Electronic states of magnetic quantum dots

Ramin M Abolfath^1,2,3, Pawel Hawrylak² and Igor Žutić¹

¹ Department of Physics, State University of New York at Buffalo, Buffalo, NY 14260, USA
² Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, K1A 0R6, Canada
³ Department of Radiation Oncology, University of Texas Southwestern Medical Center, Dallas, TX 75390, USA

E-mail: ramin@babylon.phy.nrc.ca

Received 15 June 2007
Published 28 September 2007

Recent experimental [1]–[9] and theoretical [10]–[15] studies reveal that semiconductor quantum dots (QDs) [16, 17] are desirable for tailoring magnetism which persists at much high temperatures than in their bulk counterparts [18]. These nanoscale realizations of dilute magnetic semiconductors (DMS) [19] may be suitable for a versatile control of spin and magnetism with potential applications in information storage technology and spintronics [20]. An important property of the carrier-mediated ferromagnetism in bulk-like DMS is the optical and electrical control of the magnetic ordering by changing the number of carriers [21]–[23]. However, because of the strong interplay between quantum confinement and the many-body electron–electron (e–e) Coulomb interactions in QDs, it may be possible to achieve and control the magnetic ordering of carrier spin and magnetic impurities, with a fixed number of carriers [15].

In this paper, we present a theoretical analysis to describe the quantum states of the electrons confined in II–VI quantum dots doped with Mn. In (II,Mn)VI materials, Mn does not change the number of electrons (N) because it is isoelectronic with group-II elements. In QDs, however, additional carriers are controlled by either chemical doping or by an external electrostatic potential applied to the metallic gates. Even for a small number of interacting electrons, the study of QDs doped with large numbers of Mn ( > 10) becomes computationally inaccessible to exact diagaonalization techniques [14, 15]. For this reason, we employ here a mean field theory of magnetic QDs in which Mn acts like an external magnetic field and the weak coupling between electrons and magnetic impurities can be described as spin–spin exchange interaction [15]. Because electrons are confined in all three dimensions in nanometre-sized QDs, we anticipate strong e–e Coulomb interaction. We use a finite temperature local spin density approximation (LSDA) [24] to incorporate e–e Coulomb interaction in this work.

The wavefunction of electrons in QDs can be expanded in terms of its planar with and subband ξ(z) wavefunction. The effective two-dimensional (2D) Kohn–Sham (KS) equations , in LSDA can be obtained by integrating ξ(z) [25], assuming that the first subband is filled. Here ε_iσ are the KS eigenenergies, σ = + 1 and –1 for spin up (↑) and down (↓) and H is the KS Hamiltonian which can be expressed as

Here is the Planck constant, m* is the electron effective mass, and V_QD = V₀ exp(–ρ²/δ²) is the 2D Gaussian confining potential of the quantum dot (1D parabolic confinement is chosen along the z-axis). In equation (1), V_H and V^σ_XC are Hartree and spin dependent exchange-correlation potentials [24], while γ accounts for reduction of Coulomb strength due to screening effects of the gate electrodes [26], and

J_em = J_sdn_mM is the e–Mn exchange coupling, J_sd is the exchange coupling between electron and single Mn, n_m is the spatial-averaged density of Mn, and M = 5/2 is the spin of Mn. B_M(x) is the Brillouin function [27] in which the argument x contains the Boltzmann constant k_B, T the absolute temperature, and

the effective field seen by the Mn [15]. The first term in describes the mean field of the direct Mn–Mn antiferromagnetic coupling [12], with

and

is the spin-resolved electron density, f(ε) = 1/{exp[(ε–μ)/k_BT] + 1} is the Fermi function, and μ is the temperature-dependent chemical potential, and is the self-consistent solution of which ensures that N = N_↑ + N_↓ is an integer number.

To obtain solutions of equations (1)–(5), we consider (Cd, Mn)Te QD with V₀ = –128 meV, and δ = 15.9 nm, a perpendicular width of 1 nm, J_sd = 0.015 eV nm³, n_m = 0, 0.025 and 0.1 nm^–3, corresponding to the approximate number of Mn, N_Mn = 0, 45, 180, J_em = 0, 0.94, 3.75 meV, and J^AF_eff = 0, 0.005, 0.02 meV, respectively. For CdTe, considered here, a*_B = 5.29 nm and Ry* = 12.8 meV are the effective Bohr radius and Rydberg energy, while the additional material parameters are m* = 0.106 and ε = 10.6 [15].

The shell structure of the 2D parabolic (a) and the 2D Gaussian (b) potentials are shown in figure 1. The corresponding confining potentials are shown in figure 1(c) where the dashed and bold lines represent the 2D parabolic and 2D Gaussian potentials. The distinction between the parabolic and Gaussian potentials can be experimentally resolved by far infra red (FIR) spectroscopy or transport measurements of single particle states in external magnetic field ([16, 17] and references therein). The energy-gap between s-, p- and d-orbitals is characterized by ω₀. For a 2D Gaussian potential, the characteristic energy associated with the lateral confinement, ω₀, is calculated by expanding V_QD in the vicinity of the minimum which yields , with the strength . Because of the circular symmetry of the Gaussian potential, the z-component of the angular momentum l_z is a good quantum number. As a result of the geometrical symmetry of the confining potential, the states in a p-shell with l_z = ±1 are degenerate. In contrast to the 2D parabolic potential ([16, 17] and references therein), where the dynamical symmetry of the confining potential implies occurrence of accidental degeneracies of the single particle levels with different |l_z|, the single particle levels in a d-shell are not completely degenerate. Degenerate levels d ₊  and d_– are separated by an energy-gap (1.5 meV) from the d₀-level, where the indices ±, 0 refer to angular momentum l_z = ±1, 0. We note that the ordering of the single particle s-, p- and d-eigenenergies in the Gaussian potential are similar to the ordering of the first few single particle eigenenergies in a square potential.

In the absence of direct antiferromagnetic coupling between Mn and e–e Coulomb interaction, an identification of the magnetic states of the QDs based on the present mean field theory can be expressed in terms of two energy scales: the single particle energy-gap Δ, and the effective e–Mn exchange coupling J_em. The former is the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). It is a function of the number and spin of electrons and the geometrical and dynamical symmetries of the QD confining potential. Therefore, Δ, which depends on the quantum confinement, can be tuned and controlled by the electric voltage applied to the metallic gates. On the other hand, J_em, the strength of the effective Zeeman energy, , which polarizes the spin of electrons, is independent of the quantum confinement. Thus the magnetic phase diagram of the QDs can be effectively described in terms of Δ and J_em. In the absence of magnetic field we can then distinguish between spin-polarized and spin-unpolarized states in QDs, conventionally also referred to as the ferromagnetic (FM) and antiferromagnetic (AFM) states [12]–[14]. For example, the ground state of the electrons exhibits FM ordering if J_em > Δ. The dependence of the single particle eigenenergies of the d-shell with N = 8 on h_sd is shown in figure 2. At h_sd = 0 the valence electrons with spin up and down occupy single particle level d₀ following the Pauli exclusion principle, and the ground state of N = 8 exhibits AFM ordering with N_↑ = N_↓ = 4. With increasing h_sd the gap between spin up and down opens up. At h_sd = Δ = 1.5 meV, where Δ = E_d + (d–)–E_d0, the first spin flip occurs. Because of well separated p- and d-orbitals the second spin flip occurs at much higher energies as h_sd = ω₀≈7J_em.

In LSDA, electrons are replaced by independent quasi-particles (electrons dressed by e–e interaction), where KS eigenenergies are the energy levels of the quasi-particles. In this picture, e–e interaction rearranges the structure of quantized levels and affects the shell structure of the QDs. In this limit, the quasi-particle gap Δ* (renormalized Δ) is the energy difference between HOMO and LUMO of the KS eigenenergies. It is an appropriate parameter, which can supersede the single particle gap Δ in identification of the magnetic states of the QDs.

The KS levels in the d-shell of the 2D Gaussian potential with n_m = 0 and N = 8, 9 and 10 are shown in figure 3(a). Because of circular symmetry of the confining potential, the energy levels of d ₊  and d_– are degenerate. The energy of d₀ which was lower than d ₊  and d_– due to the nonparabolicity of the Gaussian potential, increases by e–e Coulomb interaction, such that it becomes the highest energy level in the d-shell, and Δ* changes sign. In contrast to noninteracting electrons, the ordering of the KS levels in s-, p- and d-shells is closer to the one in circular potential than in the square potential. In the absence of magnetic impurities, as it is shown in figure 3(a), N = 8 forms a half-filled shell with s_z = 1, and the electron polarization P = 2/8, and N = 9 shows an open shell with s_z = 1/2, and P = 1/9. Because of d-shell overturning, N = 10 forms a closed shell with s_z = 0 and P = 0/10. Here the energy difference between the KS eigenenergies of the electrons with spin up and down is zero, and Δ*≈0.6 meV ( < J_em). With increasing e–Mn coupling to J_em = 3.75 meV, corresponding to 5% doping and within a range of spin polarization, only in the d-shell, we find no magnetic transition for N = 8, since it was already fully polarized due to the spin Hund's rule. For N = 9 and N = 10 we find transitions to s = 3/2 and s = 1, respectively. The evolution of quantized energies as a function of effective Zeeman energy and Δ* is shown in figure 4. For example Δ* < 0 corresponds to d-levels of the 2D Gaussian potential with γ = 1, whereas Δ* = 0 (b) and Δ* > 0 (c) correspond to d-levels in the 2D parabolic and 2D Gaussian potentials with γ = 0 (Δ* = Δ).

We next turn to calculating the spatially-averaged Mn-magnetization per unit area A, for the 2D parabolic and 2D Gaussian potentials as a function of N for zero (γ = 0), and full (γ = 1) e–e Coulomb interaction, shown in figure 5. This calculation illustrates how deviation from nonparabolicity of the confining potential may lead to changes in the magnetic properties of QDs. Many qualitative features indeed coincide for both of the confinements.

Figure 5. The averaged magnetization per unit area Mz as a function of number of electrons N at T = 1 K and Mn-density nm = 0.1 nm–3 (a), and nm = 0.025 nm–3 (b) for non-interacting (γ = 0, empty triangles) and interacting (γ = 1, filled triangles) electrons. At N = 8, nm = 0.025 nm–3, in a 2D Gaussian confining potential the ground state of the QD switches between FM and AFM states by e–e Coulomb interaction.

A comparison in figure 5(a) for n_m = 0.1 nm^–3 confirms that there is similarity in magnetization for both Gaussian and parabolic confinements. Here, J_em > Δ*(Δ) for γ = 1(0), and we consider all the electron numbers up to a full d-shell (N≤12). However, as we show in figure 5(b), a nonparabolic confinement can introduce additional magnetic transitions in QDs. Here n_m = 0.025 nm^–3 corresponds to magnetic doping of 1.25%. At N = 8 and γ = 0 in the 2D Gaussian potential, we find J_em < Δ and a transition to the AFM state. In contrast, at γ = 1, we find J_em > Δ*, and thus the FM state is stable.

Our findings illustrate the importance of Coulomb interactions and quantum confinement for the magnetic ordering of carrier spin and magnetic impurities in (II,Mn)VI QDs. We show that with electrostatic control of nonparabolicity of the confining potential, it is possible to control the magnetization of QDs even with a fixed number of electrons. While the choice of 2D Gaussian confinement is a convenient form to parameterize nonparabolic effects, other deviations from the parabolic confinement could also influence the magnetic ordering in QDs [28].

Acknowledgments

This work is supported by the US ONR, NSF-ECCS CAREER, the NRC HPC project, CIAR, the CCR at SUNY Buffalo and the Center for Nanophase Materials Sciences, sponsored at ORNL by the Division of Scientific User Facilities, US DOE.

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