Shubnikov-de Haas oscillations in Digital Magnetic Heterostructures

Shubnikov-de Haas oscillations in Digital Magnetic Heterostructures

Henrique J. P. Freire; J. Carlos Egues

Departamento de Física e Informática, Instituto de Física de São Carlos, Universidade de São Paulo, C.P.369, 13560-970 São Carlos-SP, Brazil

ABSTRACT

In this paper we theoretically investigate the magnetic-field and temperature dependence of the Shubnikov-de Haas oscillations in group II-VI modulation-doped Digital Magnetic Heterostructures. We self-consistently solve the effective-mass Schrödinger equation within the Hartree approximation and calculate the electronic structure and the magneto-transport properties. Our results show i) a shift of the Shubnikov-de Haas minima to lower magnetic fields with increasing temperature, and ii) an anomalous oscillation which develops when two opposite Landau levels cross near the Fermi energy. Both of these are consistent with recent magneto-transport measurements in such heterostructures.

1 Introduction

Digital Magnetic Heterostructures (DMHs) are semiconductor heterostructures where magnetic monolayers are incorporated using the digital-alloy technique [1, 2]. The s-d exchange interaction in such systems is responsible for a magnetic-field- and temperature-dependent giant spin splitting of the electronic bands [3], thus giving rise to strong spin-dependent effects. This giant splitting is up to two orders of magnitude larger than the ordinary Zeeman effect and is easily observed by magneto-photoluminescence [4]. The successful achievement of high doping carrier densities in such magnetic layered structures [5] has enabled magneto-transport and magneto-photoluminescence measurements in these systems [1, 5]. More recently, the s-d exchange enhanced spin-splitting has been used to align opposite spin Landau levels near the Fermi level [6, 7], revealing striking features in the electronic structure and magneto-transport properties of such two-dimensional electron systems.

In Ref. 6 magnetization and magneto-transport measurements in n-ZnSe/(Zn,Cd,Mn)Se DMHs were performed in the quantum Hall regime for a wide range of temperatures and carrier densities. In this paper we focus our analysis on two particular features of the experimental results, namely i) a shift of the Shubnikov-de Haas (SdH) peaks to lower magnetic fields as the temperature is raised and ii) an anomalous peak in the r_xx data near B ~ 3.2 T. We self-consistently solve the effective-mass Schrödinger equation within the Hartree approximation. Our results reproduce the reported temperature dependence of the SdH peaks and the anomalous oscillation. In the following sections we present our model and results.

2 Model and Approach

The system we study consists of a modulation doped digital magnetic quantum well [6]. One 1/16 monolayer of MnSe is inserted every 7 monolayers in a 105 Å ( ~ 35 monolayers) wide Zn_0.87Cd_0.13Se quantum well. The well is surrounded by two 120 Å intrinsic ZnSe spacers, each followed by a 200 Å n-doped ZnSe layer. The measured low-field two-dimensional density is n_s = 2.8 × 10¹¹ cm^–2. The strong Coulomb repulsion between confined electrons, which deforms the potential profile, is treated within the Hartree approximation. We self-consistently solve the effective-mass Schrödinger equation

where m = 0.145m₀ is the electron effective mass, i = 1,2,¼ is the subband index, and s_z = ±1 denotes the spin components. The effective potential is

where v₀(z) is the the quantum well profile with band offset of 210 meV [8]; v_b(z) is the Mn barrier profile with an assumed height of 800x_p meV [8], x_p being the nominal Mn concentration; ( z;B,T) is the s-d exchange contribution given by [10]

where N₀a = 0.26 eV is the s-d exchange constant, x(z) is an effective Mn concentration profile [11, 12], B_5/2 is the spin-5/2 Brillouin function, and T + T₀ is an effective temperature; v_h(z) is the Hartree potential which is calculated by solving the Poisson equation. Each Landau level energy is then given by

where n is the Landau level index,

w_c is the cyclotron energy, and g^* = 0.4 [13] is the effective Landé factor of the Zeeman term.

We use gaussian broadened Landau levels with a density of states (DOS) for each spin subband given by

where G is the broadening width. Using this DOS together with the Fermi function f(e) = {1 + exp[(e – m)/k_BT]}^–1 we self-consistently calculate the chemical potential by solving the 2D density

for m.

We calculate the longitudinal conductivity using [14] –xx

where is the width of the extended states region within each G-broadened Landau level. The transversal conductivity is given by the Drude model s_xy = –en_s/B. To calculate the resistivities we simply invert the conductivity tensor r = s^–1.

3 Results

The electronic structure and magneto-transport calculations were performed at four different temperatures T. We use x_p = 0.093 and n_s = 2.8 × 10¹¹ cm^–2 (the low field measured areal density [6]) to obtain a good agreement with the experimental findings. For the other parameters we use T₀ = 1.55 K, G = 0.36B^1/2 meVT^–1/2, = 0.25 meV, and = 0.05 meV, the same values as in Ref. 6.

Figure 1 shows the theoretical magnetic field and temperature dependencies of r_xx. In accordance with the experimental results [6], our calculated SdH oscillations i) shift to lower magnetic fields as the temperature increases, and ii) show an anomalous peak at B_c ~ 3.2 T.

The temperature dependence of the Mn magnetization [Brillouin function in Eq. (3)] is responsible for the SdH shift, as pointed out in Ref. 6. As the temperature rises, the s-d enhanced spin splitting decreases and the spin down energy increases (Fig. 2), crossing the Fermi energy at a lower B field. Note that temperature also plays a role via the Fermi distribution, which smooths the SdH oscillations as T increases (Fig. 1). However, its effect on the SdH shift within the investigated temperature range is small when compared to the s-d exchange contribution.

The anomalous peak that develops at B_c ~ 3.2 T (Fig. 1, T = 0.36 K) is a result of a Landau level crossing near the Fermi energy [Fig. 2(a)]. At T = 0.36 K, just below B_c, a SdH oscillation takes place while the Fermi energy crosses the |3, ñ level. Meanwhile the |0, ñ is increasing in energy and becomes degenerate with |3, ñ . As the magnetic field increases, the Fermi energy crosses the spin up level just before it becomes completely empty, resulting in the small SdH peak. This effect is suppressed for higher temperatures as the Fermi function increases the overlap between levels thus making them indistinguishable and giving rise to the wider peaks at T = 2.5 K and T = 4.2 K in Fig. 1.

4 Discussions and Conclusions

We have done a self-consistent calculation of the electronic structure and magneto-transport properties of a modulation-doped DMH. Our results not only reproduce the experimental findings, but also justify the phenomenological model used in Ref. 6. Since the quantum well is very deep, the main effect of the Hartree contribution is to increase the subband energy by ~ 57 meV, leaving the subband structure almost unchanged.

Note that the value of the fitting parameter x_p = 0.093 is greater than the nominal one x_p = 0.0625. A possible explanation for this discrepancy is that we have neglected the contribution of the exchange-correlation energy of the 2D electron gas. Its inclusion may enhance the spin splitting [15] thus allowing the use of a smaller value for x_p, closer to the experimental one [16].

It is worth mentioning that neither the experimental nor our theoretical results show any sign of an itinerant ferromagnetic phase transition [17] at the Landau level crossings. Reference 7 suggests that the effects observed by Knobel et al. [6] take place in a regime above a critical temperature T_c. Here we note, in addition, that B_c = 3.2 T corresponds to a filling factor of 3.6, which deviates from the integer filling factor favoring quantum Hall ferromagnetism [18]. The inclusion of exchange and correlation effects should allow us to study the possibility of quantum Hall ferromagnetic states in these magnetic heterostructures [16].

Acknowledgments

This work was supported by FAPESP (Brazil). HJPF acknowledges Klaus Capelle for useful discussions and R. Knobel for providing additional details on the samples used in Ref.[6].

References

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[12] Since x_p is very small the effective Mn concentration is rather insensitive to the actual profile so that x = x_p(1 – x_p)⁴ is a good first approximation to it. We actually use a segregation profile with characteristic length of 5Å [11]. A diffusion profile gives similar results.

[13] R. G. Knobel, Ph.D. thesis, Pennsylvania State University, 2000.

[14] T. Ando and Y. Uemura, J. Phys. Soc. Japan 36, 959 (1974).

[15] H. J. P. Freire and J. C. Egues, Braz. J. Phys. 32, 327 (2002); 33, 166(E) (2003).

[16] A similar calculation including exchange and correlation effects will be presented elsewhere.

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[18] T. Jungwirth and A. H. MacDonald, Phys. Rev. B 63, 035305 (2000).

Received on 8 March, 2003

Publication Dates

• Publication in this collection
  31 Aug 2004
• Date of issue
  June 2004

History

• Received
  08 Mar 2003