Universal behavior of the electron g-factor in GaAs/AlGaAs quantum wells

Abstract

The Zeeman splitting and the underlying value of the g-factor for conduction band electrons in GaAs/Al_xGa_{1-x}As quantum wells have been measured by spin-beat spectroscopy based on a time-resolved Kerr rotation technique. The experimental data are in good agreement with theoretical predictions. The model accurately accounts for the large electron energies above the GaAs conduction band bottom, resulting from the strong quantum confinement. In the tracked range of optical transition energies E from 1.52 to 2.0eV, the electron g-factor along the growth axis follows closely the universal dependence g_||(E)= -0.445 + 3.38(E-1.519)-2.21(E-1.519)^2 (with E measured in eV); and this universality also embraces Al_xGa_{1-x}As alloys. The in-plane g-factor component deviates notably from the universal curve, with the degree of deviation controlled by the structural anisotropy. Comment: 8 pages, 6 figures

Full text

Page 1

arXiv:physics/0610216v1 [physics.optics] 24 Oct 2006
Universal behavior of the electron g-factor in GaAs/AlGaAs quantum wells
I. A. Yugova1,2,†, A. Greilich1, D. R. Yakovlev1,3, A. A. Kiselev4, M. Bayer1,
V. V. Petrov2, Yu. K. Dolgikh2, D. Reuter5and A. D. Wieck5
1Experimentelle Physik II, Universit¨ at Dortmund, 44221 Dortmund, Germany
2Institute of Physics, St.-Petersburg State University, St.-Petersburg, 198504, Russia
3A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
4Department of Electrical and Computer Engineering,
North Carolina State University, Raleigh, North Carolina 27695-7911, USA and
5Angewandte Festk¨ orperphysik, Ruhr-Universit¨ at Bochum, D-44780 Bochum, Germany
(Dated: February 2, 2008)
The Zeeman splitting and the underlying value of the g-factor for conduction band electrons in
GaAs/AlxGa1−xAs quantum wells have been measured by spin-beat spectroscopy based on a time-
resolved Kerr rotation technique. The experimental data are in good agreement with theoretical
predictions. The model accurately accounts for the large electron energies above the GaAs conduc-
tion band bottom, resulting from the strong quantum confinement. In the tracked range of optical
transition energies E from 1.52 to 2.0 eV, the electron g-factor along the growth axis follows closely
the universal dependence g||(E) ≈ 0.445 + 3.38(E − 1.519) − 2.21(E − 1.519)2(with E measured
in eV); and this universality also embraces AlxGa1−xAs alloys. The in-plane g-factor component
deviates notably from the universal curve, with the degree of deviation controlled by the structural
anisotropy.
PACS numbers: 78.47.+p, 75.75.+a, 73.21.Fg
I. INTRODUCTION
The Lande or g-factor of carriers in a solid is one of
the fundamental properties of this material1,2,3. For con-
duction band electrons in semiconductors and semicon-
ductor heterostructures it may deviate strongly from the
free electron g-factor in vacuum g0= +2.0023 due to the
spin-orbit interaction, e. g. it is -0.44 in GaAs, -1.64 in
CdTe and can be as large as -51 in the narrow band gap
semiconductor InSb5.
Invention of spintronics has increased the interest in
spin manipulation in semiconductor heterostructures2,4
and therefore in control of the carrier g-factors.
GaAs/AlxGa1−xAs heterostructures are very suitable for
this purpose as with increasing carrier confinement the
electron g-factor’s absolute value decreases and even
crosses zero. Therefore, the Zeeman splitting can be fully
suppressed by a proper choice of structure design param-
eters and/or external conditions like strain, temperature,
electric field, and orientation of the structure in external
magnetic field6,7,8.
The electron g-factor in GaAs/AlxGa1−xquantum well
(QW) structures has been measured by various experi-
mental techniques such as spin-flip Raman scattering9,
optical orientation10,11,12,13, optically detected magnetic
resonance14,15, spin quantum beats in emission16,17,18,19,
in absorption20, or in Kerr rotation21,22. However, most
of these studies have been limited to certain widths of the
quantum wells and only in a few papers the well width
dependence of the g-factor has been reported10,17,19,22.
The first concise analysis of the electron g fac-
torinQWs wasperformed
of the Kane model23, followed by more detailed
considerations24,25,26,27. The model calculations predict
in theframework
a strong variation of the g-factor value, including a
sign reversal in the GaAs/AlxGa1−xheterosystem, and
of its anisotropy with well width. Both quantities are
controlled by the strength of electron confinement in
the QWs determined mostly by the well width and
barrier height. These theoretical predictions were
further substantiated by experimental data19,22,28,29.
The published results are commonly presented as a
dependence of the g-factor value on the QW width, i.e.
a set of different dependencies corresponding to different
barrier heights (controlled by the Al content) is required
to cover the whole range of possible QW structures.
In CdTe/Cd1−xMgxTe heterostructures, whose band
structure is similar to that of GaAs/AlxGa1−x struc-
tures, to a good approximation a universal dependence
of the electron g-factor on the heterostructure band gap
(i.e., on the energy of the optical transition between the
ground states of confined electrons (e1) and holes (hh1))
has been reported30: the g-factor was sensitive mostly
to the value of the band gap itself, but not to the way
how this gap has been obtained by the structure’s design
parameters.
The goal of this paper is to check experimentally and
theoretically, whether this universality can be extended
to GaAs/AlxGa1−xheterosystems, and if yes, what the
origins of this universality are. We present experimental
results for the time-resolved pump-probe Kerr rotation,
which allow us to determine the transverse component
g⊥of the electron g-factor with high accuracy from the
frequency of the spin precession in an external magnetic
field. Model calculations accounting for the k · p inter-
action between lowest conduction band and the upper
valence bands have been performed for the longitudi-
nal and transverse components of the electron g-factor

Page 2

2
in structures with the Al content varied from 0 to 0.45.
We found that for the longitudinal component a univer-
sal dependence is strongly supported. At the same time,
the transverse component deviates notably from a uni-
versal curve, with the degree of deviation controlled by
the structural anisotropy.
II.EXPERIMENTAL DETAILS
The GaAs/AlxGa1−x
grown by molecular-beam epitaxy on (100) oriented
GaAs substrates. The sample parameters are collected
in Table I. Samples #1 and #2 consist of several sin-
gle QWs of different width separated by 50 nm thick
AlxGa1−xAs barriers to prevent electronic coupling be-
tween the wells. Samples #3, #4 and #5 contain only
one QW. All structures are nominally undoped, but
due to residual doping of the barriers presence of back-
ground electrons in the QWs has been established from
the observation of emission of negatively charged exci-
tons. The background electrons density does not exceed
5 × 109cm−2.
For optical measurements the samples were immersed
in pumped liquid helium at a temperature of 1.6 K, and
magnetic fields up to 7 T were applied perpendicular to
the structure growth axis (Voigt geometry). The struc-
tures were characterized by means of photoluminescence
(PL) excited by 532 nm laser light with an excitation
density below 0.3 W/cm2.
The electron g-factor was evaluated from the frequency
of the electron spin quantum beats corresponding to the
Larmor precession frequency in magnetic field. A pump-
probe technique with polarization sensitivity based on
time-resolved Kerr rotation was used for detection of the
spin beats (see e. g. Refs. 2,31). A Ti:Sapphire laser
emitting 1.8 ps pulses at a repetition rate of 75.6 MHz
was tuned in resonance with the QW exciton transition.
The pump beam was circularly polarized by means of an
elasto-optical modulator operating at 50 kHz. The exci-
tation density was kept as low as possible in the range
from 0.2 to 1 W/cm2. The probe beam was linearly po-
larized, its intensity was about 20% of the pump beam.
The rotation angle of the linearly polarized probe pulse
reflected from the sample due to the Kerr rotation, was
detected by a balanced diode detector and a lock-in am-
plifier. The time-resolved Kerr rotation signal as func-
tion of the delay between probe and pump pulses gives
the evolution of the electron spin coherence generated by
the pump.
heterostructures have been
III. EXPERIMENTAL RESULTS
A typical photoluminescence spectrum obtained for
sample #2 containing four single QWs of different thick-
ness is shown in Fig. 1. The emission from the GaAs
buffer layer is presented by the dashed line. The lumi-
nescence from the QWs is dominated by recombination
of excitons, whose energy increases for the narrow QWs
due to confinement. The low energy shoulders of the ex-
citonic lines are due to the negatively charged excitons
(trions) recombination32. Trions consisting of two elec-
trons and a hole are formed by a background electron
and a photogenerated exciton.
1.521.54 1.561.58 1.60


x0.1
5.1 nm
8.8 nm
13 nm
17.2 nm
PL Intensity (arb. units)
Energy (eV)
GaAs buffer
T = 1.6 K
FIG.
GaAs/Al0.33Ga0.67As
QWs of different widths (sample #2). The emission from the
GaAs buffer layer is shown by the dashed line.
1:Photoluminescence
structure
spectrumofa
containing four single
05001000
Time (ps)
15002000 2500
-1
0
1
2
3
4
012
B (T)
34
0.00
0.02
0.04
0.06
0
5
10
15


Kerr rotation (arb. units)
1.544 eV
B = 1 T

Zeeman splitting (meV)
|g⊥| = 0.26 ± 0.005
ωL (GHz)
FIG. 2: Time-resolved Kerr rotation for a 10 nm QW (sam-
ple #3) in a magnetic field of 1 T. The black line gives the
experimental data, the thick gray line is a fit after Eq. (2) to
the data using the parameters ωL = 3.76 GHz and τ = 880 ps.
The inset shows the Zeeman splitting ∆E (left scale) and the
frequency of spin beats ωL (right scale) as function of mag-
netic field. T = 1.6 K.
An example of time-resolved spin quantum beats in a
10 nm wide QW detected in a magnetic field of B = 1 T
is shown in Fig. 2. The experimental data are plotted
by the black line. The excitation energy of 1.544 eV is

Page 3

3
TABLE I: Parameters of the studied GaAs/AlxGa1−xAs QW structures.
Sample, Al content
#1 (p343), x=0.33
QW width (nm)
14.3
10.2
7.3
4.2
17.2
13
13
13
8.8
5.1
10
8.4
GaAs buffer
8.8
GaAs buffer
GaAs buffer
GaAs buffer
GaAs buffer
transition
e1 − hh1
e1 − hh1
e1 − hh1
e1 − hh1
e1 − hh1
e1 − hh1
e1 − lh1
e2 − hh2
e1 − hh1
e1 − hh1
e1 − hh1
e1 − hh1
Laser energy (eV)
1.530
1.543
1.569
1.622
1.527
1.535
1.542
1.597
1.555
1.600
1.544
1.559
1.559
1.555
1.515
1.529
1.543
1.553
|g⊥|
0.34 ± 0.01
0.27 ± 0.01
0.13 ± 0.01
0.06 ± 0.004
0.40 ± 0.01
0.33 ± 0.01
0.33 ± 0.01
0.32 ± 0.01
0.20 ± 0.005
0.00 ± 0.04
0.26 ± 0.005
0.17 ± 0.01
0.43 ± 0.005
0.21 ± 0.01
0.44 ± 0.001
0.44 ± 0.001
0.43 ± 0.006
0.42 ± 0.004
#2 (p340), x=0.34
#3 (p337), x=0.28
#4 (11302), x=0.25
#5 (e294), x=0.32e1 − hh1
resonant with the exciton transition. The periodic oscil-
lations of the Kerr signal are due to the precession of co-
herently excited electron spins about the magnetic field,
with the oscillation period TLgiven by the spin splitting
in the conduction band ∆E. Therefore, the Larmor pre-
cession frequency ωL= 2π/TLallows precise determina-
tion of the transverse component of the electron g-factor
g⊥by
∆E = µBg⊥B = ?ωL, (1)
where µB is the Bohr magneton. Note that the sign of
the g-factor cannot be determined, but only its absolute
value. For that purpose, we fit the spin-beat dynamics
by form for an exponentially damped oscillation. In case
of a single frequency and a single decay time, which gives
an appropriate description for most of the studied struc-
tures, this form is given by the following equation:
y(t) = Aexp(−t/τ)cos(ωLt),(2)
with an amplitude A. τ is the decay time of spin coher-
ence, which for an electron spin ensemble corresponds to
the spin dephasing time T∗
2
sult of a fitting to the experiment is given in Fig. 2 by the
thick grey line. Here we exclude from the analysis the ini-
tial 15-30 ps after the pump pulse, which are contributed
by the fast relaxation of holes18,34,35.
In the inset, the determined Zeeman splitting is plot-
ted as a function of magnetic field. The corresponding
values of the Larmor frequency are also given on the right
vertical axis. The slope of the B-linear dependence gives
|g⊥| = 0.26 ± 0.005. The decay time of the spin beats in
Fig. 2 is 880 ps and is considerably longer than the radia-
tive decay of resonantly excited excitons which does not
exceed 100 ps in GaAs/AlxGa1−xAs QWs36. Therefore,
we conclude that the detected spin beats are provided by
2,33. An example for the re-
background electrons whose spin coherence is photogen-
erated by the trion formation33.
0 1000 20003000
1.521.541.56


Beats amplitude
Energy (eV)
1.553 eV
x10
|g⊥,2| = 0.42
|g⊥,1| = 0.21
|g⊥| = 0.43
0.44
Kerr rotation (arb. units)
Time (ps)
0.44
1.543 eV
1.529 eV
1.515 eV
B = 1 T




GaAs
QW
FIG. 3: Time-resolved Kerr rotation of an 8.8 nm QW (sam-
ple #5) measured at different excitation energies. The upper
curve (1.533 eV) has been scaled to show clearly the beats
from the GaAs buffer layer superimposed by the QW signal.
The experimental data are shown by the narrow black lines
and fit results by the forms discussed in the text are given by
the thick gray lines. Inset: Relative intensities of the beats
from the QW (closed circles) and the GaAs buffer layer (open
circles). T = 1.6 K.
Not in all cases the Kerr signals can be described by
a single frequency and/or single decay time. Figure 3 il-

Page 4

4
lustrates a more complicated case observed in sample #5
for a 8.8 nm wide QW. The excitation energy of 1.515 eV
is resonant with the exciton transition of the GaAs buffer
layer and the observed oscillations with |g⊥| = 0.44 are
in accord with the known g-factor value of bulk GaAs
g(GaAs)= −0.445. The decay of the spin beats shows two
characteristic times. The fast one of 360 ps may be asso-
ciated with the exciton lifetime and the long one, which
exceeds 3 ns describes the dephasing of the background
electrons in the buffer layer37. The resonant excitation
into the exciton states of GaAs results in the largest am-
plitude of the Kerr signal. However, pronounced spin
beats can be observed when the excitation energy is de-
tuned from the resonance condition and shifted further
up by 27 meV to the energy of 1.543 eV, which is still
below any QW resonance.
energies is provided by the exciton-polariton reflection
spectrum38, which is governed by the quantization of po-
laritons in the GaAs buffer layer. The spin beats period is
independent of the excitation energy, as it is determined
by spin oriented carriers, which relax from the excited
states to the bottom of the conduction band and precess
there with |g⊥| = 0.44. The amplitude of the Kerr signal,
however, decreases, as shown by the open circles in the
inset.
An increase of the laser energy to 1.553 eV brings it
in resonance with the QW exciton. The beats period
increases about twice corresponding to |g⊥,1| = 0.21. The
spin beats are damped with a decay time τ = 410 ps. At
longer delays exceeding 1000 ps the oscillation picture
becomes irregular, suggesting that at least two periodic
processes overlap. The signal was fitted by an equation
accounting for two frequencies with different decay times:
The Kerr signal at these
y(t) = A1exp(−t/τ1)cos(ωL,1t + ϕ1)
+A2exp(−t/τ2)cos(ωL,2t + ϕ2). (3)
As one can see from the fit in Fig. 3 shown by the grey
lines, the experiment at longer delays can be described
by spin beats of QW electrons with |g⊥,1| = 0.21 super-
imposed with |g⊥,1| = 0.42 beats. The latter beats can
be attributed to electrons precessing in the GaAs buffer.
The relative intensities of the amplitudes A1and A2are
given in the inset of Fig. 3.
To obtain further insight, a 13 nm wide QW in sample
#2 has been excited resonantly with three optical tran-
sitions corresponding to the ground state of the heavy-
hole exciton (e1−hh1), of the light-hole exciton (e1−lh1)
and of the exciton related to the second confined electron
and levels (e2 − hh2). For all cases spin beats with al-
most the same period corresponding to |g⊥| = 0.33 have
been found (see Table I). As the value of the electron
g- factor should strongly depend on the electron energy,
we therefore conclude that the Kerr signal is provided
by electrons which relax to the bottom of the conduc-
tion band shortly after the photogeneration and precess
there. It is remarkable that the e1 electron precession
can be accessed through the e2 − hh2 optical transition.
05001000


4.2 nm
5.1 nm
7.3 nm
8.8 nm
13 nm
17.2 nm
x10
1.620 eV
1.600 eV
1.567 eV
1.555 eV
1.536 eV
1.527 eV
|g⊥| = 0.06
∼ 0
0.13
0.20
0.33
Kerr rotation (arb.units)
Time (ps)
0.40
x10
B = 2 T
FIG. 4:
thicknesses (samples #1 and #2) measured in a magnetic
field of 2 T. Well width, excitation energy and g-factor value
are indicated for each signal. T = 1.6 K.
Time-resolved Kerr rotation in QWs of different
After having given some insight into the general fea-
tures of the experimental data and their analysis, we turn
to the problem of the g-factor dependence on the carrier
confinement conditions. Time-resolved Kerr rotation sig-
nals detected for the laser energy resonant with the ex-
citon optical transitions in QWs of different width are
given in Fig. 4. One can see that the spin beats fre-
quency, which is proportional to |g⊥| of the conduction
band electrons, decreases with decreasing well width. Os-
cillations cannot be resolved in a 5.1 nm QW with the
exciton energy at 1.600 eV. A further decrease of the QW
width down to 4.2 nm restores the spin beat oscillations.
The determined |g⊥| values are given in the figure and
are collected also in Table I.
The experimental values for |g⊥| are plotted in Fig. 5
as function of excitation energy. Most of the data, ex-
cept for some results for a 13 nm QW and the GaAs
buffer which were measured for non-resonant excitation,
were detected when the laser was resonant with the exci-
ton state e1−hh1. Measurements on different structures
confirm the monotonic decrease of the g-factor absolute
value with increasing energy leading to a sign reversal
at 1.600 eV. Our results are in good agreement with the
data of Snelling et al.10obtained by the optical orienta-
tion technique. They are also in good agreement with
model calculations for Al contents of x = 0.3 and 0.35
shown by the solid and the dashed lines, respectively.
Details of these calculations are given below in Sec. IV.
We note that the calculated dependencies are plotted as
function of energy of the optical transition between the
confined carrier levels e1 and hh1 without accounting for
the exciton binding energy. The latter may shift these

Page 5

5
1.52 1.56
Energy (eV)
1.60
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
13 nm QW


GaAs buffer
#1
#2
#3
#4
#5
Ref. [10]
In-plane electron g-factor, g⊥
x = 0.35
0.30
FIG. 5:
electron g-factor on excitation energy. Symbols are experi-
ment, solid and dashed lines are calculations for Al contents
of x = 0.3 and 0.35, respectively. Horizontal dashed lines are
guides to the eye. T = 1.6 K.
Dependence of the transverse component of the
curves to lower energy by about 8 meV in wide QWs of
about 20 nm and by about 13 meV in 4 nm QWs39.
IV. CALCULATION OF THE ELECTRONIC
g-FACTOR AND COMPARISON WITH
EXPERIMENT.
Here we present results of model calculations for the
longitudinal and transverse components of the electronic
g-factor in GaAs/AlxGa1−xAs QWs for a wide range of
well widths from 1 to 30 nm and Al contents 0 < x <
0.45, which control the barrier height. In Fig. 6 the calcu-
lated g-factors are shown as function of the e1-hh1 optical
transition energy. This provides a convenient comparison
with experiment, as the g-factor is addressed in depen-
dence of an easily measurable quantity.
To prepare Fig. 6, it was necessary, first, to calculate
accurately the lowest quantized states of electrons and
heavy holes for each set of heterostructure parameters,
and, second, to evaluate the g-factor tensor for the cal-
culated electron state. To achieve advanced accuracy in
the energy levels, we applied the Kane multiband Hamil-
tonian, accounting exactly for the coupling between the
lowest conduction band Γ6and the upper valence bands
Γ8 and Γ7, and retaining also all remote band terms
that notably affect the dispersion of the relevant quasi-
particles in conduction and valence band. Details of the
computational procedure were presented elsewhere40,41.
Then by directly following the prescriptions in Ref. 26,
two independent components of the g-factor tensor at the
bottom of the first electron subband in the QW structure
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.671.68 1.69
0.05
0.06
0.07
⊥


T=1.6 K
0.4
2.01.9 1.8
1.7
Energy (eV)
1.6
AlGaAs
g
- - - - g

0.15
0.45
0.2
0.3
0.35
0.25
x=0.1
Electron g-factor
1.5


FIG. 6:
lines) components of the electron g-factor as function of the
energy of the optical transition in GaAs/AlxGa1−xAs QWs,
calculated for various barrier compositions and different QW
widths. Open circles show experimental data for AlxGa1−xAs
alloys taken from Ref. 44. Inset details dependences for the
longitudinal g-factor, which closely follow the dependence for
AlxGa1−xAs.
Longitudinal (solid lines) and transverse (dashed
TABLE II:
Al0.35Ga0.65As.
procedure for the AlxGa1−xAs alloys. The data are taken
from Ref. 5.
Band structure parameters for GaAs and
Also included is the type of interpolation
GaAs
0.341
28.9
0.067 m0
-0.44
0.45 m0
Al0.35Ga0.65As
0.32
26.7
0.09 m0
0.58
0.45 m0
Interpolation
basic, linear
basic, linear for Pcv
composite
composite
constant
∆SO (eV)
2P2
mbulk
gbulk
mhh
cv/m0 (eV)
can be explicitly calculated: the in-plane g-factor g⊥us-
ing Eq. (6) in Ref. 26 and the g-factor along the growth
direction g||with help of Eq. (10) in Ref. 26.
The parameters characterizing the band structure of
the AlxGa1−xAs alloy are known to a great definiteness.
Numeric values for all applicable quantities are collected
in Table II for GaAs and Al0.35Ga0.65As (the data are
taken from Ref. 5). The following scheme for parameter
evaluation has been adopted for an arbitrary Al com-
position x: for most of the parameters (named basic in
Table II), we use a simple linear interpolation (extrapola-
tion for x > 0.35). This group contains the interband mo-
mentum matrix element Pcvand the spin-orbit splitting
∆SO in the valence band. Rigorously speaking, higher
accuracy can probably be achieved if also the bowing
constants for the interpolation curves were known, but
for the studied range 0 < x < 0.45 linear interpolations
should be sufficient. As the only notable exception, we
use the common interpolation equation with bowing term

Page 6

6
for the AlxGa1−xAs band gap5:
Eg(x) = 1.519 + 1.04x + 0.47x2[eV ].(4)
Some parameters are obviously model derivatives and
we denote them composite: the bulk electron effective
mass mbulk and the bulk g-factor gbulkare defined by a
subset of basic quantities. These composite parameters
are not expected to follow a linear interpolation law, so
we apply a different approach: when we need the value
of a composite parameter for some alloy composition, we
directly calculate it from interpolated basic parameters.
For example42,
gbulk= g0−4
3
P2
m0
cv
∆SO
Eg(Eg+ ∆SO)+ δgremote
(5)
where m0is mass of free electron in vacuum and δgremote
is a correction due to the higher lying bands. First, we
use the specific numerical values of gbulk from Table II
and Eq. (5) to evaluate δgremote (which is treated here
as basic and interpolated when necessary).
apply Eq. (5) again to calculate gbulk for arbitrary al-
loy composition from a complete set of interpolated ba-
sic parameters. The heavy hole effective mass mhh is
taken to be independent of the alloy composition. A ra-
tio ∆EC/∆EV = 60/40 for the offsets in the conduction
and valence bands was taken for the calculations43.
Let us now describe in detail the results summarized
in Fig. 6. Here the two independent components of the
g-factor tensor are shown for the electron confined in
QWs with different barrier composition x as function of
the e1 − hh1 optical transition energy. The components
for magnetic field applied parallel (g||) and perpendicular
(g⊥) to the structure growth axis are given by the solid
and dashed lines, respectively. For each composition of
the barrier material g⊥> g||. The lowest energy for the
optical transition is achieved for an infinitely wide QW,
for which the energy asymptotically approaches that of
bulk GaAs (1.519 eV at T = 1.6 K) and both g tensor
components converge to meet the electron g-factor value
in the bulk: g(GaAs)= −0.44. Therefore, for each x value
the curves for the two g- factor components form a petal
with the root corresponding to the principal band gap
and the g-factor of bulk GaAs and the tip corresponding
to the values of the AlxGa1−xAs barriers (limit of a very
narrow QW). As the band gap and the g-factor both
grow monotonously with the Al composition, the petal
size also increases and its tip draws the line gbulk(Eg) cor-
responding to the bulk g-factor value for a range of alloy
compositions (shown in Fig. 6 by a thick solid line, open
circles are experimental data for AlxGa1−xAs alloys44).
Implicitly, the thick solid line in Fig. 6 is defined by
Eq. (5): assuming linear interpolations for Pcv, ∆SO
and δgremote, and the composition dependence for Eg(x)
given by Eq. (4), gbulk(Eg) can be easily recoveredanalyt-
ically as a series expansion. For the GaAs/AlxGa1−xAs
heterosystem, the first terms in the expansion are5:
Then, we
gbulk(Eg) ≈ −0.445+3.38(Eg−1.519)−2.21(Eg−1.519)2,
(6)
with Egmeasured in eV.
Quite remarkably, the growth direction component
g||(Ee1−hh1) follows very closely the AlxGa1−xAs de-
pendence gbulk(Eg) for an arbitrary barrier composition.
The inset in Fig. 6 shows a close-up of the energy de-
pendence of g||for different x. It illustrates that the g||
values for the whole range of x from 0 up to 0.45 basically
coincide, besides minor deviations, with the dependence
for bulk AlxGa1−xAs.
Although g⊥ follows generally the same trend, it de-
viates notably from the alloy dependence. This devia-
tion is caused by the structural anisotropy in the struc-
ture: therefore it depends on the barrier height and
strength of spatial confinement. However, for the studied
GaAs/AlxGa1−xAs QWs the maximum deviation of g⊥
from the bulk dependence is always moderate, and never
exceeds 0.3. Also, in the immediate vicinity of the petal
root (corresponding to the case of very weak spatial con-
finement and negligible penetration of the electron state
into barriers) some universality with respect to different
barrier compositions can be spotted for g⊥.
V.DISCUSSION AND CONCLUSIONS
In order to assess the meaning and the validity of the
universal dependence of the g-factor on the heterostruc-
ture band gap, let us first consider a hypothetical alloy
heterosystem AxB1−x. We assume that the system can
be accurately described by the Kane model, in which the
two basic parameters (i) the interband matrix element
Pcvand (ii) the spin-orbit splitting of valence band ∆SO
are plain independent of the composition (consequently,
these quantities will be equal in the well and barrier lay-
ers). Another assumption is that (iii) the valence band
offset is exactly zero. Analysis shows, that for such a re-
markable heterosystem, a truly universal dependence of
the electron g-factor on the e1 − hh1 energy in the QW
structure is expected. Moreover, no g-factor anisotropy
would be expected. We would like to note that similar
universality should also reappear when the barriers in the
heterostructures are extremely high, preventing consid-
erable penetration of the electron wave function into the
barriers.
In reality, conditions (i)-(iii) are not satisfied, so that
the degree of the parameter mismatch (along with the ex-
tent of the penetration of a confined electron into the bar-
rier layers) governs the deviations from the universal be-
haviour when one changes barrier composition and QW
width. The matrix element Pcvis modified only moder-
ately from one semiconductor to another and the modu-
lation in the value of ∆SOis almost negligible in cation-
substituted materials, including the AlxGa1−xAs alloys.
However, it can be considerable in anion-substituted
solid solutions5. Condition (iii) appears to be most vul-
nerable, as the valence band offset accounts for about
40% of the difference in band gaps for the well and
barrier materials in GaAs/AlxGa1−xAs heterostructures,

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7
but some approximate universality can be reasonably ex-
pected. The profound universality in the behavior of the
growth direction component of the electron g-factor in
GaAs/AlxGa1−xAs QWs, even though it is not exact,
obviously exceeds these expectations. A detailed numer-
ical analysis shows that it appears due to a fragile and
delicate balance in the dependence of the material pa-
rameters on composition: when the band gap increases
with Al content, slight reductions of Pcvand ∆SOcounter
play and mostly compensate the effect of moderate but
nonzero valence band offset in the type I band alignment
at the heterointerface, the particular experimental value
of mhhalso helps.
To conclude the discussion, approximate universalities
in the g-factor behavior can be expected in general for
cation-substituted alloy heterostructures with a type I
band alignment and small-to-moderate valence band off-
sets. Heterosystems with these properties are known to
include a number of important III-V and II-VI ternary
semiconductors. However, one should be cautious about
an indiscriminate extension of these conclusions to arbi-
trary materials.
In summary, the electronic g-factor has been studied
experimentally and theoretically in GaAs/AlxGa1−xAs
QWs for a wide range of well widths and Al contents. The
results are represented as a g-factor dependence on the
energy of the e1 − hh1 optical transition in QWs. A re-
markable universality has been established for the g ten-
sor components along the structure growth axis g||. The
deviation of the transverse components g⊥from this uni-
versal behavior is controlled by the structure anisotropy.
ACKNOWLEDGMENTS
We appreciate fruitful discussions with E. L. Ivchenko
and with I. V. Ignatiev.
Harley for providing us additional information about
his samples.This work has been supported by the
BMBF program ”nanoquit”, by the ISTC grant 2679
and by the RFBR grant 06-02-17157. Research stays
of IAY in Dortmund have been financed by the Deutsche
Forschungsgemeinschaft via Graduiertenkolleg 726 ”Ma-
terials and Concepts for Quantum Information Process-
ing” and grants Nos. 436 RUS 17/98/05 and 436 RUS
17/69/06. AAK acknowledges partial financial support
from DARPA and ONR.
We are thankful to R. T.
†Electronic address: irina yugova 05@mail.ru
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