R. Rinaldi, P. V. Giugno, and R. Cingolani
Unità INFM-Dipartimento Scienza dei Materiali, Università di Lecce, Lecce, Italy
H. Lipsanen, M. Sopanen, and J. Tulkki
Optoelectronic Laboratory, Helsinki University of Technology, 02150 Espoo, Finland
J. Ahopelto
VTT Electronics, Otakaari 7B, 02150 Espoo, Finland
An unprecedentedly well resolved Zeeman effect has been observed when confined carriers moving along a closed mesoscopic path experience an external magnetic field orthogonal to the orbit plane. Large Zeeman splitting of excited higher angular momentum states is observed in the magnetoluminescence spectrum of quantum dots induced by self-organized InP islands on InGaAs/GaAs quantum well. The measured effect is quantitatively reproduced by calculations including the vertical quantum well confinement and strain induced, nearly parabolic, lateral confinement, together with the magnetic interaction.
 |
, where n is the principal quantum number
and m is the angular momentum quantum number. The
splitting occurs when the magnetic field is parallel to
the symmetry axis of the structure. This is attributed
to the breaking of the degeneracy of states with different
values of the angular momentum quantum number
induced by the external magnetic field (Zeeman splitting)
[7-9].
State-of-the-art quantum dot samples were fabricated
by self-organized growth of InP islands on the topmost
barrier of an 
quantum well (QW)
[10]. The propagation of the strain field induced by the
InP islands through the GaAs barrier layer results in a
parabolic potential in the QW plane (see inset in Fig. 1).
The samples consisted of a 6.5 nm
QW and
a 5 nm GaAs cap layer, covered by InP islands of 20 nm
height and 80 nm diameter. The resulting depth of the
lateral confining potential was about 70 meV for electrons
and 25 meV for heavy holes [11].
In Fig. 1 we show the cw photoluminescence spectra as
a function of the excitation power density. The spectra
show up to five intersubband transitions involving quantized
electron and hole states in the dots. At the lowest
excitation intensity (514 nm line of an argon laser, 
)
the spectrum exhibits three well resolved
bands due to radiative transitions involving the first
three quantized states. The luminescence line at 1.439 eV
is the lowest excitonic line of the InGaAs quantum well.
This signal arises from the sample area between adjacent
islands. With increasing excitation intensity, we observe
the filling of higher-energy quantized states, up to the fifth
level. Under all pumping conditions, the separation of
the recombination lines is rather constant (about 17 meV).
The energy position of the observed transitions is consis-tent
with the calculated confinement energies, according
to Ref. [12]. Clearly, these dots evidence strong three-dimensional
confinement properties.
In Fig. 2(a) we report the cw magnetoluminescence
spectra obtained under 
excitation in-tensity
and with the magnetic field parallel to the growth
axis of the structure szd. The zero field spectrum ex-hibits
four well resolved lines due to intersubband re-combination
of carriers from the four lowest QD energy
levels. With increasing magnetic field the recombination
lines corresponding to excited states broaden and split into
two or more lines of different amplitude and width. This
is clearly observable in Fig. 2(a), where the split transi-tions
arising from the single lines at 0 T are connected
by dashed lines. Note that changing the magnetic field
orientation by 90± (B parallel to the quantum well layer),
the excited transitions in the spectra exhibit neither split-ting
nor appreciable shift with increasing field, as shown
in Fig. 2(b).
 |
As shown by us recently [12], the confinement ener-gies
of our quantum dot system can be reproduced by
the Luttinger-Kohn model. The effective potential for
electrons and heavy holes includes the QW confinement
potential (vertical confinement) and the strain induced deformation
potential (lateral confinement). Because of the
axial symmetry, the envelope wave functions can be written as
,
where the principal quantum num-ber
n specifies the number of radial modes and m the z
component of angular momentum
. The states
are labeled 
for
where states with the same
are degenerate at zero field. Therefore our ground state is 1
, the first excited state
, and the second
excited state 
and 
(nearly de-generate).
Because of the decoupling of heavy and light hole states the Schrödinger equation for electrons (e) and holes (h) in external magnetic field reads
(1)
where isotropic electron mass is assumed for the electron
.
The radial and vertical heavy hole masses are given in terms of Luttinger
parameters by 
and
, respectively [14].
is the rectangular
band edge con-finement due to the quantum well, and
is the approximately
harmonic strain potential calculated by the theory of elasticity [12].
 |
the Zeeman
effect is determined by
,
where 
is the angular momentum. The total Zeeman effect, linear in
B, is thus given by
(2)
The diamagnetic shift determined by the HD term in the Hamiltonian is given by
(3)
The diamagnetic shift is quadratic in the field up to a few teslas before obtaining a linear asymptotic limit at very high field strengths [14]. If Coulomb interaction is taken into account, i.e., electrons and holes form excitons, the total diamagnetic shift of the exciton will be smaller than that forecast by Eq. (3), due to the additional confinement of the exciton wave function in the dot. This effect has been neglected in the present calculations [15].
Furthermore, the atomic part of the "atomic" Zeeman
splitting should be taken into account (this is often called
spin splitting, although it also involves the interaction of
the orbital magnetic momentum with the magnetic field).
This effect is related to the electron magnetic moment of
the atomic parts of the Bloch wave functions. For electrons
we have 
where 
and 
ranges between –0.598 and –0..948 [14]. For
holes (at lowest order) we have
where 
, and
is the atomic g factor.
However, even with the higher order corrections, [14], this effect
is of the order of 1 meV, comparable to the quantum
well case. This cannot be resolved in the luminescence
spectra, and can safely be neglected here.
The results of the theoretical analysis are shown by
the lines in Fig. 3, where we plot the transition energies
calculated by means of Eqs. (2) and (3) for states of
different m value, split by the Zeeman effect. The
selection rules for these transitions are
and
[16].
Figure 3(a) clearly shows the lifting of the degeneracy
of the 
levels. The experimentally observed splitting
of this level as a function of the magnetic field is shown in
the inset. The dependence is clearly linear, as expected,
for the Zeeman splitting of an axially symmetric quantum
dot. With increasing excitation power, it is possible to
observe the splitting of several filled states (up to the
fifth luminescence peak consisting of
and
transitions).
It is worth noting that the
states
do not split and exhibit only a weak diamagnetic shift in
the field [see dashed lines in Figs. 3(a) and 3(b)]. The
fine structure of the fan plots becomes more and more
complicated with increasing the quantum number n.
The measured diamagnetic shift of the ground level
(2 meV at 8 T) is smaller than that obtained by Eq. (3)
(4 meV at 8 T for free carriers), due to the neglect of
excitonic effects. If we add the experimental diamagnetic
shift to the Zeeman term [rather than Eq. (3)], we find
an even better agreement between theory and experiment.
This indicates that excitonic effects could play some role
in our experiment. A quantitative analysis of these effects
will be presented in a forthcoming study [17]. In the
opposite configuration (B perpendicular to the growth
axis) [Fig. 2(b)], the average magnetic energy due to the
angular part of the wave function is zero because the angle
between the z component of the angular momentum and
the magnetic field direction is
. In this configuration
the magnetic field direction is no longer a symmetry axis
for the Hamiltonian, and the magnetic field induced shift
of the photoluminescence bands is very small because
the orbit of carriers crosses the rectangular well potential
boundary along the z direction. Very recently, Bayer
et al. [6] have reported the magnetoluminescence spectra
of QD’s fabricated by shallow etching. However, their
interpretation was based on a Zeeman splitting given by
(for 
transitions) which is by far larger
than our experimental data and our Eq. (2).
In conclusion, we have presented the first well resolved
quantitative measurement and theoretical interpretation
of a fundamentally new phenomenon, the breaking of
the degeneracy of quantum dots states with different
angular momentum quantum numbers induced by an axial
magnetic field. This Zeeman effect is peculiar to quantum
confined nanostructures.
We gratefully acknowledge D. Cannoletta, A. Mel-carne, and M. Corrado for expert technical help.
we estimate,
on the basis of peak intensities, that there are approximately
6 8 electrons and 6 8 holes in a quantum dot. Inset:
schematic diagram of the strain-induced confining potential of
the dots.
growth axis and 
growth axis and
. The dashed lines in
(a) connect split transitions arising from single bands at 0 T.
(symbols).
Transitions with the same n value are indicated by the same
symbol. Curves are calculated according to Eqs. (2) and (3).
The dashed lines indicate unsplit
transitions. Continuous
lines indicate split 
transitions associated with
states. Note that symbols associated with
transitions change slope at high fields due to anti-crossing
of different states. Inset: magnetic field dependence
of the Zeeman splitting of the 1
transition. (b) The same as (a) but for