Electron g-Factor in Diluted Magnetic Semiconductor Quantum Well with Parabolic Potential in the Presence of Rashba Effect and Magnetic Field

1 Introduction

Spin-dependent phenomena have recently attracted considerable and continuous attention as they are the key concepts in modern spintronics. The spin-orbit coupling, which couples the electron spin and its orbital motion, has been the subject of several theoretical and experimental researches [1].

A spin field-effect transistor, proposed by Datta and Das [2], is based on the fact that spin precession can be controlled by an external field with the help of the spin-orbit interaction. In a crystal with bulk inversion asymmetry (BIA), the energy bands are spin splitted for a given direction of the wave vector k. In semiconductor heterostructures, the spin splitting may also occur [3] as a result of the structural inversion asymmetry (SIA), as it was discussed in an early paper by Rashba [4, 5].

In addition, the spin splitting of the subbands can be enhanced by introducing magnetic ions (Mn) in the quantum well, quantum wire, and quantum dot structures, for example, Hg_1-yMn_yTe diluted magnetic semiconductor (DMS) structures. DMS provides tailoring the spin splitting and the spin polarization with an interesting possibility due to strong s-d exchange interaction between the carriers and the local magnetic ions. The spin splitting in the DMS system can be tuned by changing the external magnetic field [6, 7].

DMSs are among the best candidates of combining semiconductor electronics with magnetism. The Landé g-factor in DMS is a function of the applied magnetic field, the temperature, and the molar fraction y [8]. For instance, for low field values g(H→ 0)=–0.5 in CdTe and g(H→ 0)=+100 in Cd_0.98Mn_0.02Te at helium temperature. Recently, the incorporation of Mn ions into the crystal matrix of different II-VI semiconductors has led to successful fabrication of DMS quantum dots and magnetic DMS hybrid structures [9–11].

Low dimensional DMS systems are special interests.

For most experimental and theoretical realizations and studies, quantum dots (QDs) can be described as effectively two-dimensional (2D) systems in a confining potential which is usually modelled as hard-wall or parabolic confinement [12]. In Gharaati and Khordad [13], effective electron g*-factor in the InAs quantum wire under an applied magnetic field and the Rashba effect were investigated.

In Hashimzade et al. [14], the electrical conductivity of an electron gas in parallel electric and magnetic fields directed along the plane of a parabolic quantum well across the profile of the potential was studied. General expression for the electrical conductivity being applicable for any magnitudes of the magnetic field and degeneration level of the electron gas, taking account of electronic scattering with spin flip was found.

In Mehdiyev et al. [15], the effect of the magnetic field, the Rashba SO interaction, the s-d exchange interaction, and finite temperature on the conductance of a DMS hollow cylindrical wire have been studied.

The spin transport properties of an n-type Hg_0.98Mn_0.02Te M2DEG have been investigated in Sanders et al. [16]. The giant spin–orbit splitting has been studied separately by changing temperature and varying the gate voltage, respectively.

2 Theory

In this paper we are presenting a theoretical study of the Landé g factor in a diluted magnetic semiconductor quasi-2D electron gas of a finite thickness with Rashba spin-orbital coupling and in plane magnetic field.

The Hamiltonian for quasi-2D gas of electrons moving under an external in plane magnetic field in the presence of Rashba spin-orbital interactions by imposing a parabolic confining potential in the transverse direction and the s-d exchange Heisenberg interaction between the conduction electrons and Mn ions, which can be taken into account by adding an appropriate exchange term is given by

(1)H = (p^+ecA→)22m +  mω022x2 + 12gμBHσz + αℏ[σ→ × (p^ + ecA→)]z + Hex (1)

where p^ = −iℏ∇→ is the impulse operator, e electron charge and A the magnetic vector potential. m and g are the effective mass and Landé factor for electrons,  σ→ is the vector of Pauli matrices, and α the Rashba parameter, which describes the strength of the spin-orbital coupling, An external magnetic field is chosen to be along z axis H→ = (0,0,H) under the gauge A→ = (0,Hx,0). The second term in Hamiltonian is the confining potential in x-direction, approximated as a parabola with a frequency ω₀, which is a characteristic parameter of the electron gas thickness. The third term in (1), Zeeman splitting energy,  μB = eℏ2m0, is the Bohr magneton of a free electron with mass m₀. The last term in (1) is the s-d exchange of Heisenberg interaction between the conduction electrons and Mn ions, which can be taken into account by adding an appropriate exchange term to the total Hamiltonian [6]

(2)Hex = ∑Ri→J(r→ − Ri→)Si→σ→ (2)

where Si→ and σ→ are spin operators of the i-th magnetic ion and electrons, J(r→ − Ri→) is electron-ion exchange integral, and vectors r→ and Ri→ define the coordinates of the electron and Mn ions. The sum runs over all Mn ions. We will use the mean-field approximation inserting the mean value of Mn spin in z direction 〈S_z〉 instead of the corresponding operator and ascribing spin x 〈S_z〉 to every crystal site, where x is the mole fraction of Mn. In this approximation, the exchange Hamiltonian can be rewritten in the form

(3)Hex = 12〈Sz〉N0Js-dσz = 3Aσz (3)

where N₀ is the density of unit cells and J_s-d is constant which describes the exchange interaction according to the s-d exchange integral.

The thermodynamic average 〈S_z〉 of the z component of a localized Mn spin in the approximation of non-interacting magnetic moments is determined by the empirical expression

(4)〈Sz〉 = −SB5/2(SgMnμBHkBT) (4)

where B5/2(SgMnμBHkBT) is the Brillouin function, S=5/2 corresponds to the spins of the localized electrons of Mn ions, ^g_Mn=2 is the g-factor of Mn, and k_B- is the Boltzmann constant.

In order to solve time independent Schrödinger equation with a spinor, one expresses the electron wave functions with spin-up f_↑(x) and spin-down f_↓(x) as

(5)Ψn(x,y,z) = (fn↑(x)fn↓(x))ei[kyy+kzz] (5)

which yields

(6)(d2dx2 − (ky + eHℏcx)2 − m2ω02ℏ2x2 − mℏ2gμBHσz + 2mEℏ2 − (kz)2 − 2mℏ23Aσz − 2mαℏ2(σx(ky + mωcℏx) + iσy(ddx))) (fn↑(x)fn↓(x)) = 0 (6)

Applying the change of variables

(7)ξ = mΩℏ(x − x0) (7)

where

(8)x0 = −ℏkyωcmωc2Ω2, Ω =  ω02 + ωc2, ωc = eHmc (8)

The wave function can be written as

(9)fn ↑(x) = β1e−ξ2/2Hn(ξ), fn ↓(x) = β2e−ξ2/2Hn + 1(ξ) (9)

where β₁ and β₂ are constants and H_n(ξ) is the Hermite polynomial and n is the quantum number. Substituting these functions into (6) and using the recurrence and normalization relations for Hermite polynoms, we found that the coefficients β₁ and β₂ satisfy the eigenvalue equation

(10)(ϵ1 − 2n − 1)β1 − 2(1 + n)RℏΩ(ωcΩ + 1)β2 = 0 (10)

(11)−β1RℏΩ(ωcΩ + 1) + (ϵ2 − 1 − 2n − 2)β2 = 0 (11)

where

(12)ε1 = −ℏmΩky2ω02Ω2 − 1ℏΩg(E)μBH − 6AℏΩ + 2EnℏΩ − ℏmΩ(kz)2 (12)

(13)ε2 = −ℏmΩky2ω02Ω2 + 1ℏΩg(E)μBH + 6AℏΩ + 2EnℏΩ − ℏmΩ(kz)2 (13)

(14)R = mα2ℏ2 (14)

Imposing the condition of nontrivial solutions on this set of equations, (10) and (11) yield the eigenvalues given for spin-up states

(15)En+ = ℏ22mky2ω02Ω2 + ℏ22m(kz)2 + (1 + n)ℏΩ − ℏΩ2(−1 + gμBHℏΩ + 6AℏΩ)2 + 2(1 + n)RℏΩ(1 + ωcΩ)2 (15)

If we use the wave function

(16)fn ↑(x) = β1e−ξ2/2Hn − 1(ξ), fn ↓(x) = β2e−ξ2/2Hn(ξ) (16)

We obtain for spin-down states

(17)En− = ℏ22mky2ω02Ω2 + ℏ22m(kz)2 + nℏΩ + ℏΩ2(−1 + gμBHℏΩ + 6AℏΩ)2 + 2nRℏΩ(1 + ωcΩ)2 (17)

Equations (15) and (17) can be rewrite as follows:

(18)En± = ℏ22mky2ω02Ω2 + ℏ22m(kz)2 + (12 ± 12 + n)ℏΩ∓ ℏΩ2 (−1 + gμBHℏΩ + 6AℏΩ)2 + 2(12 ± 12 + n)RℏΩ(1 + ωcΩ)2 (18)

The total spin splitting energy separation δ can be expressed as:

(19)δ = En+ − En− = ℏΩ − 12(−ℏΩ + gμBH + 6A)2 + 2(1 + n)RℏΩ(1 + ωcΩ)2−12(−ℏΩ + gμBH + 6A)2 + 2(n)RℏΩ(1 + ωcΩ)2 (19)

For high Landau numbers n the total spin splitting energy separation, one arrives at a good approximation [16]:

(20)δ = ℏΩ − (−ℏΩ + gμBH + 6A)2+Δ2 (20)

Δ is the splitting at B=0 and ω₀=0. The effective electron g*-factor can be obtained by the following expression

(21)g∗ = δμBH (21)

3 Results and Discussion

In this section, we present our numerical results on the effective g*-factor of the electrons in a Hg_1-yMn_yTe DMS quantum well with parabolic confinement. We use the following set of bulk parameters: m_e=0.047m₀, where m₀ is the free electron mass. Other parameters are also used in our calculations: In Liu et al. [18], Figure 3 indicated Δ=12.4 meV and N₀J_s-d=0.4 eV mentioned in Yang et al. [19] and g=–20 (is the g-factor of the band electrons without exchange term) are taken from Gui et al. [20].

In Figure 1 we plot the effective g*-factor as a function of Mn concentration. (Figure 1 was plotted with (19) and (20) calculated δ values. In Figure 1 red line shows the calculated values in (20). The value of the magnetic field is fixed at H=1T and α=160 meV·nm. By looking at this figure we notice that the effective g*-factor decreases with increasing Mn concentration. This characteristic behaviour arises from the fact that the s-d exchange interaction energy is directly proportional to Mn concentration.

Figure 1

The electron effective g*-factor as a function of Mn concentration.

In Figure 2 the variation of the effective g*-factor of electrons with the Rashba spin-orbital interaction parameter α, for H=1T and y=0 for the lowest Landau level is shown n=1. By increasing the strength of the SOC, the effective g*-factor decreases.

Figure 2

The electron effective g*-factor as a function of Rashba spin-orbital interaction parameter α.

In Figure 3, we plot the effective g*-factor of electrons versus the temperature. The temperature dependence of the exchange interaction is determined by the average spin of the magnetic ion 〈S_z〉, which shows the behaviour of the Brillouin function. By increasing the temperature, the electron effective g*-factor increases.

Figure 3

The electron effective g*-factor as a function of the temperature.

The magnetic field dependence of the effective g*-factor shown in Figure 4. In Figure 4 an increase of the effective g* factor with the strength of the applied magnetic field is observed.

Figure 4

The electron effective g*-factor as a funciton of magnetic field.

Figure 5 shows the effective g-factor in DMSs quantum well as a function of confinement length l0 = ℏmω0 for α=160 meV·nm. The magnetic field was taken as H=1T. From Figure 5 we observed that the effective g-factor decreases as the confinement length increases.

Figure 5

The electron effective g*-factor as a function of confinement length l0 = ℏmω0.

4 Conclusion

In summary, the effect of the magnetic field, the Rashba SO interaction, the s-d exchange interaction, and finite temperature on the effective g*-factor of a DMS with parabolic confinement quantum well have been studied. We investigated the electron effective g*-factor as a function of spin-orbital coupling parameter in magnetic fields, as well as the magnetic field and temperature dependence of Landé factor for fixed Mn concentrations and Rashba parameter.