Spin-orbit splitting of valence subbands in semiconductor nanostructures

Abstract

We propose the 14-band $\mathbf k \cdot \mathbf p$ model to calculate
spin-orbit splittings of the valence subbands in semiconductor quantum wells.
The reduced symmetry of quantum well interfaces is incorporated by means of
additional terms in the boundary conditions which mix the $\Gamma_{15}$
conduction and valence Bloch functions at the interfaces. It is demonstrated
that the interface-induced effect makes the dominating contribution to the
heavy-hole spin splitting. A simple analytical expression for the interface
contribution is derived. In contrast to the 4$\times$4 effective Hamiltonian
model, where the problem of treating the $V_z k_z^3$ term seems to be
unsolvable, the 14-band model naturally avoids and overcomes this problem. Our
results are in agreement with the recent atomistic calculations [J.-W. Luo et
al., Phys. Rev. Lett. {\bf 104}, 066405 (2010)].

Full text

Spin-orbit splitting of valence subbands in semiconductor nanostructures
M. V. Durnev, M. M. Glazov, and E. L. Ivchenko
Ioffe Physical-Technical Institute of the RAS, 194021 St. Petersburg, Russia
We propose the 14-band k·pmodel to calculate spin-orbit splittings of the valence subbands in
semiconductor quantum wells. The reduced symmetry of quantum well interfaces is incorporated
by means of additional terms in the boundary conditions which mix the Γ15 conduction and valence
Bloch functions at the interfaces. It is demonstrated that the interface-induced effect makes the
dominating contribution to the heavy-hole spin splitting. A simple analytical expression for the
interface contribution is derived. In contrast to the 4×4 effective Hamiltonian model, where the
problem of treating the Vzk3
zterm seems to be unsolvable, the 14-band model naturally avoids and
overcomes this problem. Our results are in agreement with the recent atomistic calculations [J.-W.
Luo et al., Phys. Rev. Lett. 104, 066405 (2010)].
PACS numbers:
I. INTRODUCTION
As follows from the time inversion symmetry and the
Kramers theorem, the electronic states in centrosymmet-
ric systems are at least doubly-degenerate. On the other
hand, in the three-, two- and one-dimensional systems
lacking a center of space inversion, the degeneracy of free
Bloch single-electron states is removed with exception
of particular points and directions of the Brillouin zone.
The removal of spin degeneracy occurs due to spatially
antisymmetric part of the single-particle periodic Hamil-
tonian H(r,ˆ
p)with allowance for the spin-orbit interac-
tion. In terms of the effective Hamiltonian H(k)the in-
teraction appears as a spin-dependent contribution odd
in the electron wave vector k. This contribution is re-
sponsible for a number of fascinating and important ef-
fects being actively studied nowadays, see, e.g., Refs. [1–
8].
In nano- and heterostructures, the quantum confine-
ment strongly modifies free-carrier dispersion. Partic-
ularly in quantum wells (QWs), the parabolic conduc-
tion band turns into series of two-dimensional subbands
shifted, in parallel, along the energy axis. In the absence
of inversion center, the spin-orbit interaction splits each
subband with the splitting described by linear and, some-
times, cubic kterms in the 2×2 effective Hamiltonian.9,10
Because of the more complex valence band structure, the
dispersion of holes is also much more complicated than
that in the conduction band. Rashba and Sherman11
were the first to calculate the spin splitting of the top-
most heavy- and light-hole subbands in QWs grown along
zk[001] from zinc-blende lattice semiconductors by us-
ing the bulk effective Hamiltonian (four-by-four matrix)
consisting of the conventional Luttinger Hamiltonian and
spin-dependent terms of the order of k3. They imposed
the simplest conditions for the four-component hole enve-
lope wave function, ψ= 0, on the boundaries of the QW
and obtained k-linear terms in the effective Hamiltoni-
ans of two-dimensional hole subbands. Direct extension
of the procedure developed in Ref. 11 for the realistic
models of quantum confinement, particularly, including
the effects of finite barrier, deems impossible owing to
the presence of k3
zspin-dependent term in the bulk 4×4
Hamiltonian. This term makes consistent matching of
the valence-band wave functions really challenging, since
it leads to a k2
zcontribution to the velocity operator ˆvz
and, therefore, to a singularity of the flux ∝ˆvzψat the
interface.
Here we develop the 14-band k·pmodel to calculate
spin splittings of hole subbands in QWs, which allows
us to avoid the k3
z-term problem. Moreover, we propose
additional terms in the boundary conditions for the 14-
component envelope which naturally describe the inter-
face heavy-light hole mixing arising due to anisotropy of
chemical bonds at the interfaces.12–17 The developed k·p
approach presents an independent alternative to atom-
istic calculations of the spin-orbit splittings in QWs.18,19
The paper is organized as follows: Sec. II presents the
symmetry analysis of the spin-dependent terms in the
valence band Hamiltonian, Sec. III outlines the 14-band
k·pmodel and the spin-orbit splittings in the bulk semi-
conductor as well as boundary conditions for the QW
structures; numerical results and analytical approxima-
tions are presented in Sec. IV, and Sec. V contains brief
conclusions.
II. SYMMETRY CONSIDERATIONS
A. Bulk zinc-blende-lattice crystals
We begin the symmetry analysis by reminding that,
in a bulk zinc-blende-lattice semiconductor, the expan-
sion of spin-dependent part H(3)
cof the electron effective
Hamiltonian in the conduction band Γ6starts from the
nonzero cubic term
H(3)
c=γc(σκ
κ
κ),(1)
where γcis the band parameter, σand κ
κ
κare pseudovec-
tors composed of the Pauli matrices in the coordinate
system xk[100],yk[010],zk[001] and the cubic com-
binations κx=kx(k2
y−k2
z)etc. The expansion of the
4×4 effective Hamiltonian in the Γ8valence band starts
arXiv:1311.5747v1 [cond-mat.mes-hall] 22 Nov 2013

2
from the first order term H(1)
v= (4k0/√3)V k, where
Vx={Jx, J 2
y−J2
z}s,{A, B}s= (AB +BA)/2,Jαare
the angular momentum matrices in the basis of spherical
harmonics Y3/2,m.20,21 Although the Tdpoint symmetry
allows this term, the coefficient k0is nonzero only due to
the k·pmixing between the valence states and the re-
mote Γ3states, it is quite small and may be neglected for
the GaAs-based systems.20,21 In the Γ8band, the cubic-
kterm contains three linearly independent contributions,
as follows,
H(3)
v=γv(Jκ
κ
κ)
+1
2δγv"a2X
α
J3
ακα+a3X
α
Vαkαk2
α−1
3k2#,
(2)
where, for further convenience, two of the three coeffi-
cients are presented as products of the parameter δγv
which has the dimension of γc, γvand dimensionless fac-
tors a2/2and a3/2.20 It is noteworthy that the first term
is non-relativistic in its origin, it is symmetry-allowed for
the Γ15 band. Two terms in the second line of Eq. (2) are
relativistic and one can see that the last summand con-
tains the “dangerous” contribution proportional to k3
z.
B. Quantum well structures
In the following symmetry analysis we consider only
symmetrical QW structures grown along the crystallo-
graphic axes [001], [110] or [111] and having the point
symmetries (i) D2d, (ii) C2vand (iii) C3v, respectively.
The Cartesian coordinates are conveniently chosen along
the axes (i) xk[100], y k[010], z k[001] or x1k[1¯
10], y1k
[110], z1k[001], (ii) x2k[1¯
10], y2k[001], z2k[110], and
(iii) x3k[11¯
2], y3k[¯
110], z3k[111]. In a QW struc-
ture the Γ8valence band is split into the heavy- and
light-hole-like states. In the following we choose the ba-
sic states at the Γ-point (kxj=kyj= 0) transforming
under the symmetry operations as the Bloch functions
ψ(1)
hh ≡ |Γ8,−3/2i=↓(Xj−iYj)/√2,(3)
ψ(2)
hh ≡ |Γ8,3/2i=− ↑ (Xj+ iYj)/√2
or
ψ(1)
lh ≡ |Γ8,1/2i= [2 ↑ Zj− ↓ (Xj+ iYj)]/√6,(4)
ψ(2)
lh ≡ |Γ8,−1/2i= [2 ↓ Zj+↑(Xj−iYj)]/√6.
Here ↑,↓are the spin-up and spin-down two-component
columns, Xj,Yjand Zjare the periodic orbital Bloch
functions in the chosen coordinate system xj, yj, zj(j=
1,2,3), and for simplicity we omit the index jin the spin
columns.
1. Growth direction [001]
The states in the heavy-hole subbands hh1, hh3. . .
and light-hole subbands lh2, lh4. . . transform accord-
ing to the Γ6spinor representation of the point group
D2dwhereas the eigenstates lh1, lh3. . . and hh2, hh4. . .
form the bases for the Γ7representation. In the method
of invariants21, the 2×2 matrix effective Hamiltonian in
each hole subband is decomposed into a linear combina-
tion of four basis matrices. Since both direct products
Γ6×Γ6and Γ7×Γ7reduce to the same direct sum of
irreducible representations Γ1+ Γ2+ Γ5, the basis ma-
trices can be chosen common for the subbands of Γ6and
Γ7symmetries. If the basic functions of the spinor rep-
resentations are chosen in the form (3), (4), then the set
of basic matrices includes the identity matrix (Γ1rep-
resentation), pseudospin matrix σz1(Γ2representation)
and two pseudospin matrices σx1, σy1transforming as the
pair of wave vector components ky1, kx1(Γ5representa-
tion). This allows one to write the linear-kterm in the
effective Hamiltonian as
H[001]
n=β(n)
1(σx1ky1+σy1kx1) = β(n)
1(σxkx−σyky),(5)
where n=hhν or lhν,ν= 1,2. . . , and β(n)
1are the
subband parameter. In Eq. (5), the effective Hamilto-
nian is presented in the two coordinate systems x, y, z
and x1, y1, z1relevant for the (001) structures. We stress
that the same form of the effective Hamiltonian for the
heavy-hole and light-hole subbands results from the spe-
cial order of the Bloch functions in Eqs. (3) and (4).
2. Growth direction [110]
Both heavy- and light-hole states transform according
to the same spinor representation Γ5of the group C2v.
Among components kx2, ky2,σαjonly kx2and σz2trans-
form according to equivalent representations. As a result,
the linear-kterm has the form
H[110]
n=β(n)
2σz2kx2,(6)
with β(n)
2being the subband parameters.
3. Growth direction [111]
The pair of functions (3) and the states hhν transform
according to the reducible representation Γ5+ Γ6of the
C3vpoint group. The direct product (Γ5+ Γ6)×(Γ5+
Γ6) = 2Γ1+ 2Γ2does not contain the Γ3representation
which means that the k-linear splitting of the heavy-hole
states is symmetry-forbidden. The first non-vanishing
spin-dependent contribution to the heavy-hole effective
Hamiltonian is cubic in kand has the form22
H[111]
hhν =γ(ν)
1σx3ky3k2
y3−3k2
x3(7)
+γ(ν)
2σy3kx3k2
x3−3k2
y3+γ(ν)
3σz3ky3k2
y3−3k2
x3

3
with three independent parameters γ(ν)
1,γ(ν)
2and γ(ν)
3.
By contrast, k-linear terms are allowed in the dis-
persion of light-hole subbands. Indeed, the functions
(4) and the light-hole states lhν transform according
to the two-dimensional representation Γ4. The product
Γ4×Γ4= Γ1+Γ2+Γ3contains a Γ3representation mean-
ing that the k-linear light-hole splitting is described by
H[111]
lhν =β(ν)
3(σx3ky3−σy3kx3),(8)
with a single parameter β(ν)
3.
III. 14-BAND MODEL
A. Energy spectrum and spin splittings in bulk
semiconductor
The 14-band k·pmodel, called sometimes 5-level k·p
model or the 14×14 extended Kane model, displays the
full symmetry of a zinc-blende-lattice crystal and de-
scribes in detail the electron dispersion in the vicinity of
the Γpoint in materials.9,20,23–26 The model includes the
Γ8vand Γ7vvalence bands formed from the orbital Bloch
functions X,Y,Z(Γ15 representation in the coordinate
system x, y, z), the lowest conduction band Γ6cformed
from the invariant orbital function S(Γ1symmetry) and
the higher conduction bands Γ8cand Γ7coriginating from
the Γ15-symmetry orbital functions X0,Y0,Z0. For the
spinor Γ-point Bloch functions |Ni(N= 1 . . . 14), we use
the notations
|Γ6c, mi,|Γ8v, m0i,|Γ7v, mi,|Γ8c, m0i,|Γ7c, mi,(9)
where m=±1/2and m0=±1/2,±3/2, the Γ6basis is
taken in the form ↑S,↓S, and the Γ8basis is given by
Eqs. (3) and (4), the Γ7basic functions are also taken in
the canonical form.10 The model contains eight param-
eters of the 14-band model, namely, the band gap Eg,
the energy distance E0
gbetween the Γ7cand Γ6states,
spin-orbit splittings ∆and ∆0of the valence and higher
conduction bands, interband matrix elements of the mo-
mentum operator
P= i ~
m0hS |ˆpx|Xi ,(10)
P0= i ~
m0hS |ˆpx|X0i,
Q= i ~
m0hX0|ˆpy|Zi ,
and, finally, the interband matrix element of the spin-
orbit interaction between the valence and higher conduc-
tion bands defined by
∆−= 3hΓ8c, m0|Hso|Γ8v, m0i=−3
2hΓ7c, m|Hso|Γ7v, mi,
(11)
TABLE I: Analytic expressions for the effective mass of an
electron in the Γ6conduction band and the Luttinger param-
eters γ1,γ2and γ3for the Γ8vband.9
m0
me
=2m0P2
~2Eg1−1
3
∆
Eg+ ∆ −2m0P02
~2E0
g1−2
3
∆0
E0
g+ ∆0
γ1=2m0
3~2P2
Eg
+Q2
Eg+E0
g
+Q2
Eg+E0
g+ ∆0
γ2=m0
3~2P2
Eg
−Q2
Eg+E0
g
γ3=m0
3~2P2
Eg
+Q2
Eg+E0
g
where Hso =~2([σ× ∇U(r)]p)/4m2
0c2is the spin-
orbit Hamiltonian, U(r)is the spin-independent single-
electron periodic potential, cis the speed of light, and
m0is the free-electron mass. Note that hereafter we ig-
nore the difference between the generalized momentum
operator πand the operator p=−i~∇because the
k·(π−p)correction is usually negligibly small.27 The
14×14 Hamiltonian matrix H(14)
N0Nis a sum of the diag-
onal matrix elements E0
NδN0Nand off-diagonal matrix
elements linear either in ∆−or in k. As frequently used
in the simplified multiband k·pmodels,9we ignore the
free-electron term (~2k2/2m0)δN0N, which in the case of
QW structures reduces the number of boundary condi-
tions at an interface from 28 to 14 and simplifies numer-
ical calculations.
The diagonalization of the 14-band Hamiltonian yields
the electron energy spectrum in the bulk material. Ta-
ble I summarizes four fundamental band parameters, the
electron effective mass in the Γ6conduction band and the
three Luttinger parameters for the Γ8vband, calculated
in the second order of the k·pperturbation theory. Ta-
ble II shows three different parametrizations of the 14-
band model used in Refs. 20, 25 and 28. One can see
that these parametrizations provide close values of the
conduction-band effective mass and Luttinger parame-
ters but as demonstrated below quite different values of
k-dependent spin-orbit splittings.
Following Ref. [20] we present the coefficients γcand γv
in Eqs. (1) and (2) as sums γc=γc0+δγcand γv=γv0+
a1δγv/2, respectively, which allows one to separate the
k·pthird-order contributions (γc0, γv0) from the fourth
order contributions (δγc, δγv), which include one order in
∆−. The third order contributions were found in Ref. [20]
with the result
γc0=−4
3P P 0Q∆(E0
g+ ∆0)+∆0Eg
EgE0
g(Eg+ ∆)(E0
g+ ∆0),(12)
γv0=4
3P P 0QEg+E0
g+ ∆0/2
Eg(Eg+E0
g)(Eg+E0
g+ ∆0).(13)
Note that, as compared with Eqs. (18) and (20) of
Ref. [20], we use here the different sign for Q, include
the factor i~/m0in our definition of Pand P0, and re-
fer E0
gnot to the Γ8cbut to Γ7cband. The fourth-order

4
TABLE II: Three parametrizations of k·pmodel for GaAs used in the literature. Energy gaps are given in eV, matrix elements
P,P0and Qare given in eV˚
A. Conduction band effective mass and Luttinger parameters are calculated after expressions given
in Tab. I.
Parametrization Eg∆E0
g∆0∆−P P 0Q me/m0γ1γ2γ3
Ref. [28] (I) 1.52 0.341 3.02 0.2 -0.17 9.88 0.41 8.68 0.063 8.51 2.08 3.53
Ref. [20] (II) 1.52 0.34 2.93 0.17 -0.1 10.3 3.3 6 0.062 7.51 2.7 3.4
Ref. [25] (III) 1.52 0.341 2.97 0.17 -0.061 10.31 3 7.7 0.061 8.42 2.48 3.63
TABLE III: The constants of spin-orbit splitting calculated after Eqs. (12)–(15) (in eV˚
A3), corrections to the parameters
ajcalculated after Eq. (17), and spin-orbit constants for kk[110] calculated after Eqs. (18) (in eV˚
A3) for three different
parameterizations introduced in Tab. II.
Parametrization γc0δγcγcγv0δγvδa1δa2δa3γlh γhh
(I) -2.39 -22.0 -24.4 6.65 76.7 0.883 -0.431 -0.258 96.4 13.2
(II) -13.9 -10.6 -24.5 39.5 34.7 0.396 -0.193 -0.116 79.3 5.84
(III) -16.0 -8.13 -24.1 45.7 26.9 0.645 -0.315 -0.189 80.5 8.79
correction to γcreads
δγc=4
9∆−QP2(3E0
g+ 2∆0) + P02(3Eg+ ∆)
EgE0
g(Eg+ ∆)(E0
g+ ∆0).(14)
The fourth-order corrections δγvai/2in the Γ8valence
band are conveniently written as (a(0)
i+δai)δγv/2(i=
1,2,3), where
δγv=−4
9
∆−P2Q[∆ + 2∆0+ 3(Eg+E0
g)]
∆Eg(Eg+E0
g+ ∆)(∆0+Eg+E0
g).(15)
and
a(0)
1=13
4, a(0)
2=−1, a(0)
3= 1 .(16)
The terms proportional to a(0)
irepresent ∆−P2Qcon-
tribution, they were derived in Ref. [20]. Equation (15)
differs from Eq. (21a) in Ref. [20] by extra ∆in the nu-
merator and denominator. In addition to a(0)
ithere are
other contributions proportional to ∆−Q3which are dis-
regarded in Ref. [20] and can be presented in the form
δai=ci
Q2Eg
P2(Eg+E0
g),(17)
c1=41
12 , c2=−5
3, c3=−1.
These contributions may play an important role because
Pand Qare of the same order of magnitude. It is worth
to mention that additional contributions ∝∆−P02Qare
negligibly small as compared to those in Eq. (17).
For completeness, below we write down the expressions
for the coefficients of the k3splittings ∆hh(k) = γhh k3,
∆lh(k) = γlh k3of the heavy- and light-hole subbands for
the particular direction kk[110]:
γhh =1
2
γv 1−3ξ−1
p1+3ξ2!+δγv" 1−2ξ
p1+3ξ2!
+1
2δa1 1−3ξ−1
p1+3ξ2!+1
8δa2 7−21ξ−13
p1+3ξ2!
+1
4δa3 1 + ξ
p1+3ξ2!#
,(18a)
γlh =1
2
γv 1 + 3ξ−1
p1+3ξ2!+δγv" 1 + 2ξ
p1+3ξ2!
+1
2δa1 1 + 3ξ−1
p1+3ξ2!+1
8δa2 7 + 21ξ−13
p1+3ξ2!
+1
4δa3 1−ξ
p1+3ξ2!#
,(18b)
where the ratio ξ=γ3/γ2characterizes the valence band
warping. Table III summarizes the values of spin-orbit
splitting constants for the conduction and valence bands
calculated, again for different parameterizations of the
k·pmodel. It is seen that the fourth-order contribu-
tions are comparable with and can be even larger than
the third order ones. Moreover, the inclusion of the
corrections δaiin Eqs. (18) significantly increases the
spin splitting of heavy-hole states. For example, in the
parametrization (I) (Ref. 28) the omission of δaiterms
yields γhh ≈4.6eV˚
A3while the corrected value is larger
by almost a factor of 3.

5
B. Boundary conditions and electronic states in
QWs
Let us now apply the 14-band model to calculate the
energy spectrum in a symmetric QW grown along the
zk[001] direction. In addition to the basis (9), we
use another set of basic functions |l, si(l= 1 . . . 7, s =
±1/2), where |l, 1/2i=↑Rlfor the spin s= 1/2and
|l, −1/2i=↓Rlfor s=−1/2, and Rlare the orbital
Bloch functions S,X,Y,Z,X0,Y0,Z0. This basis con-
sisting of products of the up- and down-spinors and the
orbital functions is more convenient for the formulation
and analysis of boundary conditions at the interfaces.
Within the 14-band approach each electron state Ψn,j in
a QW is described by 14 envelope functions fnj,ls in the
expansion
Ψnj =eikkρ
√SX
ls
fnj,ls(z)|l, si.(19)
Here zis the growth axis, kkis the in-plane wave vec-
tor with two components kx, ky,Sis the normaliza-
tion area, the subscript ndenotes the subband, e.g.,
n=e1, hh1, lh1etc., and jis a pseudospin index enu-
merating two states in each subband ndegenerated in
the Γ-point (kk= 0). The energy spectrum Enj (kk)in
the n-th electronic subband in k-space is obtained from
the numeric solution of the Schr¨odinger equation
H(14) kk,ˆ
kzΨnj =Enj (kk)Ψnj ,(20)
where kzis replaced by a differential operator ˆ
kz=
−i∂/∂z acting on the envelopes fnj,ls(z).
Equation (20) should be supplemented with the bound-
ary conditions. The key requirement is associated with
the conservation of the particle flux. For a 14-component
envelope fls the flux is given by
S=1
~X
l0s0ls
f∗
l0s0
∂H(14)
l0s0,ls(k)
∂kfls ,(21)
or explicitly one has, e.g., for the flux x-component
Sx=i
~hPˆ
f†
Xˆ
fS−ˆ
f†
Sˆ
fX+P0ˆ
f†
X0ˆ
fS−ˆ
f†
Sˆ
fX0
+Qˆ
f†
Z0ˆ
fX−ˆ
f†
Xˆ
fZ0+ˆ
f†
X0ˆ
fZ−ˆ
f†
Zˆ
fX0i ,(22)
where two-component spinor envelopes
ˆ
fl="fl,1/2
fl,−1/2#
are introduced for brevity.
Since the k·pHamiltonian contains kzterms only of
the first order, it is enough to impose one condition per
envelope fnj,ls. In what follows we assume that the inter-
band matrix elements P,P0,Q, and ∆−are the same in
the QW and barrier materials, so that solely the diagonal
elements E0
N, i.e. the positions of bands at k= 0, expe-
rience discontinuities at heterointerfaces. In this case the
simplest and intuitively natural set of boundary condi-
tions conserving the flux could be merely the continuity of
all envelopes at the interface, see Eq. (22). However, such
boundary conditions do not account for the reduced mi-
croscopic symmetry of a single interface described by the
C2vpoint group and caused by the anisotropy of chem-
ical bonds in the (001) plane.12–14,29 Therefore, we are
interested in boundary conditions as simple as possible
but those which conserve the flux and make allowance
for the interface heavy-light hole mixing. We recall that
in calculations based on 4×4 Luttinger Hamiltonian and
four-component envelope Φthis kind of state mixing is
described by an extra term in the boundary conditions13
ΦA= ΦB,(23)
ˆ
M−1
A∂Φ
∂z A
=ˆ
M−1
B∂Φ
∂z B
+2
√3
tl-h
a0m0{JxJy}sΦ,
where the matrix Mis diagonal and comprises the val-
ues of heavy-hole, mhh =m0/(γ1−2γ2), and light-hole,
mlh =m0/(γ1+ 2γ2), effective masses in the [001] direc-
tion, tl-his a real coefficient, and a0is the lattice con-
stant. In order to include the hole mixing effect into the
14-band k·pmodel, we provide the minimal generaliza-
tion of natural boundary conditions for envelopes fnj,ls as
the requirement of continuity of the five envelopes fnj,ls
corresponding to Rl=S,X,Y,Zand Z0and the follow-
ing discontinuity of the remaining envelopes, as follows,
ˆ
fnj,X0A=ˆ
fnj,X0B+˜
tˆ
fnj,XB,(24)
ˆ
fnj,Y0A=ˆ
fnj,Y0B+˜
tˆ
fnj,YB,
where ˜
tis a real dimensionless interface-mixing parame-
ter. One can readily see that the proposed conditions (24)
are in agreement with the flux continuity. The bound-
ary conditions for the 4×4model can be obtained from
Eqs. (24) taking into account that the Qkzoff-diagonal
matrix elements in the 14×14 Hamiltonian couple ˆ
fnj,X0
with ˆ
fnj,Yand ˆ
fnj,Y0with ˆ
fnj,X. As a result we arrive in
the linear-kzapproximation at the second boundary con-
dition (23) with the heavy-light hole mixing coefficient
tl-h≡2m0a0
√3~2Q˜
t . (25)
In what follows we use tl-has an independent parameter
of our theory. It is worthwhile to stress that an inclusion
of other extra terms in the 14-band boundary conditions
modifies Eqs. (23) to a more complicated form of Ref. 30.
IV. RESULTS AND DISCUSSION
Figures 1 and 2 display the main results of our 14-
band model calculation, with the generalized boundary

6
k || [100] k || [110]
(b)
Spin-orbit splitting (meV)
0
0.2
0.4
0.6
0.8
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
hh1
h+
h−
e1
(d)
Spin-orbit splitting (meV)
0
0.2
0.4
0.6
0.8
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
parametrization (I)
GaAs/Al0.35Ga0.65As
a = 100 Å
tl-h = 0
hh1
h+ (lh1)
h− (hh2)
(a)
Hole energy (meV)
−50
−40
−30
−20
−10
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
GaAs/Al0.35Ga0.65As
a = 100 Å
tl-h = 0
parametrization (II)
hh1
h+
h−
(c)
Hole energy (meV)
−40
−30
−20
−10
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
FIG. 1: Dispersion (a, c) and spin splitting (b, d) of valence subbands for GaAs/Al0.35Ga0.65 As QW. The calculations are done
for two parametrizations (see Tab. II): (I) [panels (a) and (b)] and (II) [panels (c) and (d)]. The spin splitting of conduction
subband e1is presented in (b) and (d) for comparison.
conditions (24), of valence band structure in a 100-˚
A-
thick GaAs/Al0.35Ga0.65As QW. The energy dispersion
of three topmost valence subbands is obtained neglecting,
Fig. 1, and taking into account, Fig. 2, the heavy-light
hole interface mixing. Each figure contains four panels.
Panels (a), (b) are calculated for the parametrization (I),
see Tab. II, panels (c), (d) represent the parametriza-
tion (II). Solid and dashed lines in panels (a), (c) show
the subband dispersion, while the wave vector depen-
dence of the spin splittings is depicted in panels (b) and
(d). As mentioned above we assume the same values of P,
P0,Q,∆,∆0and ∆−for the well and barrier materials,
the band gap of AlxGa1−xAs solid solution is taken from
the quadratic equation31 Eg(x) = Eg(0)+1.04x+0.46x2.
The standard ratio 2/3 is used for the Γ8v- and Γ6c-band
off-sets, ∆Evand ∆Ec. The off-set ∆Ec0for the higher
conduction band is chosen to equal −∆Evsince all pro-
posed parametrizations give practically the same value
for the sum Eg+E0
gin GaAs and AlAs.
We begin the discussion from comparison of the Γ-
point positions of the valence subbands calculated by us-
ing the 14-band model and 4×4 Luttinger Hamiltonian
in the absence of interface mixing, tl-h= 0. In the four-
band Luttinger model the heavy- and light-hole envelope
functions and Γ-point energies Ehhν ,Elhν (ν= 1,2. . . )
are found from
−~2
2
d
dz
1
mhh
d
dz +V(z)Φhhν =Ehhν Φhhν ,
−~2
2
d
dz
1
mlh
d
dz +V(z)Φlhν =Elhν Φlhν ,(26)
where V(z)is the confinement potential determined by
the valence band offsets, and functions Φ(z)satisfy the
Bastard boundary conditions given by the Eqs. (24) at
tl-h= 0. It turns out that the Γ-point energies calcu-
lated in the 14-band and Luttinger models agree with
each other within 5% accuracy. It is seen from Figs. 1(a)
and 1(c) that the two sets of parameters result in sig-
nificantly different positions of the two lower subbands
at kk= 0 despite relatively small (.30%) difference of

7
k || [100] k || [110]
GaAs/Al0.35Ga0.65As
a = 100 Å
tl-h = 0.5
parametrization (II)(c) hh1
h+
h−
Hole energy (meV)
−40
−30
−20
−10
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
GaAs/Al0.35Ga0.65As
a = 100 Å
tl-h = 0.5
parametrization (I)
(a)
hh1
h+
h−
Hole energy (meV)
−50
−40
−30
−20
−10
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
hh1
h+
h−
(b)
Spin-orbit splitting (meV)
0
1
2
3
4
5
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
k || [100] k || [110]
(d)
Spin-orbit splitting (meV)
0
1
2
3
4
Wave vector (106 cm-1)
−1.5 −1.0 −0.5 0 0.5 1.0 1.5
FIG. 2: Dispersion (a, c) and spin splitting (b, d) of valence subbands for GaAs/Al0.35Ga0.65As QW with account for the
interface mixing of heavy and light holes [tl-h= 0.5corresponding to ˜
t≈0.07 for the set (I) and ˜
t≈0.1for the set (II)]. The
calculations are done for two parametrizations (see Tab. II): (I) [panels (a) and (b)] and (II) [panels (c) and (d)].
the Luttinger parameters. For the parameter set (II),
these subbands are much closer in energy than for the
set (I). Moreover, the calculation carried out within the
Luttinger Hamiltonian model for the parametrization (II)
gives the crossing between the hh2and lh1states at
a≈95 ˚
A, see Ref. 34 for details. For the 100 ˚
A-thick
QW, the pure-state energies still lie very close to each
other, the mixing is remarkable and we use for these Γ
states in Figs. 1 and 2 the notation h+, h−instead of lh1
and hh2, respectively.
The heavy-light hole mixing is crucial both for the h±
states and the spin-orbit splitting of the valence sub-
bands. In the 4×4 effective Hamiltonian approach one
could expect to get the hh2-lh1mixing by including
the Vzˆ
k3
zterm of Eq. (2). The matrix Vzindeed mixes
the |Γ8,3/2istate with the |Γ8,−1/2istate as well as
|Γ8,−3/2iwith |Γ8,1/2i. The inclusion of the last term
in Eq. (2) into the effective Hamiltonian makes however
the problem unsolvable, neither rigorously nor approxi-
mately, even for the infinitely high barriers in which case
the boundary conditions reduce to vanishing of the en-
velopes at the both interfaces. Firstly, strictly speaking,
the order of differential equations increases, and addi-
tional unphysical solutions appear. Secondly, an attempt
to take the operator Vzˆ
k3
zinto account as a perturbation
encounters the problem of a nonhermitian nature of the
ˆ
k3
zoperator, hhh2|ˆ
k3
z|lh1i 6=hlh1|ˆ
k3
z|hh2i∗, defined on the
space of envelopes satisfying Eqs. (26) and vanishing at
the interfaces. It is worth to mention that, due to the spe-
cial reason,37 the linear-kterms in the valence subbands
hhν, lhν related to the Vzˆ
k3
zoperator and found as a first-
order correction to the energy spectrum in Ref. [11] are
finite and can be compared with the 14-band calculations,
see below. For barriers of finite height the inconsistency
related to the Vzˆ
k3
zoperator seems insurmountable for
finding both the Γ-point energies and the linear-kdisper-
sion. On the other hand, the 14-band k·pmodel under
study allows one to derive the cubic terms of Eq. (2) for
bulk materials and to compute the QW valence eigen-
states comprising an admixture of the Bloch functions

8
|Γ8,3/2iand |Γ8,−1/2ior |Γ8,−3/2iwith |Γ8,1/2iat
the point kx=ky= 0.
The curves in Fig. 2 are calculated for the same set of
parameters as in the previous figure, with one exception:
Now the interface mixing parameter ˜
tin the boundary
conditions (24) is nonzero and corresponds to a reason-
able value of the parameter tl-h= 0.5related with ˜
tby
Eq. (25). Comparing Figs. 1 and 2 we observe striking
effects of the interface mixing. The splitting between the
h+and h−states in Fig. 2(c) tremendously increases.
Furthermore, all the spin splittings in Fig. 2 are enhanced
by about an order of magnitude. The positions of the h+
and h−states at the Γ-point can perfectly be evaluated
in the framework of Luttinger Hamiltonian and general-
ized boundary conditions (23). In the first order in tl-h
the role of extra term in the boundary conditions (23)
can be reduced to an effective matrix element
∆l-h=tl-h~2
m0a0
Φhh2(zi)Φlh1(zi)(27)
that mixes the lh1and hh2states at kk= 0. Here zi
is the coordinate of the right-hand interface. The mixed
state energies are given by
Eh±=Ehh2+Elh1
2±sEhh2−Elh1
22
+ ∆2
l-h.(28)
At the crossing point, where Ehh2=Elh1, the splitting
between the h+and h−eigenstates equals 2|∆l-h|and
each of them is an equal admixture of the hh2and lh1
pure states. The comparison of Figs. 1(c) and 2(c) shows
that, in parametrization II, the “bulk” hh2-lh1mixing
inside the QW layer is by an order of magnitude smaller
than that due to the interface effect.
GaAs/AlAs
a = 85 Å
parametrization (II)
parametrization (I)
hh1
pseudopotential calculations (J.-W. Luo et al.)
|β(hh1)
1| (meV Å)
0
50
100
Interface mixing parameter tl-h
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
FIG. 3: The results of calculations for heavy-hole (hh1) spin
splitting as a function of the interface mixing strength for a
GaAs/AlAs 85 ˚
A well. The dashed line indicates the result
of pseudo potential calculations obtained for the same QW in
Ref. 19.
The 14-band model developed here automatically pro-
vides the spin-orbit splitting of conduction and valence
subbands for kk6= 0. Like the energy dispersion pre-
sented in panels (a,c), the spin splittings are given for
two directions of the in-plane wavevector kkk[100] and
kkk[110]. It is noteworthy that the anisotropy of spin
splittings becomes pronounced at kk∼106cm−2. One
can see in Figs. 1(b,d) that, even in the absence of in-
terface mixing, the k-linear spin splitting of hh1sub-
band is comparable with that for e1conduction subband
also shown (by triangles). The other notable feature is
a huge linear-in-kspin splitting of hh2and lh1(or h+
and h−) subbands which is particularly pronounced for
the set of parameters (II) where these states are close in
energy. For this set of parameters the linear terms for
h±states markedly exceed those for e1electron and hh1
valence subbands. Such a behavior was uncovered by
Rashba and Sherman11 in the model of infinite barriers:
It is caused by (i) the heavy-light-hole mixing by the off-
diagonal elements of Luttinger Hamiltonian proportional
to ˆ
kz(kx±iky)and (ii) Vzˆ
k3
zterm in the bulk spin-orbit
Hamiltonian (2). The heavy-light-hole interface mixing
results in the considerable enhancement of spin splittings
of the valence subbands but has only a small effect on the
spin splitting in the e1conduction subband.32
To analyze further the effect of interface mixing, we
present in Fig. 3 the absolute value |β(hh1)
1|as a function
of tl-hfor a 85 ˚
A-thick GaAs/AlAs QW. The spin split-
ting constant β(hh1)
1in Eq. (5) vanishes at a particular
value of tl-h≈0.2 either 0.5 for the parametrizations (II)
and (I), respectively, where the bulk-inversion asymme-
try and interface-inversion asymmetry contributions to
the splitting cancel each other. Here, it is worth to re-
mark that the cancellation takes place for positive values
of tl-h, while for tl−h<0the absolute value of the split-
ting monotonously increases with an increase of |tl−h|.
The sign of this parameter lies beyond the limits of this
work. Atomistic calculation of the spin-orbit splitting
performed in Ref. [19] predicted huge values of β(hh1)
1for
the hh1subband reaching 115 meV˚
A for a GaAs/AlAs
structure with the GaAs layer thickness of 85 ˚
A, as shown
in Fig. 3 by the horizontal dashed line. For this particu-
lar structure the 14-band model yields the same value of
hh1spin-orbit splitting assuming tl-h= 1.2÷1.6. Rel-
atively high values of interface-mixing parameter meet
the expectation of a monotonous increase of tl-hwith
the content xof the heteropair GaAs/AlxGa1−xAs. The
dependence of β(hh1)
1on the QW width ais shown in
Fig. 4(a). One can see that for the both sets of param-
eters the interface mixing dominantly contributes to the
spin splitting. In the selected well-width range, β(hh1)
1is
a monotonous function of a. For smaller values of the
width this coefficient reaches a maximum and then de-
creases with the increasing penetration of the wavefunc-
tion into the barriers. For completeness, the variation
of spin-splitting coefficients β(h±)
1with ain the h±sub-
bands is included in Fig. 4(a), see the inset.

9
tl-h = 0.5
parametrization (I)
h−
h+
|βh±
1| (meV Å)
10
50
100
Well width (Å)
50 100 150
tl-h = 0
parametrization (I)
parametrization (II)
|β(hh1)
1| (meV Å)
0
50
100
Well width (Å)
60 80 100 120 140
interface Eq. (30)
infinite barriers
|β(hh1)
1| (meV Å)
0
50
100
Well width (Å)
60 80 100 120 140
(a) (b)
FIG. 4: Spin-orbit k-linear term β(n)
1for the hh1subband in a GaAs/Al0.35Ga0.65 As QW. (a) 14-band numerical calculation
is shown for two sets of parameters (solid and dashed lines) and for two values of interface mixing parameter: tl-h= 0 and
tl-h= 0.5. The inset represents the results for h+and h−subbands at tl-h= 0 for the parameterization (I). (b) Analytical
calculation of β(hh1)
1. Three bottom curves are obtained in the limit of infinitely-high barriers from Eq. (8) of Ref. 11: the solid
curve represents the parametrization (I), the dotted and dashed curves are calculated for the parametrization (II) neglecting
and taking into account the corrections δaiin Eq. (17), respectively. Two top curves demonstrate the interface-induced spin
splitting according to Eq. (30) with tl-h= 0.5.
The numerical results presented above can be inter-
preted in terms of three independent contributions to
the spin-orbit splitting of the valence subbands. First
one is similar to that in the conduction band and, for the
heavy-hole subband hh1, originates from the PαJ3
ακα
and (Vxkx+Vyky)k2
zterms in the spin-orbit Hamilto-
nian for the Γ8band, Eq. (2), averaged over the size-
quantization wavefunction. The second contribution re-
sults from the interference of the ∝Vzk3
zterm in Eq. (2)
and off-diagonal elements Hof the Luttinger Hamilto-
nian. In evaluation of this second contribution one en-
counters the “dangerous” k3
zmatrix element which can
be calculated only in the limit of infinite barriers. In this
limit, the sum of two contributions is given by Eq. (8)
of Ref. 11. The third contribution to the k-linear split-
ting of the heavy-hole subband arises from the interface-
induced heavy-light-hole mixing and becomes dominant
for |tl-h|&1. It can be evaluated within the 4-band
model using the Luttinger Hamiltonian and the bound-
ary conditions Eq. (23) and taking into account that for
tl-h= 0 the heavy-hole wave functions can be presented
as33
Ψ±3/2= Φhh1(z)|Γ8,±3/2i±i(kx±iky)Slh(z)|Γ8,±1/2i.
Here the admixture of |Γ8,±1/2istates is considered in
the first order in kk, and the function Slh is found from34
−~2
2
d
dz
1
mlh
d
dz +V(z)−Ehh1Slh(z) =
=−√3~2
m0a0γ3
d
dz s
Φhh1(z).(29)
Here Ehh1is the energy of hh1subband in the Γ-point,
Eq. (26), and as before the curly brackets assume sym-
metrization of operators. Allowance for the tl-h6= 0 in
Eq. (23) gives rise to the interface inversion asymmetry
contribution to the hh1subband, which in the first order
in heavy-light hole interface mixing reads15,35
β(hh1)
1;int =2tl-h~2
m0a0
aΦhh1(zi)Slh(zi).(30)
The results of analytical calculations of the above con-
tributions to the hh1spin splitting are presented in
Fig. 4(b). The two sets of curves are depicted: the set
of three bottom curves corresponds to the “bulk” con-
tribution calculated in the limit of infinite-barrier well,
Ref. 11, and the set of two top curves represents the
interface-induced contribution calculated after Eq. (30),
for tl-h= 0.5and the finite barriers corresponding to a
GaAs/Al0.35Ga0.65 As QW. For calculation of the “bulk”
contribution, bottom dashed and solid curves, we used
parameters aiwith inclusion of corrections Eq. (17). For
comparison, the dotted curve in Fig. 4(b) is calculated
for ai=a(0)
iaccording to Eq. (8) of Ref. 11 for the set of
parameters (II). From the bottom curves in Fig. 4(a) and
Fig. 4(b) it is clearly seen that the infinite-barrier model
strongly overestimates the “bulk” contribution to the spin
splitting found within the 14-band model for tl-h= 0.
This can be attributed mainly to (i) significantly larger
values of the ˆ
k2
zoperator averaged over the heavy-hole en-
velope Φhh1found in the infinite-barrier well compared
to the case of finite barriers, and (ii) the overestimation
of Vzk3
zeffect.

10
TABLE IV: Valence-band spin splittings for a
100 ˚
A GaAs/Al0.35Ga0.65 As QW.
n β(n)
1(meV˚
A) γ(n)
1(eV˚
A3)γ(n)
2(eV˚
A3)
tl-h= 0 hh112.2 −82 −31
h+13.5 −153 54
h−67 −140 −78
tl-h= 0.5hh136 55 29
h+186 −412 −677
h−230 −475 −475
Similar analytical procedure can also be used to cal-
culate the interface induced k-linear spin-orbit splitting
of the h±subbands. The detailed discussion of the en-
ergy spectrum for this pair of subbands will be presented
elsewhere, here we resort to a simple resonant approxi-
mation which neglects all energy bands but h+and h−.
In this case, the dominant contribution results from the
interface-induced mixing of the heavy- and light- hole
states and reads17
β(h±)
1=±2√3~2
m0
∆l-hDlh1nγ3ˆ
kzoshh2E
p(Ehh2−Elh1)2+ 4∆2
l-h
,(31)
Dlh1nγ3ˆ
kzoshh2E=ZΦlh1(z)nγ3ˆ
kzosΦhh2(z)dz .
Equation (31) closely reproduces the results of numerical
calculation of β(h±)
1for parametrization (II) where the
subbands h±are particularly close in energy.
Above we have paid the main attention to the k-linear
spin splitting of valence subbands. However, it follows
from the 14-band calculations presented in Fig. 1(b, d)
and Fig. 2(b, d) that, at kk∼106cm−1, cubic in kterms
begin to play essential role resulting in the anisotropy
of the spin splitting. The k3contribution of n-th hole
subband in [001]-grown QWs contains two independent
parameters γ(n)
1and γ(n)
236
H[001]
n=γ(n)
1σx1k3
y1+σy1k3
x1
+γ(n)
2σx1k2
x1ky1+σy1k2
y1kx1.(32)
The parameters of Hamiltonian Eq. (32) extracted from
numerical simulation of a 100˚
A-GaAs/Al0.35Ga0.65 As
QW are listed in Tab. IV. It is worth to stress that (i)
interface mixing makes a significant contribution both to
k-linear and k3terms in the valence band effective Hamil-
tonian, and (ii) the k-linear terms given by β(n)
1indeed
exceed by far the k-linear terms in the bulk valence-band
Hamiltonian.
V. CONCLUSIONS
To conclude, we have presented here the 14-band k·p
model extended to allow for the reduced microscopic sym-
metry of QW interfaces which makes it possible to calcu-
late the spin-orbit splitting of hole subbands in QWs. We
proposed a simple boundary condition, Eq. (24), which
takes into account heavy-light hole mixing at the inter-
face due to anisotropic orientation of interface chemical
bonds. Main contributions to the hole spin splitting are
identified. The developed model has been applied to cal-
culate the valence-band spin splittings in (001) QWs, but
it can be used as well for QWs of any crystallographic
orientation including the (110) and (111) orientations.
The results of numerical calculations are well described
by the developed analytical theory. Moreover, we have
demonstrated that the large values of the spin splitting
for the topmost heavy-hole subband predicted in Ref. 19
on the basis of atomistic calculations can be ascribed to
the relatively strong interface-induced mixing of heavy-
and light-hole states.
Acknowledgments
We are grateful to M.O. Nestoklon and E.Ya. Sherman
for valuable discussions.
This work was supported by RFBR, Dynasty Founda-
tion, as well as EU project POLAPHEN.
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37 The spin-orbit splitting of the hh1subband contains sym-
metrized combinations hhh1|k2
z|lhνi∗+hlhν|k2
z|hh1iwhich
are meaningful for the envelope functions found in the limit
of infinite barriers.