Solid State Communications 143 (2007) 395–398www.elsevier.com/locate/ssc

Self-consistent solutions of the Curie temperature inferromagnetic/ferroelectric hybrid double quantum wells

S.H. Parka,∗, S.W. Ryua, J.J. Kima, W.P. Honga, H.M. Kima, N. Kimb, J.W. Kimb, T.W. Kangb,S.J. Leeb, S.N. Yic

a Department of Electronics Engineering, Catholic University of Daegu, Hayang, Kyeongsan, Kyeongbuk 712-702, South Koreab Quantum-functional Semiconductor Research Center, Dongguk University, 26 Pil-Dong-3Ga, Joong-Gu, Seoul 100-715, South Korea

c Department of Semiconductor Physics, Korea Maritime University, Busan 606-791, South Korea

Received 25 May 2007; accepted 16 June 2007 by H. AkaiAvailable online 22 June 2007

Abstract

The ferromagnetic transition temperature as a function of the electric field for a ferromagnetic/ferroelectric hybrid double quantum well(HDQW) system is investigated by using a self-consistent method. These results are also compared with those obtained from the flat-band model.The self-consistent solution shows that, the change in Curie temperature for the dipole right and the dipole left occurs near Fg = 17 and 10meV/nm, respectively. This effect is caused by asymmetry of electrostatic potential due to screening charges. On the other hand, the results ofthe flat-band model shows that the change in Curie temperature occurs at lower bias voltage than that of the self-consistent model. In the case ofthe flat-band model, the ratio of ferromagnetic transition temperatures Tc/Tco has a constant value and is independent of Fd . On the other hand,Tc/Tco of the self-consistent calculation shows a significant dependence on the Fd value.c© 2007 Elsevier Ltd. All rights reserved.

PACS: 85.60.Bt; 85.30.De; 85.30.Vw; 78.20.Bh

Keywords: A. Ferromagnetic; A. Ferroelectric; A. Hybrid double quantum well; D. Curie temperature

1. Introduction

Since the diluted magnetic semiconductor (DMS) has beengenerally known as one of the most promising candidates forspintronic device materials, many experimental and theoreticalstudies have been performed on various DMS materials andstructures. Because the spintronic devices should ultimately beoperated at room temperature, much effort has been focused onincreasing the ferromagnetic transition temperature Tc of DMSabove room temperature. Among many materials, ZnMnO isconsidered to have Tc above 300 K with 5% Mn per unit celland 3×1020/cm3 holes according to a theoretical prediction [1].Recently, observation of the ferroelectric properties wasreported in Li-doped ZnO bulk samples. The reason for theferroelectric property is attributed to the following: when the

∗ Corresponding author.E-mail addresses: shpark@cu.ac.kr (S.H. Park), leesj@dongguk.edu

(S.J. Lee).

0038-1098/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2007.06.016

size of the dopant Li atom (0.6 A) is smaller than the hostZn atom (0.74 A), then the Li atoms can occupy off-centrepositions, thus locally inducing electric dipoles, thereby leadingto ferroelectric behaviour [2].

On the theoretical side, a ferromagnetic/ferroelectric hybriddouble quantum well (HDQW) system using ZnMnO andZnLiO layers has been studied as a new spintronic device [3].Here, the ferromagnetic transition temperature of the HDQWsystem was investigated as a function of the applied electricfield considering spontaneous electric polarization in the ZnLiOwell and the flat-band model was used assuming a regimeof low carrier density of 1 × 1011/cm2 in the wells. On theother hand, there has been little work on the carrier densitydependence of the ferromagnetic transition temperature in aHDQW system. In particular, in the case of the high carrierdensity, the self-consistent treatment is expected to be veryimportant to obtain exact solutions for the wave functions.

In this research, we investigate the ferromagnetic transitiontemperature as a function of the electric field for a HDQW

396 S.H. Park et al. / Solid State Communications 143 (2007) 395–398

system using a self-consistent method. The band structuresand wave functions are obtained by solving the Schrodingerequation for electrons and the 3×3 Hamiltonian for holes [4,5].These results are also compared with those obtained fromthe flat-band model. The HDQW system has a structureof MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO andlower LiZnO wells and MgZnO barriers. We assume that theelectron and hole are simultaneously injected into the wells.The ZnO well contains an additional Mn-doped p-type layerat the middle of the well or at the upper edge of the well.The lower well LiZnO is the p-type ferroelectric well withspontaneous polarization. The dimension of the structure arechosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and the cappinglayer is 20 nm. The confinement potential is V0 = 263 meVwith 20% of Mg per unit cell in ZnMgO barriers.

2. Theory

The self-consistent band structures and wave functions areobtained by iteratively solving the Schrodinger equation forelectrons, the block-diagonalized Hamiltonian for holes, andPoisson’s equation [4,5]. The total potential profiles for theelectrons and the holes are

Vc(z) = Vcw(z) − |e|φ(z),

Vv(z) = Vvw(z) − |e|φ(z),(1)

where Vcw(z) and Vvw(z) are the square-well potentials for theconduction band and the valence band, respectively, and φ(z)is the screening potential induced by the charged carriers andsatisfies the Poisson equation

ddz

(ε(z)

ddz

)φ(z) = −|e|[p(z) − n(z)], (2)

where ε(z) is the dielectric constant. The electron and the holeconcentrations, p(z) and n(z), are related to the wave functionsof the n-th conduction subband and the m-th valence subbandby

n(z) =kT me

π h2

∑n

| fn(z)|2 ln(1 + e[E f c−Ecn(0)]/kT ) (3)

and

p(z) =

∑σ=U,L

∑m

∫dk‖

k‖

2π

∑v

|gσ(ν)mk‖

(z)|2

×

(1

1 + e[E f v−Evm (k‖)]/kT

), (4)

where me is the effective mass of electrons, h is Planck’sconstant divided by 2π , n and m are the quantized subbandindices for the conduction and the valence bands, E f c andE f v are the quasi-Fermi levels of the electrons and theholes, respectively, Ecn(0) is the quantized energy level of theelectrons, Evm(k‖) is the energy for the m-th subband in thevalence band, σ denotes the upper (U) and the lower (L) blocksof the Hamiltonian, k‖ is the in-plane wave vector, ν refers tothe new bases for the Hamiltonian, and fn(z) and gσ(ν)

mk‖(z) are

the envelope functions in the conduction and the valence bands,respectively. The potential φ(z) is obtained by integration [6]:

φ(z) = −

∫ z

−L/2E(z′)dz′, (5)

where

E(z) =

∫ z

−L/2

1ε(z)

ρ(z′)dz′. (6)

The procedures for the SC calculations consist of the followingsteps: (i) Start with the potential profiles Vc and Vv withφ(0)(z) = 0 in Eq. (1); (ii) solve the Schrodinger equation(for electrons) and the block-diagonalized Hamiltonian (forholes) with the potential profiles φ(n−1)(z) in step (i) to obtainband structures and wave functions; (iii) for a given carrierdensity, obtain the Fermi-energies by using the band structuresand the charge distribution by using the wave functions; (iv)solve Poisson’s equation to find φ(n)(z); (v) check if φ(n)(z)converges to φ(n−1)(z). If not, set φ(n)(z) = wφ(n)(z) + (1 −

w)φ(n−1)(z), n = n + 1; then, return to step (ii). If yes, theband structures and the wave functions obtained with φ(n−1)(z)are solutions. An adjustable parameter w (0 < w < 1) istypically set to 0.5 at low carrier densities. With increasingcarrier densities, a smaller value of w is needed for rapidconvergence.

We write Tc in the form [7,8]

Ts =S(S + 1)J 2

pd

12kB

m∗

π h2

∫dz|g(z)|4c(z), (7)

where c(z) is a magnetic ion distribution function, Jpd is theexchange integral of carrier–spin exchange interaction, and Sis a Mn ion spin. We calculate the change in the fourth powerof growth direction envelope functions, |g(z)|4, of carriers atthe lowest energy subband in the HDQW as a function of theapplied electric fields.

3. Results and discussion

Fig. 1 shows the HDQW system having a structure ofMgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO andlower LiZnO wells and MgZnO barriers. We assume that theelectron and the hole are simultaneously injected into the wells.The ZnO well contains an additional Mn δ-doped p-type layerof a 0.5 monolayer at the distance of 2 nm toward the well fromthe interface between the ZnO well and the MgZnO barrier.The lower well LiZnO is the p-type ferroelectric well withspontaneous polarization. The dimension of the structure arechosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and thecapping layer is 7 nm. In numerical calculations, we choosephysical parameters for p-type ZnO/MgxZn1−xO/LiyZn1−yOwith x = 0.2 and y = 0.05. The parameters for Zn1−xMgxOare obtained from linear combinations of the parameters forZnO and MgO [9]. However, many of the material parametersfor MgO are not well known, so we assumed the parametersto be equal to those of ZnO as a first approximation in thecase of a lack of published data because the Mg composition

S.H. Park et al. / Solid State Communications 143 (2007) 395–398 397

Fig. 1. HDQW system having a structure of MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO and lower LiZnO wells and MgZnO barriers.

in the ZnMgO barrier is relatively small (x = 0.2). Also, theparameters for Zn1−yLiyO are assumed to be equal to those ofZnO because of a small composition of Li. We used 65/35 asthe ratio between the conduction- and the valence-band offsets(1Ec/1Ev) in the ZnO/ZnMgO heterostructure.

Fig. 2 shows the potential profile and the wave functions(HH1) at zone centre (k‖) for dipole down (left) and dipoleup (right) obtained by a self-consistent calculation (solid lines).For comparison, we plotted those (dashed lines) obtained bythe flat-band model. Here, the depolarizing field is Fd =

15 meV/nm and the electrostatic potential at the interface dueto screening charges is assumed to be V1 = 0.02 eV. The self-consistent solutions are obtained at the sheet carrier density ofN2D = 1 × 1012/cm2. In the case of a low bias voltage Fg ,the wave functions are shown to be located on the right sidewell. On the other hand, in the case of a high bias voltageFg = 2 meV/nm, the wave functions of the self-consistentsolution for both dipole down (left) and dipole up (right) casesare located in the right side while those of the flat-band modelare located on the left side. This is because of the band-bendingeffect due to the free-carrier screening, which makes the thebottom in the potential well to exist on the right side.

Fig. 3 shows the dependence of the ratio of ferromagnetictransition temperatures Tc/Tco on bias voltage Fg appliedacross the Mn-doped HDQW for dipole down (left) and dipoleup (right) obtained by a self-consistent calculation (solid lines).Here, Tco is Tc for the flat-band model at a bias voltage Fg =

10 meV/nm with V1 = 0.02 eV and Fd = 15 meV/nm.The self-consistent solutions are obtained at the sheet carrierdensity of N2D = 1 × 1012/cm2. For comparison, we plottedthose (dashed lines) obtained by the flat-band model. The self-consistent solution shows that, in the case of the dipole right, thechange in Curie temperature occurs near Fg = 17 meV/nm.Also, by reversing the direction of spontaneous polarization,the change in Curie temperature is shown to occur near Fg =

10 meV/nm. This effect is caused by asymmetry of electrostaticpotential due to screening charges. On the other hand, theresults of the flat-band model shows that the change in Curietemperature occurs at lower bias voltage than that of the self-

Fig. 2. Potential profile and the wave functions (HH1) at zone centre (k‖) fordipole down (left) and dipole up (right) obtained by a self-consistent calculation(solid lines). For comparison, we plotted those (dashed lines) obtained by theflat-band model.

Fig. 3. Dependence of the ratio of ferromagnetic transition temperaturesTc/Tco on bias voltage Fg applied across the Mn-doped HDQW for dipoledown (left) and dipole up (right) obtained by a self-consistent calculation(solid lines). Here, Tco is Tc for the flat-band model at a bias voltage Fg =

10 meV/nm with V1 = 0.02 eV and Fd = 15 meV/nm.

consistent model. Therefore, it is important to consider the self-consistent treatment to obtain more exact result at higher carrierdensity.

Fig. 4 shows the dependence of the ratio of ferromagnetictransition temperatures Tc/Tco on the depolarizing field Fd ofthe Mn-doped HDQW for dipole down (left) and dipole up(right) obtained by a self-consistent calculation (solid lines)with the applied voltage Fg = −10 meV/nm across the sample.The self-consistent solutions are obtained at the sheet carrierdensity of N2D = 1 × 1012/cm2. For comparison, we plottedthose (dashed lines) obtained by the flat-band model. In the caseof the flat-band model, we know that the ratio of ferromagnetictransition temperatures Tc/Tco has a constant value and isindependent of Fd in an investigated range of the Fd value. Thisis because, with the applied voltage Fg = −10 meV/nm, thewave functions for the flat-band model are located in the leftside for both dipole down (left) and dipole up (right) cases.However, Tc/Tco of the self-consistent calculation shows a

398 S.H. Park et al. / Solid State Communications 143 (2007) 395–398

Fig. 4. Dependence of the ratio of ferromagnetic transition temperaturesTc/Tco on the depolarizing field Fd of the Mn-doped HDQW for dipole down(left) and dipole up (right) obtained by a self-consistent calculation (solid lines)with the applied volage Fg = −10 meV/nm across the sample.

significant dependence on the Fd value. For example, in thecase of the dipole down (left), the transition of Tc/Tco from 1.0to 0.0 is observed near Fd = 10 meV/nm. On the other hand,its transition occurs near Fd = −3 meV/nm for the case ofthe dipole up (right). This means that, for a given the Fd value,smaller applied voltage Fg is needed to obtain the transiton toTc/Tco = 1.0 for the dipole down case, as shown in Fig. 3.

In summary, the ferromagnetic transition temperature as afunction of the electric field for a HDQW system is investigatedby using a self-consistent method. The band structures andwave functions are obtained by solving the Schrodingerequation for electrons and the 3 × 3 Hamiltonian for holes.These results are also compared with those obtained from theflat-band model. The self-consistent solution shows that, the

change in Curie temperature for the dipole right and the dipoleleft occurs near Fg = 17 and 10 meV/nm, respectively. Thiseffect is caused by asymmetry of electrostatic potential due toscreening charges. On the other hand, the results of the flat-bandmodel shows that the change in Curie temperature occurs atlower bias voltage than that of the self-consistent model. In thecase of the flat-band model, the ratio of ferromagnetic transitiontemperatures Tc/Tco has a constant value and is independent ofFd . On the other hand, Tc/Tco of the self-consistent calculationshows a significant dependence on the Fd value.

Acknowledgements

This work was supported by the Korea Science andEngineering Foundation through the Quantum-functionalSemiconductor Research Centre at Dongguk University.Partially, this work was supported by the second stage of BK21.

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