A. WIXFORTH, C. GAUER, AND J.P. KOTTHAUS
Sektion Physik der LMU München, D-80539 München, Germany
M. KUBISA, AND W. ZAWADZKI
Polish Academy of Sciences, Warsaw, Poland
B. BRAR, AND H. KROEMER
Department of Electrical and Computer Engineering, UC Santa Barbara, CA 93106, USA
We report on intersubband resonances in InAs/AlSb quantum wells subjected to a magnetic field B parallel to the layer of the two-dimensional electrons. In this geometry, both spin-conserving and spin-flip transitions can be excited simultaneously and are energetically well separated by the depolarization shift as depolarization effects solely influence the spin-conserving transitions. Moreover, they are found to cause a pronounced narrowing of the lineshape as compared to the single-particle picture. We also find that the spin-flip transitions remain split even at B=0T. This splitting is attributed to the bulk inversion asymmetry of InAs as it lifts the spin degeneracy for finite values of the in-plane wavevector k.
In the past, inelastic light scattering has been proven to be a powerful experimental method to investigate intersubband transitions and in particular many-body effects because charge- and spin-density excitations can be observed simultaneously [1]. For absorption spectroscopy, however, the dipole selection rules demand a strong spin-orbit interaction for the observation of spin-flip intersubband transitions (SF-IST). Therefore, SF-IST could so far only be investigated in 2D hole systems [2] with strong band-mixing and spin-orbit interaction. Under the influence of finite in-plane magnetic fields, however, such spin-flip excitations can also be excited in an electronic system [3].
Until now, most of the theoretical and experimental work has focused on magnetic fields applied perpendicular to the plane of the two-dimensional electron gas (2DEG) and fundamental phenomena such as cyclotron resonance [4] or the quantum Hall effect [5] have been described in great detail for this configuration. On the other hand, the so-called Voigt geometry (VG) , where the magnetic field is oriented parallel to the 2DEG, has received only limited attention in the literature [6,7]. Here, we would like to present results of intersubband resonance experiments in VG which offer novel and unexpected insights into the optical properties and spin effects on the bandstructure of semiconductor quantum wells.
We wish to review some of our results where both spin-conserving (SC) and spin-flip (SF) transitions can be observed in the same experiment. From the energetic separation between the SC- and the SF-IST we can directly deduce the depolarization shift. In addition, we find that depolarization effects drastically reduce the nonparabolicity-induced line-broadening of the single particle excitation. In VG, the in-plane magnetic field is found to break the polarization selection rule for intersubband transitions stating that only the electric field component in the growth direction couples to the resonance [8]. We show both experimentally and theoretically that - independent of the underlying bandstructure - the oscillator strength is proportional to the square of the magnetic field B providing a new and accurate method for the determination of intersubband matrix elements [9,10]. The excitation of SF-IST, on the other hand, can be only understood if one accounts for the specific bandstructure. We show that the lack of inversion symmetry together with a narrow band gap and a high carrier density is a prerequisite for its observation. All these requirements are perfectly met by the InAs/AlSb quantum wells investigated here and make them ideal candidates for our experiments. This material combination results in unusually deep quantum wells (2.1eV at the [[Gamma]]-point) with Te serveing as an effective dopant. Hence, very high electron concentrations of up to 8.1012cm-2 can be achieved [11]. Moreover, the InAs band-gap of only 0.42eV [12] is considerably smaller than in the widely studied AlGaAs/GaAs-system.
The samples described here were grown on a GaAs substrate followed by a sequence of buffer layers [15] to accommodate the large lattice mismatch between the substrate and the active layers. The 20 period multi-quantum well consists of 15 nm InAs layers and 10 nm wide d -doped Te:AlSb barriers. To minimize band-bending effects [16] the topmost AlSb barrier was chosen to be 55nm wide. The whole structure was capped by a 5 nm wide layer of GaSb to prevent the oxidization of the underlying AlSb. Hall effect measurements at low temperatures revealed a carrier density of Ns = 2.5.1012cm-2 per well. Mid-infrared spectra were taken at T=4K using a rapid-scan Fourier transform spectrometer with the samples mounted either in Voigt- or in multiple reflection path geometry. Experimentally, we determined the relative change in the transmission T(B!=0)/T(B=0) of unpolarized light for different in-plane magnetic fields B or alternatively, ratioed against a reference substrate.
In Fig. 1 we show a transmission spectrum taken in the conventional IST configuration where the sample is tilted by an angle with respect to the optical axis. This tilt provides the necessary finite perpendicular component of the incoming radiation as it is required for the excitation of an ISR. The resonance position of about 135mV agrees quite well with our calculations and also the oscillator strength exhibits the typical dependence on the tilt angle as expected for an IST [21]. The lower traces of Fig. 1 are measured in VG with in-plane magnetic fields between B=2T and B=13T and referenced to zero magnetic field. We observe two well separated resonances - one at around E=100meV (I) and a second at E=135meV (II) which both gain oscillator strength at higher magnetic fields. The lineshapes of the resonances I and II are drastically different: Line I is much broader and strongly asymmetric and the degree of asymmetry is magnetic field dependent whereas line II does not change its shape with magnetic field. The energetic position of line I shifts to higher energies from about E=94meV at B=2T to E=109meV at B=13T, whereas the position of resonance II remains almost constant. As shown in the inset, the integrated absorption of resonance II, being proportional to the oscillator strength, varies as the square of the magnetic field. From this findings we can readily identify line II as the depolarization shifted IST.
Before we turn to the interpretation of the unexpected line I, we would like to comment on the fact that the VG mediates the excitation of an IST by in-plane polarized light . As to the co-ordinate system used throughout this report, we choose the electric confinement to be in the y-direction and the magnetic field in the z-direction. For B=0T and x-polarized light the polarization selection rule forbids resonant excitation of the electrons. In the limit of very high magnetic fields, however, the cyclotron radius is smaller than the well width and the electrons will exhibit a bulk-like cyclotron motion [4,5] and x-polarized light may couple to this motion. Although the optical response at intermediate magnetic fields - when the cyclotron radius is comparable to the well width - can also be understood from an entirely classical argument [17], we want to give here an alternative explanation as it provides insight into an interesting physical problem:
Figure 1: The upper trace shows a transmission spectrum of our 150Å InAs/AlSb multi-quantum well ratioed against a reference substrate with the unpolarized radiation incident under an oblique angle of 57[[ring]]. The resonance strength follows the same polarization dependence as expected for intersubband transitions. In the magnetic field ratio of the same sample we show spectra T(B!=0)/T(B=0) at in-plane magnetic fields B=2T, 4T, 6T, 8T, 10T, 11T, 12T and, 13T. Both resonances I and II vanish at zero magnetic field and their absorption increases at higher fields. The inset shows that the integrated absorption of line II depends quadratically on the magnetic field [3].
The electronic wavefunction for the n-th subband can be written as
[5] meaning that the interaction Hamiltonian has to be a function of y in order
for the intersubband matrix element to be non-zero. With the gauge
and x- polarized ([[alpha]]x) light we find for the interaction Hamiltonian
.
For B!=0, it becomes y-dependent , resulting in a matrix element proportional
to the magnetic field
(1)
For the polarization [[alpha]]z parallel to the magnetic field the IST matrix element remains zero even at finite fields. From the above arguments also follows that the integrated absorption and therefore the oscillator strength of the IST varies as B2. For GaAs/AlGaAs quantum wells, we could show before that this B2 dependence provides an alternative and accurate way to determine the optical matrix elements of the IST [9]. In the case of InAs, however, a quantitative analysis requires that the band mixing be taken into account.
To this end we consider an 8-band
model where the interaction between the conduction band with the light-hole,
heavy-hole and split-off levels modifies the electronic dispersion as described
in [18]. In this framework we determine the oscillator strength of the
spin-conserving IST by calculating the matrix elements for the electric dipole
excitations with the interaction Hamiltonian in the matrix form
.
A0 is the amplitude of the vector potential, Pi the canonical momentum with i
(x,y,z)
and HKane corresponds to the 8-band matrix Hamiltonian. With Eg=0.42eV,
[[Delta]]=0.38eV and the Kane energy 2m0[[kappa]]2=22.9eV [12] the
multi-band calculation yields a matrix element of
=31.0Å
for a 150Å wide InAs/AlSb quantum well. In the experiment as described in
Fig. 1a (B=0T), we obtain a somewhat lower value of 24.5Å. Evaluating (1)
for the magnetic field induced IST the matrix element
is found to be 25.0Å - almost the same value as in the conventional
absorption measurement. This very good agreement proves that in the Voigt
configuration an accurate determination of the matrix elements is indeed
possible [11].
For an in-plane magnetic field
all magneto-electrical subbands [18] in the quantum well are spin-split at
finite magnetic fields as depicted in the inset of Fig. 2. At low fields the
negative g-factor of InAs results in an energetic decrease of the
-levels
and in an increase of the
-levels
(Zeeman regime). At higher B the diamagnetic shift as described in [19]
dominates and raises the eigenenergies irrespective of the spin-orientation.
According to the inset of Fig. 2 two spin-conserving and two spin-flip
transitions between
and
are possible. This implies that the magnetic field induced depolarization
shifted (i.e. spin-conserving) IST described above must be regarded as the
superposition of the two possible spin-conserving excitations. Also shown are
the single particle magnetic field dispersion of these transitions for k||=0.
The transition energy of the two spin-conserving excitations differs only by
about 3meV at the maximum field of B=13T and has a very weak magnetic field
dependence. At high values of k|| (kF=4x106cm-1 in our
sample), however, all resonance energies are strongly reduced (about 35meV at
k||=kF) as a result of the non-parallel subband dispersions.
To account for the many-body phenomena we neglect the exciton effect as it gives only a small contribution and concentrate on the depolarization which influences the resonances drastically at the high carrier densites. We generalize the approach of Allen et al. [20] to calculate the depolarization integral in the 8-band model [21] and its magnetic field dependence. As described in [22, 23] the depolarization not only leads to an increase of the transition energies as compared to the single-particle picture but also to a drastic reduction of the nonparabolicity-induced line-broadening. This explains both the narrow symmetric lineshape of resonance II and the failure to resolve the two spin-conserving transitions. Furthermore, the slight decrease of their resonance energies can be understood as a result of the magnetic field dependence of the depolarization integral which has the dimension of a length [21]. The magnetic field reduces the spatial extent of the wavefunctions leading to a diminished depolarization and thereby to a reduced transition energy.
While resonance II in Fig. 1 could be explained in terms of a coupling between the cyclotron motion and the intersubband resonance, we interpret line I as a combined spin-flip intersubband resonance. For bulk InSb [23] it is well known that strong mixing between the conduction and valence bands may induce combined spin-flip and cyclotron resonance excitations. In particular for narrow gap III-V semiconductors this optical mode - being actually dipole-forbidden - can be observed because of the strong band-band interaction. In our sample, however, this mechanism is too weak to accout for the experimental resonance strength.In addition to this band-mixing the lack of inversion symmetry can also give rise to spin-flip excitations [25, 26]. As our structure is doped symmetrically we can neglect the electric-field induced inversion asymmetry [26]. We therefore conclude that the bulk inversion asymmetry is the only source for spin-flip excitations in our sample.
Figure 2: Transition energies between the two lowest subbands in the single particle picture. The spin-conserving transitions differ only very little in energy while the spin-flip excitations are energetically well separated. For B!=0T all subbands are spin-split as shown in the inset and a total of four transitions are possible. The energy levels rise at high fields since the diamagnetic shift dominates the Zeeman effect. (from reference [3])
In the 8-band Hamiltonian it leads to additional terms of the schematic form
G(PiPj+PjPi), where G is a material constant and Pi,j the canonical momentum
[27]. As a first approximation, we neglect the influence of these terms on the
eigenenergies but we do consider them in the computation of the 8-component
wavefunction and the electron-photon interaction Hamiltonian. Not being aware
of the theoretical G value for InAs we take
as in InSb [28]. In this model we find that the matrix elements for the
spin-flip processes increase strongly as a function of the magnetic field and
the transition probability of the spin-flips becomes snon-zero only for large
values of the wavevector k, as realized in our samples. The matrix elements of
both spin-conserving transitions, on the other hand are roughly equal and about
twice as large as for the spin-flip transitions. Analogous to the
spin-conserving resonances, the SF-IST are only excited by the radiation
component [[alpha]]x.
Spin-flip transitions are not subjected to the depolarization effects as the macroscopic charge distribution remains constant [1]. Therefore SF-IST are lower in energy than their spin-conserving counterparts. For our sample the energetic difference for B->0T between lines I and II represents to a good approximation the calculated depolarization shift, further supporting our interpretation of line I. Also, the B->0T position of resonance I is in very close agreement with the computed k||-dependent single particle transition energies. Line I is much broader than line II at all magnetic fields as the nonparabolicity-induced broadening is compensated by the depolarization field [21]. From these arguments it follows that resonance I can indeed be regarded as the superposition of the two SF-IST sketched in the inset of Fig. 2.
After these qualitative considerations we calculated the expected transmission
spectrum where we assumed a fixed level broadening of
[3]. While the depolarization shift separates the spin-flip from the
spin-conserving transitions, neither the two spin-conserving nor the two
spin-flip resonances could be resolved. Either line turned out to be a
superposition of two excitations with roughly equal oscillator strength. As
found experimentally, the absorption of the spin-conserving and spin-flip
resonances is about the same. The linewidth of the high energy spin-conserving
transitions is drastically reduced by the depolarization field and its
resonance position is almost field-independent. The low energy spin-flips, on
the other hand, form one broad and rather symmetric line. All in all, the
oscillator strengths and the energetic positions of our calculation agree very
well with the experimental observations. However, the predicted lineshape of
resonance I does not exhibit the low-energy shoulder found in the experiment.
This asymmetry cannot be understood by the Zeeman spin-splitting between
and
as the shoulder does not disappear for B->0T which is in contrast to the
above
model.
Figure 3: (a) Transmission spectra of the 150Å InAs/AlSb multi-quantum well in the multiple reflection path geometry for B<=8T with [[Delta]]B=1T. Line II is interpreted as a superposition of the spin-conserving transitions while resonance I is now resolved into the two possible spin-flip excitations denoted Ia and Ib. The inset is a sketch of the experimental configuration. (b) Resonance positions fro lines Ia and Ib as a function of the in-plane magnetic field B. With increasing B the two lines merge as the effective g-factor goes to zero.
To improve the spectral resolution of our experiment we thus mounted our sample in the multiple reflection path geometry as sketched in the inset of Fig. 3a [29]. As before, however, the magnetic field is oriented parallel to the 2DEG and the experimental curves were referenced against B=0T. In contrast to the previous experiment, a splitting of line I into two resonances Ia and Ib corresponding to the two spin-flip transitions is now clearly resolved. As shown in Fig. 3b, this splitting increases with decreasing magnetic fields and reaches approximately [[Delta]]E=17meV for B-> 0T. This experimental result is diametrically opposed to our calculations shown in Fig. 2. The discrepancy between theory and experiment can be resolved if the effect of the bulk inversion asymmetry on the energy levels is properly included. For finite values of the wavevector k it leads to a lifting of the degenerate spin-levels even at B=0T [30], i.e. the electrons are subject to an additional internal magnetic field Bint. This field Bint originates from crystal electric fields which are Lorentz-transformed in the frame of the moving electrons. The electronic dispersion now contains a term proportional to k3 with the parameter [[gamma]] describing the strength of the inversion asymmetry [30]. The spin-dependent k3-term of the electronic energy leads to a splitting [31] given by
(2)
Here, the value of ky is determined by the well width L with
for
and
for
[33,34]. The value of [[gamma]] has been calculated to be
130eVÅ3 [34]. For k~kFermi we find reasonable agreement with
the experimental splitting of [[Delta]]E=17meV. Alternatively, we can calculate
[[gamma]] from the oscillator strength of the spin-flip transitions given by
the parameter
.
Following Cardona et al. [30] we approximate
(3)
and find [[gamma]] =150eVÅ3 which is in good agreement with both the theory and the value deduced from the zero-field spin-splitting.
At finite magnetic fields the total spin-splitting is a combination of the Zeeman effect and the internal magnetic field induced by the bulk inversion asymmetry. For an AlGaAs/GaAs quantum well in the Faraday configuration it has been predicted that the total splitting vanishes at a finite external magnetic field [13]. Our experiment seems to suggest that also a parallel magnetic field can compensate the internal field since the separation of lines Ia and Ib diminishes for high B. However, a detailed description of this effect is rather involved and and is the subject of present investigations [35].
To summarize, we have observed two types of intersubband optical transitions in InAs/AlSb quantum wells with a magnetic field applied parallel to the 2DEG: Spin-conserving excitations are induced by purely electromagnetic effects which can be interpreted as a coupling of the cyclotron motion to the intersubband transition. Intersubband excitations involving a spin-flip, on the other hand, are a consequence of the specific InAs band-structure. The bulk inversion asymmetry together with the narrow band gap of InAs make the oscillator strength of theses resonances non-neglegible at finite magnetic fields. The simultaneous observation of the spin-conserving and spin-flip resonances allows us to measure directly the depolarization shift separating both excitations. Moreover, our experimental results clearly demonstrate that the depolarization effect strongly reduces the broadening of intersubband resonances as a consequence of conduction band nonparabolicity. In the multiple reflection path geometry we are able resolve the two spin-flip transitions at low magnetic fields. From their energetic separation extrapolated to B->0T we deduce the inversion asymmetry parameter [[gamma]] which is found to be in good agreement with theoretical predictions and our results as deduced from the oscillator strength of the spin-flips.
Acknowledgements
We gratefully acknowledge valuable discussions with U. Rössler, and F. Pikus . The work in Munich was sponsored by the Volkswagen Stiftung, M.K. and W. Z. acknowledge the financial support of the Polish Comittee for Scientific Research under grant No. 2P03B13911. The Santa Barbara group gratefully acknowledges support from the Office of Naval Research and from QUEST, the NSF Science and Technology Center for Quantized Electronic Structures (Grant DMR 91-20007). The Munich - St. Barbara cooperation is also supported by the European Community via grant EC-US 015:9826.
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