Sektion Physik der Ludwig-Maximilians-Universität München,

D-80539 München, Germany



Department of Electrical and Computer Engineering, UC Santa Barbara, CA 93106, USA

The spin-splitting of the cyclotron resonance in quasi-two-dimensional electron systems has puzzled the community for quite some time. It is expected in systems that exhibit a certain band-non parabolicity, i.e., where both the effective electron mass m* as well as the effective Land g-factor g* depend on the energy. This way, cyclotron resonance transitions at different Landau filling factors are no longer energetically degenerate. However, despite the expected observability, most of the realised quasi two dimensional electron systems do not exhibit a spin split cyclotron resonance, whereas the splitting is observed in three dimensions. We demonstrate a system, in which the spin splitting of the cyclotron resonance is well observed and follows exactly a simple two-band model. Here, we investigate the density dependence of the observed resonance splitting and compare it to existing theories.

InAs quantum wells between AlSb barriers provide a very promising material combination in terms of future high speed transistor and optoelectronic applications. Moreover, it was the first structure containing a quasi-two-dimensional electron system (Q2DES) exhibiting a pronounced spin-splitting of the cyclotron resonance1,2 (CR). Although expected, no such splitting had been observed before on any other 2D system, a fact that puzzled both theorists as well experimentalists over the last ten years. In this view it is even more surprising that the observed CR line splitting for the InAs system can be quantitatively explained in a very simple two-band model including the band-nonparabolicity of this narrow-gap material.

The samples investigated here are nominally undoped and 15 nm wide InAs wide quantum wells sandwiched between AlSb barriers that have been grown by molecular beam epitaxy on a semi-insulating GaAs substrate. To accommodate for the rather pronounced lattice mismatch between the substrate and the active layers, a sophisticated buffer layer sequence has been employed3. The carrier densities range in the Ns=1...5.1012 cm-2, and the mobilities u as extracted from magneto transport measurements are about u=20...80 T-1.

In Fig. (1a), we depict a calculated Landau fan chart together with the magnetic field dependent Fermi level for a given sample with Ns=1.4.1012 cm-2. Band-nonparabolicity has been included in the calculation using a very simple two-band model as originally derived by Kane4. We model the energy dependent effective mass m* by


Here, =0.023 m0 denotes the band-edge mass of InAs, and EG=400 meV the InAs bandgap at low temperatures. E=E1+EF is the sum of the ground state energy of the quantum well and the Fermi level energy. The inset of Fig. (1a) shows two transitions for the CR that are possible for different filling factors.

Fig. 1: left : Landau fanchart calculated acc. to eq. (1) together with the oscillating position of the Fermi level. The ground state energy E1 has been handled as a fit parameter. The carrier density is taken from magneto-transport measurements. The insets indicate possible CR transitions for two different filling factors. Right : Experimentally determined CR mass as a function of the magnetic field B. The solid lines are the result of the calculation of the nonlinear fanchart.

Fig. (1b) depicts the resonance positions of the observed CR converted into an effective cyclotron mass at a given carrier density. We observe a strong oscillation of the CR line position and splittings of the CR as a function of an external magnetic field, which can be explained by the energy dependent cyclotron mass and Landé g-factor, respectively. At even integer Landau filling factors, a strong splitting of the CR line is observed, corresponding to a large change of the cyclotron mass ([Delta]m-splitting). At odd integer filling factors, the somewhat smaller splitting can be related to a change of the effective g-factor ([Delta]g-splitting). Our experimental findings (dots) are quantitatively explained1 (solid lines) by the above two band model together with a guess for the ground state energy E1 using an Ansatz in the model of the infinitely deep quantum well. For the energy dependence of the effective g* factor, we use1 g*(E)=g0(1+aE) with g0= -15, and a=2.25.10-3 meV-1.

Here, we would like to report on our experiments where we change the carrier density Ns either by field-effect5 or by exploiting the pronounced photo-effects6 as present in our samples. For a fixed magnetic field, we are thus able to tune the Landau-filling factor by varying the carrier density. The density dependence of m* for small magnetic fields, i.e. without taking a spin- and Landau splitting into account, has been evaluated earlier by Gauer et al. (ref. 7) in great detail. There, the agreement between theory and experiment is excellent for a variety of quantum wells of different widths (see Fig. 2a). Following the calculations of Zawadzki8 they find an analytical expression for the CR mass as a function of the carrier density NS:


=(1+/ EG) denotes the ground state energy according to the model as represented by eq. (1). We thus can expect that the description of the observed CR splittings as a function of Ns using eq. (2) is very accurate, too.

In Fig. 2b, we show the result of our experiment for a fixed magnetic field and varied carrier density together with the predictions of both models (1) and (2). We observe steplike jumps of the CR line position corresponding to steplike changes of the cyclotron mass around integer filling factors. Again, [Delta]m- and [Delta]g-splittings are observed. The experimentally obtained resonance positions are indicated as open symbols, whereas the lines represent the calculations according to eq. (1) and eq. (2), respectively. The solid lines have been calculated according to eq. (1) using a constant and density independent ground state energy E1=61 meV as extracted from measurements like in Fig. 1, whereas the broken lines are the result of a quasi-self consistent calculation according to eq. (2).

Surprisingly, for high magnetic fields, where the spin- and Landau splitting of the CR is clearly observable, the excellent low field description of the behavior of the CR mass as a function of NS completely fails. It rather seems that the simplest model including a density independent E1 is a much better choice in this case.

Fig. 2: Left : Cyclotron mass as a function of the carrier density for small magnetic filds and a variety of different samples of two different quantum well widths. The symbols represent the experimental values, whereas the solid lines are the result of a calculation according to eq. (2).

Right : Experimentally obtained CR mass as a function of the carrier density NS (cuircles).for higher magnetic field where CR splittings are resolved. The solid (dashed) lines are the prediction of models (1) and (2), respectively. Surprisingly, the more 'exact' model (2) does NOT reflect the experimental findings.

The answer to this contradiction is unsolved, to date. As the carrier density is precisely known from transport data, we conclude that the 2-band model does not adequately describe the splitting. We have also calculated the Landau-levels with a full 8-band model which yields almost identical results. This indicates that coupling to the split-off band and also to more remote bands is obviously not that important as far as the energy dependent effective mass is concerned. Also, self-consistent band bending effects that might be the reason for the inconsistency can be shown to be of only very little importance. Another possible factor has been pointed out in ref. 2. Here, the authors emphasize the importance of stress effects that are related to the lattice mismatch mentioned above. However, a detailed analysis of the influence of stress on the effective mass has been performed in ref. 7 which showed that - at least for the samples studied here, no significant influence is observable. The reason why the Kane model fails to give a consistent picture thus remains unclear. It might be related to internal electric fields or the lack of inversion asymmetry in III-V-compounds giving rise to additional terms8 in the Hamiltonian which are not yet included in our analysis.

We appreciate valuable discussions with R.J. Warburton, C. Gauer, and W. Zawadzki. The work in Munich was sponsored by the Volkswagen Stiftung. The Santa Barbara group gratefully acknowledges support from the Office of Naval Research and from QUEST, the NSF Science and Technology Centre for Quantized Electronic Structures (Grant DMR 91-20007).

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and H. Kroemer, Sol. St. Comm. 86, 633 (1993)

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5. A. Simon, J. Scriba, C. Gauer, A. Wixforth, J.P. Kotthaus, C.R. Bolognesi, C. Nguyen, G. Tuttle, and H. Kroemer, Mat. Sci. Engin. B21, 201 (1993)

6. C. Gauer, J. Scriba, A. Wixforth, and J.P. Kotthaus, C. Nguyen, G. Tuttle, J.H. English, and H. Kroemer, Semicond. Sci. Techn. 8, S137 (1993)

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C. Nguyen, B. Brar, and H. Kroemer, Semicond. Sci. Techn. 9, 1580 (1994)

8. C. Gauer, M. Hartung, A. Wixforth, J.P. Kotthaus, B. Brar, and H. Kroemer

Surf. Sci. 361/362, 472 (1995)

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* Dept. of Physics, Simon Fraser University, Burnaby BC V5A 1S6, Canada,

SS Texas Instruments, Houston TX