric con-
figuration Vb = 035, where the strength of the intersubband-
induced SO coupling η is the strongest (−16.7482 meV nm)
while the the energy difference between the subband edges
∆E = E1 − E0 (0.9353 meV) is the smallest. In accord
with Eq. (52), we find an appreciable change in the bulk
effective mass m∗ near kq = 036. The ratio Eso/∆E [see
Eq. (53)] is shown in Fig. 11(c) and the ratio m¯± = m∗±/m∗
in Fig. 11(d). These intersubband-SO-induced changes in the
effective masses m∗± may have a sizable effect on the measured
mobilities and cyclotron frequencies in InSb wells.
6. SUMMARY
Starting from the 8 × 8 Kane model in heterostructures, we
have derived in some detail an effective electron Hamiltonian
which contains a new intersubband-induced SO interaction
term which arises in quantum wells with more than one quan-
tized subband. Unlike the usual Rashba SO term, the intersub-
band SO coupling here is non-zero even for symmetric wells.
For structurally asymmetric wells we have also accounted for
the Rashba-type SO interaction within each subband.
We have also outlined the projection procedure (“folding
down”) to obtain quasi-2D Hamiltonians by integrating out
the confined variables. For two subbands in asymmetric wells
we find a 4 × 4 quasi-2D Hamiltonian resembling Rashba’s,
but containing three SO couplings: the two Rashba couplings

13
Appendix A: SELF-CONSISTENT PROCEDURE
1. Effective Schro¨dinger equation
The single-particle electron Hamiltonian HQW of our quan-
tum wells [Eq. (31))] is clearly separable. The transverse mo-
tion (x,y) is free while that along the z direction is confined
by the quantum well. To solve the corresponding Schro¨dinger
equation HQWΨkqv(r) = EkqvΨkqv(r) we assume a wave func-
tion of the form
Ψkqv(r) = 〈r|kqv〉 =
1
√
A
exp(ikq · rq)ψv(z) (A1)
(A is a normalizing area) which leads to the 1D Schro¨dinger
equation
(
− ~
2
2m
d2
dz2 + Vsc(z)
)
ψv(z) =
(
Ekqv −
~2k2q
2m∗
)
= Evψv(z), (A2)
from which we obtain the quantized energy levelsEv and wave
functions ψv(z). As we shall see, the subband structure of the
well Ekqv = Ev + ~2k2q /2m∗ and the corresponding total wave
function Ψkqv(r) will be used (within a self-consistent proce-
dure) to construct the electron charge density, from which the
corresponding Hartree potential can be obtained via the Pois-
son equation.
As mentioned in Sec. 3.2 Vsc in Eq. (A2) contains not only
the structural confining potential but also the “Hartree contri-
butions” (i) the purely electronic mean-field potential (elec-
tronic Hartree potential) and (ii) the external gate potential
plus the modulation doping potential. Further down we dis-
cuss these contributions in detail. Each of these contributions
is determined from a Poisson equation with an appropriate
charge distribution and boundary condition.
a. Self-consistency
Since the electronic charge distribution ρe(z)
(∝ ∑v |〈r|kqv〉|2) depends on the detailed form of the sev-
eral potentials (modulation doping, gates, and electronic
Hartree), and these, in turn, depend on ρe(z), we have to
solve the problem self-consistently. The standard procedure
is as follows: (i) to solve Eq. (A2)) with an initial guess for
Vsc which we take to be just the structural potential plus the
external gates and modulation doping potential [i.e., in the
first run we do not include the electronic Hartree potential
Ve(z)]; (ii) to construct the electronic charge density ρe(z)
[from the eigenfunctions obtained in step (i)] and the corre-
sponding Ve(z) via Poisson equation; and (iii) to solve again
the Schro¨dinger equation with the new Vsc, which in this new
iteration includes Ve(z) (as well as the other potentials: gates,
modulation doping, and structural confinement). We repeat
this process until convergence is attained.
b. Numerics
We use the sixth-order Numerov method to solve the
Schro¨dinger equation.37,38,39,40 Poisson equation (see Sec. 2)
is solved via a semi-analytical Numerov method.41 All nu-
merical integrations are performed using a Gaussian integra-
tion method.42 In our numerical implementation we use the
dimensionless form of Eq. (A2),
d2 ˜ψv
dz˜2
= ˜Vv ˜ψv, ˜ψv = ψv(z˜), z˜ = zl , (A3)
where
˜Vv =
2π
ε1
[
Vsc(z˜) − Ev], ε1 = π~
2
m∗l2
. (A4)
We choose l = 1 nm as our length unit and ε1 as the relevant
energy scale.
2. Poisson equations for the electronic and gate plus
modulation doping potentials
The self-consistent electronic potential energy Vsc(z) =
−eφsc(z) can be split in two parts, Vsc(z) = Vwb(z) + VH(z).
Vwb(z) = Vw(z) + Vb(z) described the structural quantum-well
potential. The “Hartree” contribution VH(z) = Ve(z) + Vg(z)
arises from the electronic charge density and from the exter-
nal gates plus the modulation doping regions (symmetrically
located around the well; see Fig. 12). Figure 12 also shows
the Dirichlet boundary conditions Va and Vb, which are in fact
the external gates at the end points ±L of our system.
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
V g
(eV
)
z (nm)
I II III IV V
Va Vb
−L −Ld −Ld + w
ρa ρb
Va = 0.2
Va = 0.0
Figure 12: Schematic representation of doping layers of width w and
densities ρa and ρb plus the external gates Va and Vb. By varying the
external gates (we usually keep Va = 0 and vary Vb) we can alter the
spatial symmetry of our quantum wells, Fig. 2. The curves illustrate
the calculated gate + modulation doping potential Vg(z) for Va = Vb
(dashed line) and Va > Vb (solid line), both with ρa > ρb.
a. Gate+modulation doping potential
We can write separate Poisson equations for Vg(z) and Ve(z)
as these arise from distinct charge densities. For Vg(z) we have

14
(see Fig. 12)
d2
dz2
Vg =
e2
ǫrǫ0























0, (I) : −L ≤ z ≤ −Ld,
ρa, (II) : −Ld ≤ z ≤ −Ld + w,
0, (III) : −Ld + w ≤ z ≤ Ld − w,
ρb, (IV) : Ld − w ≤ z ≤ Ld,
0, (V) : Ld ≤ z ≤ L,
(A5)
where ǫ0 is the permittivity, ǫr is the dielectric constant43,44
and ρa,b are the doping densities. From the continuity of Vg(z)
and its first derivative and assuming the Dirichlet boundary
conditions Vg(−L) = Va and Vg(+L) = Vb, we find
Vg =























c1z + c2, (I) : −L ≤ z ≤ −Ld,
1
2 Az
2 + c3z + c4, (II) : −Ld ≤ z ≤ −Ld + w,
c5z + c6, (III) : −Ld + w ≤ z ≤ Ld − w,
1
2 Bz
2 + c7z + c8, (IV) : Ld − w ≤ z ≤ Ld,
c9z + c10, (V) : Ld ≤ z ≤ L,
(A6)
with
A = e
2ρa
ǫrǫ0
, B = e
2ρb
ǫrǫ0
, (A7)
where the constants ci are given in Appendix B. Figure 12
shows two solutions of Eq. (A5), both having ρa > ρb and
Va = Vb (dashed line) and Va > Vb.
b. Electronic Hartree potential
The electronic Hartree contribution Ve(z) is determined
from
d2
dz2
Ve(z) = − eǫrǫ0 ρe(z), (A8)
with (including spin)
ρe(z) = 2eA
∑
v,kq
|ψv(z)|2 f (Ekqv) =
em∗
π~2
kBTλe(z), (A9)
where
λe(z) =
∑
v
|ψv(z)|2 ln[1 + eβ(µ−Ev)/kBT ], (A10)
and
f (Ekqv) =
1
1 + e(Ekqv−µ)/kBT
, Ekqv =
~2k2q
2m∗
+ Ev. (A11)
We solve Eq. (A8) for Ve(z) using an accurate Numerov
scheme41 with the Dirichlet boundary conditions Ve(±L) = 0.
Similarly to the Schro¨dinger equation in Eq. (A3)), we find
it convenient here to write the Poisson equation (A8) in a di-
mensionless form
d2
dz˜2
˜Ve = − ˜λe, ˜λe =
kBT
ε1
lλe(z˜), ε2 ˜Ve = Ve(z˜), (A12)
where ε1 is the energy scale given in Eq. (A4) and
ε2 =
e2
ǫrǫ0l
. (A13)
c. Electron density and chemical potential
From the total electronic charge
∫
dV ρe(z) = enT A (A14)
we can straightforwardly [using Eq. (A9))] obtain the total
areal concentration of electrons
nT =
∑
v
nv, (A15)
with the nv’s denoting the subband occupations
nv =
m∗
π~2
kBT ln
[
1 + e(µ−Ev)/kBT
]
. (A16)
When nT is fixed (i.e., the nT -constant model), we can de-
termine the chemical potential µ from Eq. (A15),
π~2nT
m∗kBT
=
∑
v
ln
[
1 + e(µ−Ev)/kBT
]
. (A17)
Appendix B: COEFFICIENTS ci’S
Using the continuity of Vg and its first derivative together
with the (Dirichlet) boundary conditions at the end points
Vg(−L) = Va and Vg(L) = Vb, we can determine the coeffi-
cients ci’s appearing in Eq. (A6). In the regions I and V we
find
c1 = −
2Ld − w
2L
wC− − wC+ −
V−
L
, (B1)
c2 = −
1
2
(2Ld − w) wC− − L wC+ + V+, (B2)
c9 = −
2Ld − w
2L
wC− + wC+ −
V−
L
, (B3)
c10 = +
1
2
(2Ld − w) wC− − L wC+ + V+, (B4)
with
C± =
1
2
(A ± B), V± = 12 (Va ± Vb), (B5)
and A and B defined in Eq. (A7). In the modulation doping
regions II and IV, we have
c3 =
w2 − 2wLd + 2LLd
2L
C− + (Ld − w)C+ − V−L , (B6)
c4 = +
1
2
(Ld − w)2 C− + 12(L
2
d − 2wL)C+ + V+, (B7)
c7 =
w2 − 2wLd + 2LLd
2L
C− − (Ld − w)C+ − V−L , (B8)
c8 = −
1
2
(Ld − w)2 C− + 12(L
2
d − 2wL)C+ + V+. (B9)
In the central region III, we have
c5 = +
2L − 2Ld + w
2L
wC− −
V−
L
, (B10)
c6 = −
1
2
(2L − 2Ld + w) wC+ + V+. (B11)

15
∗ Electronic address: sousa@if.sc.usp.br
† Electronic address: egues@if.sc.usp.br
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26 The renormalization of the conduction wave function is crucial to
obtain an effective Pauli-like equation which properly includes the
Darwin term and other higher other corrections. Winkler discusses
this point in detail in Ref. 20 (chapters 5 and 6).
27 Here we neglect the Darwin and all higher-order terms and focus
on only the Rashba-like contributions.
28 The Kane effective mass in Eq. (32) neglects corrections from
higher bands; here we use m∗ as a parameter determined from
experiment.
29 Here we are following a procedure similar to that used by W. Za-
wadzki and P. Pfeffer, Semicond. Sci. Technol. 19 R1 (2004), that
is, we neglect the spin-dependent boundary conditions along the
growth direction. These authors have also performed a detailed
analysis considering the effects of the spin-dependent boundary
conditions (along the growth axis of the well) on the SO energy
splittings. They found that the inclusion of the spin-dependent
boundary conditions gives rise to small corrections to the cal-
culated SO energy splittings, with more sizable corrections for
heavily-doped (> 1012 cm−2) narrower band gap heterostruc-
tures (e.g. InAs/In0.8Al0.2As). However, the influence of the spin-
dependent boundary conditions on the Rashba couplings (α) is not
mentioned in their study. We believe that the inclusion of the spin-
dependent boundary conditions (along the growth) in our prob-
lem will not alter the results in any essential way (certainly not
qualitatively); For an alternate description of the Rashba effect
using a multi-band approach see U. Ekenberg and D. M. Gvozdic,
arXiv:0801.0089v1. As these authors emphasize, the contrasts be-
tween these two descriptions deserve further study.
30 For a 14×14 k ·p model which accounts for the Rashba and Dres-
selhaus terms on the same footing, see, e.g. F. V. Kyrychenko,
C. A. Ullrich and I. D’Amico, J. Mag. Magn. Mat. (to be pub-
lished); authors investigate whether the intersubband SO coupling
discussed here should be accounted for when extracting the spin
Coulomb drag from intersubband spin plasmon linewidths; they
conclude it has a negligible effect on that property.
31 R. J. Warburton, C. Gauer, A. Wixforth, J. P. Kotthaus, B. Brar
and H. Kroemer, Phys. Rev. B 53, 7903 (1996); E. L. Ivchenko
and S. A. Tarasenko, JETP 99, 379 (2004); J. B. Khurgin, Appl.
Phys. Lett. 88, 123511 (2006), have investigated optical intersub-
band couplings (dipolar approximation) in quantum wells. Even
though these authors have taken into account the spin-orbit inter-
action (via the k · p approximation) when calculating these light-
induced transitions, we emphasize that the intersubband coupling
we consider in our work [Eqs. (34)–(37) or (40)–(43)] is of a dif-
ferent nature. By looking at, e.g., the intersubband matrix element
in Eqs. (15)–(17) in the work of Ivchenko and Tarasenko, we can
see that their matrix element (i) is proportional to the amplitude
of the electromagnetic vector potential A and, more importantly,
that (ii) it is independent of the Kane matrix element P. Our in-
tersubband coupling, on the other hand, is directly proportional to
the Kane matrix element P [defined in our Eq. (10)] and does not
depend on a vector potential A (we do not treat optical absorption,
i.e., A = 0 in our work). We believe, however, that intersubband
light absorption may provide an interesting means to experimen-
tally probe and contrast the spin orbit interactions (in the single
and double wells) investigated here. This issue will be addressed
in a future work.
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16
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