Landau levels in two and three-dimensional electron gases in a wide parabolic quantum well

a27v322a

Landau Levels in Two and Three-Dimensional Electron Gases in a Wide Parabolic Quantum Well

C.S. Sergio, G.M. Gusev, J.R. Leite,

Instituto de Física da Universidade de São Paulo, SP, Brazil

E.B. Olshanetskii, A.A. Bykov, N.T. Moshegov, A.K. Bakarov, A.I. Toropov,

Institute of Semiconductor Physics, Novosibirsk, Russia

D.K. Maude, O. Estibals, and J.C. Portal

GHMF, MPI-FKF/CNRS, BP-166, F-38042, Grenoble, Cedex 9, France

Received on 23 April, 2001

Shubnikov-de Haas oscillations are measured in a wide parabolic quantum well with 6 subbands in a tilted magnetic fi eld. We find two types of oscillations. The oscillations at low magnetic field are shifted towards higher fields with tilted angles, and can be attributed to the two-dimensional Landau state at the bottom subband. The position of the second type oscillations do not shift with tilted angles indicating a three-dimensional character of the Landau state formed by the highest subbands. The bottom level in the quantum well is not overlapped with the highest subbands due to the enhanced quantum scattering time of the lowest subbands.

I. Introduction

When a magnetic field is applied in a bulk semiconductor, the free eletrons which carry the eletric charge perform an orbital motion in the plane perpendicular to the magnetic field direction. This motion becomes quantized, and equally spaced levels (the Landau levels) separated in energy by w_c are formed. The energy of the system is given by

where i = 1, 2, 3, ... is the Landau quantum number, m is the effective mass of the electron, and w_c = eB/m is the cyclotron frequency.

The electrons within one Landau leavel may be considered to behave as if they were one-dimensional. The density of states (DOS), (E), which in the abstance of a magnetic fild is a parabola given by (E) µ E^1/2dE, now becomes the sum of a set of one-dimensional densities of states, where (E) µ E^1/2dE, each starting at the bottom of a Landau level. The very sharp singularities at the bottom of each Landau level is the origin of the Shubnikov-de Haas (SdH) effect. In practice these sharp features are smeared out by scattering.

When free particles are confined to a small region of space, either by a potential barrier formed by physical boundaries of the sample, the energy levels of the particles become quantized due to the wave-like behavior of the particles. Are of the simplest example of this is a square well potential. For a square well of width w_e the energy of the bound states are given by (infinite potential barrier)

where n = 1, 2, 3, ... is the subband index. We see that the energy separation increases from the bottom to top levels with subband number.

If a magnetic field is applied perpendicular to the two-dimensional (2D) electron gas, then a total quantization of the electron levels takes place. The resulting DOS consists of a set of d-functions separated by w_c, in the absence of scattering. When scattering is present each d-function broadens into peaks with width G.

In present the work we study remotely doped 4000 Å parabolic quantum well (PQW) with intermediary density, which allow us to obtain 6 occupied subbands (for full case we have 8 subbands occupied). In order to characterize the wide parabolic well and determine the subband structure we measure SdH oscillations in a tilted magnetic field. The oscillations contain two frequencies, one depends on the tilt angle, and other does not. We attribute such behaviour to the three-dimensional (3D) Landau states formed by the 5 higher subband and 2D Landau states originated from the lowest subband.

II. Experiment and Discussion

The samples used are the GaAs/Al_xGa_1-xAs PQW grown on undoped (100) GaAs substrate by molecular-beam epitaxy. On the top of the substrate there is 10,000 Å GaAs buffer layer with 20 periods of AlAs(5 ML)GaAs(10 ML) superlattice, followed by 5000 Å Al_xGa_1-xAs with x varying from 0.07 to 0.27. The structure consists of a 4000 -Å-wide Al_xGa_1-xAs well in which x was quadratically varied between x = 0, at the center of the well, and x = 0.19, at the edges of the well. On each side, the well is bounded by Si-doped ( ~ 5.0 ×10¹¹cm^-2) Al_0.3Ga_0.7As layers, grown next to spacer layers. The thicknesses of the undoped Al_0.3Ga_0.7As space layers are 100 Å. A 100 Å GaAs cap layer was grown as final layer of the estructure.

After growth, are photolithographically defined Hall bar with dimensions 100 X 200 mm. Four-terminal resistance and Hall measurements were made down to 50 mK in magnetic field up to 17 T. The measurements were performed with an ac current not exceeding 10^-8 A. Resistance was measured for different angles q between the field and substrate plane in magnetic field using an in situ rotation of the sample.

The mobility of the electron gas in the well is m_H = 210×10³cm²/Vs, and the electron concentration is n_H = 2.5×10¹¹cm^-2 - from the Hall effect at low field.

Three dimensional pseudocharge is N₊ = 0.9×10¹⁶cm^-3 which corresponds to the classical width of the 3D electron gas w_e = n_H/N₊ = 2900 Å. We perform the numerical self-consistent calculations for PQW of width W = 4000 Å, which yields the following energies for the first 6 electric subbands (in meV): E₁ = 0.05; E₂ = 0.21; E₃ = 0.46; E₄ = 0.80; E₅ = 1.22; E₆ = 1.73; and E_F = 2.03 meV (for m = 0.075 m₀).

Fig. 1 shows the low field dependence of the SdH oscillations for different angles q. The oscillations are periodic in 1/B and contain only single frequency. The position of the oscillations are shifted, as expected or 2D electron gas, when magnetic field is tilted from the normal to the substrate. The magnetoresistance are very well described by the conventional formula for the SdH oscillations in the 2D case: [1]

where A_T = (2 p²k_B T)/(w_c) , t is a quantum lifetime, E_F(2D) is the Fermi energy of the 2D level, and R₀ represents the classical resistance in zero applied field.

From the comparison of the experimental SdH oscillations (Fig. 1, q = 0) and Eq. 3 we extract the carrier density n_s1 = 0.7×10¹¹cm^-2, which is coincident with 2D electron density obtained from the calculation for the lowest subband. Surprisingly, we don't see any contribution at this magnetic field from the second subband.

Fig. 2 shows R_xx(B) extended to the magnetic field up to 3 T. We can see 3 oscillations indicated by arrows. Surprisingly, the position of these oscillations does not depend on the tilt angle. We attribute such behaviour to the formation of the 3D Landau states. In real systems the energy levels will have finite widths because of the disorder, therefore corresponding electric subbands can overlap. Naively, it is expected that the lowest subbands will overlap first, when the width of the well increases, because the distance between levels D_ij = E_j - E_i grows up as the square of the index number. However, if the broadening of the levels G_j increases faster than d_ij = D_ij/2 the highest electric subbands are collapsed to the bulk Landau states before the lowest one. Therefore the specific features of the investigated wide PQW is a coexistence 3D and 2D electron states inside of the well. In the tilted field 2D SdH oscillations are shifted to the higher magnetic field, and can cross 3D SdH peaks, which does not depend on the tilt angle.

The theoretical expression for the SdH oscillations in 3D case is slightly different from the 2D case: [2]

Fits the experimental curve for 3D SdH oscillations to the Eq. 4 give the value E_F(3D) = 1.88 meV. From this value we find the bulk concentration for highest subbands N_(3D) = 0.7×10¹⁶cm^-3.

The density profile for the 5 higher subbands is not a constant and has a deep minimum in the center as we can see in Fig. 3. Therefore the sheet density can not be recalculated from the equation n_s = w_e ·N_(3D). The width of the self-consistent electron density profiles can be defined as:

where n(z) = å n_si |f_i (z)|², and f_i is the envelope function of the electrons in the ith subband.

The sheet density of the electrons in the 5 highest subband is (n_H - n_s1) = 1.8×10¹¹cm^-2. We obtain the self-consistent value w_eff = 2600 Å and find bulk density for the quasi-three-dimensional subband N_(3D) = (n_H - n_s1)/w_eff = 0.7×10¹⁶cm^-3, which is equal than the bulk density determined from the measurements of the 3D SdH oscillations.

Furthermore, we calculate t following the formalism of the Ando and Gold taking into account the influence of the intersubband coupling on the screening and correlation corretions. [3] We consider only two major scattering mechanisms - remote and background impurity scattering. The results of the level broadening G_i = /2t_i are, in meV: G₁ = 0.06; G₂ = 0.08; G₃ = 0.18; G₄ = 0.21; G₅ = 0.25; and G₆ = 0.40. Our empirical finding is that G₂ < d₁₂ for 2D confinement effects to be observable in botton subband. We obtain G₂» d₁₂. For highst subbands G_j > d_ij - therefore theses subbands are overlapped and form the 3D system.

III. Conclusions

In the present work we realize the system with 2D and quasi-3D electron gas coexisting in the same quantum well. We use standard analysis of SdH oscillations in the tilted magnetic field and explore the fact that 2D Landau states are sensitive to the perpendicular magnetic field. We evaluate the broadening of the levels due to remote and background impurity scattering in the presence of the intersubband scattering and find that the bottom subband is not overlapped with the highest subbands. Therefore, the 2D state belongs to the lowest subband and the 3D state, to the highest subband.

It is known that 2D and 3D systems obey several properties, which are radically different, such as localization in random potential. We believe that our system can be used for comparing such effects.

Acknowledgments

We would like to thank FAPESP for financial support.

References

[1] T.Ando, J.Phys.Soc.Jpn , 37, 1233 (1974).

[2] Landau Level Spectroscopy, Modern Problems in Condensed Matter Sciences, edited by G. Landwehr and E.I. Rashba, NORTH-HOLLAND, Volume 27.2 (1991).

[3] A Gold, Phys. Rev. B, 35, 723 (1987); P.T.Coleridge , Phys. Rev. B, 44, 3793 (1991); E.Zaremba, Phys. Rev. B, 45, 14143 (1992).

Publication Dates

• Publication in this collection
  26 Nov 2002
• Date of issue
  June 2002

History

• Received
  23 Apr 2001