*D-80539 München, Germany*

*E-mail: achim.wixforth@physik.uni-muenchen.de*

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 employed^{3}.
The carrier densities range in the Ns=1...5^{.}10^{12
}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^{.}10^{12 }cm^{-2}. Band-nonparabolicity has
been included in the calculation using a very simple two-band model as
originally derived by Kane^{4}. We model the energy dependent effective
mass m* by

_{
}

Here, _{
}=0.023
m_{0 }denotes the band-edge mass of InAs, and E_{G}=400 meV the
InAs bandgap at low temperatures. E=E_{1}+E_{F} 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. (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 explained^{1} (solid lines) by the above two band
model together with a guess for the ground state energy E_{1} using an
Ansatz in the model of the infinitely deep quantum well. For the energy
dependence of the effective g* factor, we use^{1}
g*(E)=g_{0}(1+aE) with g_{0}= -15, and
a=2.25^{.}10^{-3 }meV^{-1}.

Here, we would like to report on our experiments where we change the carrier
density N_{s} either by field-effect^{5} or by exploiting the
pronounced photo-effects^{6 }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
Zawadzki^{8} they find an analytical expression for the CR mass as a
function of the carrier density N_{S}:

_{
}

_{
}=_{}(1+_{}/
E_{G}) 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 N_{s} 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 E_{1}=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 N_{S} completely fails. It
rather seems that the simplest model including a density independent E_{1
}is a much better choice in this case.

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 terms^{8
}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).

1. J. Scriba, A. Wixforth, J.P. Kotthaus, C. Bolognesi, Ch. Nguyen,

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

2. M.J. Yang, R.J. Wagner, et al., *Phys. Rev. B*

3. G. Tuttle, H. Kroemer, and J.H. English, *J. Appl. Phys.* **67**,
3032 (1990)

4. E.O. Kane, *Journ. Chem. Solids* **1**, 249 (1957)

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)

7. C. Gauer, J. Scriba, A. Wixforth, J. P. Kotthaus, C.R. Bolognesi,

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)

present adress:

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

^{SS }Texas Instruments, Houston TX