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ferroelectric hybrid double quantum wells
Solid State Communications 143 (2007) 395–398 www.elsevier.com/locate/ssc
Self-consistent solutions of the Curie temperature in ferromagnetic/ferroelectric hybrid double quantum wells S.H. Park a,∗ , S.W. Ryu a , J.J. Kim a , W.P. Hong a , H.M. Kim a , N. Kim b , J.W. Kim b , T.W. Kang b , S.J. Lee b , S.N. Yi c a Department of Electronics Engineering, Catholic University of Daegu, Hayang, Kyeongsan, Kyeongbuk 712-702, South Korea b 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. Akai Available 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 10 meV/nm, respectively. This effect is caused by asymmetry of electrostatic potential due to screening charges. On the other hand, the results of the 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 of the 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 been generally known as one of the most promising candidates for spintronic device materials, many experimental and theoretical studies have been performed on various DMS materials and structures. Because the spintronic devices should ultimately be operated at room temperature, much effort has been focused on increasing the ferromagnetic transition temperature Tc of DMS above room temperature. Among many materials, ZnMnO is considered to have Tc above 300 K with 5% Mn per unit cell and 3×1020 /cm3 holes according to a theoretical prediction [1]. Recently, observation of the ferroelectric properties was reported in Li-doped ZnO bulk samples. The reason for the ferroelectric property is attributed to the following: when the ∗ Corresponding author.
E-mail addresses: [email protected] (S.H. Park), [email protected] (S.J. Lee). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.06.016
˚ is smaller than the host size of the dopant Li atom (0.6 A) ˚ then the Li atoms can occupy off-centre Zn atom (0.74 A), positions, thus locally inducing electric dipoles, thereby leading to ferroelectric behaviour [2]. On the theoretical side, a ferromagnetic/ferroelectric hybrid double quantum well (HDQW) system using ZnMnO and ZnLiO layers has been studied as a new spintronic device [3]. Here, the ferromagnetic transition temperature of the HDQW system was investigated as a function of the applied electric field considering spontaneous electric polarization in the ZnLiO well and the flat-band model was used assuming a regime of low carrier density of 1 × 1011 /cm2 in the wells. On the other hand, there has been little work on the carrier density dependence of the ferromagnetic transition temperature in a HDQW system. In particular, in the case of the high carrier density, the self-consistent treatment is expected to be very important to obtain exact solutions for the wave functions. In this research, we investigate the ferromagnetic transition temperature 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 structures and wave functions are obtained by solving the Schr¨odinger equation for electrons and the 3×3 Hamiltonian for holes [4,5]. These results are also compared with those obtained from the flat-band model. The HDQW system has a structure of MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO and lower LiZnO wells and MgZnO barriers. We assume that the electron and hole are simultaneously injected into the wells. The ZnO well contains an additional Mn-doped p-type layer at the middle of the well or at the upper edge of the well. The lower well LiZnO is the p-type ferroelectric well with spontaneous polarization. The dimension of the structure are chosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and the capping layer is 20 nm. The confinement potential is V0 = 263 meV with 20% of Mg per unit cell in ZnMgO barriers. 2. Theory The self-consistent band structures and wave functions are obtained by iteratively solving the Schr¨odinger equation for electrons, the block-diagonalized Hamiltonian for holes, and Poisson’s equation [4,5]. The total potential profiles for the electrons and the holes are Vc (z) = Vcw (z) − |e|φ(z),
(1)
Vv (z) = Vvw (z) − |e|φ(z),
where Vcw (z) and Vvw (z) are the square-well potentials for the conduction band and the valence band, respectively, and φ(z) is the screening potential induced by the charged carriers and satisfies the Poisson equation d d ε(z) φ(z) = −|e|[ p(z) − n(z)], (2) dz dz where ε(z) is the dielectric constant. The electron and the hole concentrations, p(z) and n(z), are related to the wave functions of the n-th conduction subband and the m-th valence subband by n(z) =
kT m e X π h¯ 2
| f n (z)|2 ln(1 + e[E f c −E cn (0)]/kT )
(3)
n
and p(z) =
X XZ σ =U,L m
×
kk X σ (ν) |gmkk (z)|2 2π v 1 , −E (k )]/kT
dkk
1 + e[E f v
vm
k
(4)
where m e is the effective mass of electrons, h¯ is Planck’s constant divided by 2π , n and m are the quantized subband indices for the conduction and the valence bands, E f c and E f v are the quasi-Fermi levels of the electrons and the holes, respectively, E cn (0) is the quantized energy level of the electrons, E vm (kk ) is the energy for the m-th subband in the valence band, σ denotes the upper (U) and the lower (L) blocks of the Hamiltonian, kk is the in-plane wave vector, ν refers to σ (ν) the new bases for the Hamiltonian, and f n (z) and gmkk (z) are
the envelope functions in the conduction and the valence bands, respectively. The potential φ(z) is obtained by integration [6]: Z z E(z 0 )dz 0 , (5) φ(z) = − −L/2
where Z
z
E(z) = −L/2
1 ρ(z 0 )dz 0 . ε(z)
(6)
The procedures for the SC calculations consist of the following steps: (i) Start with the potential profiles Vc and Vv with φ (0) (z) = 0 in Eq. (1); (ii) solve the Schr¨odinger equation (for electrons) and the block-diagonalized Hamiltonian (for holes) with the potential profiles φ (n−1) (z) in step (i) to obtain band structures and wave functions; (iii) for a given carrier density, obtain the Fermi-energies by using the band structures and 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, the band structures and the wave functions obtained with φ (n−1) (z) are solutions. An adjustable parameter w (0 < w < 1) is typically set to 0.5 at low carrier densities. With increasing carrier densities, a smaller value of w is needed for rapid convergence. We write Tc in the form [7,8] Ts =
Z 2 S(S + 1)J pd m∗ dz|g(z)|4 c(z), 12k B π h¯ 2
(7)
where c(z) is a magnetic ion distribution function, J pd is the exchange integral of carrier–spin exchange interaction, and S is a Mn ion spin. We calculate the change in the fourth power of growth direction envelope functions, |g(z)|4 , of carriers at the lowest energy subband in the HDQW as a function of the applied electric fields. 3. Results and discussion Fig. 1 shows the HDQW system having a structure of MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO and lower LiZnO wells and MgZnO barriers. We assume that the electron and the hole are simultaneously injected into the wells. The ZnO well contains an additional Mn δ-doped p-type layer of a 0.5 monolayer at the distance of 2 nm toward the well from the interface between the ZnO well and the MgZnO barrier. The lower well LiZnO is the p-type ferroelectric well with spontaneous polarization. The dimension of the structure are chosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and the capping layer is 7 nm. In numerical calculations, we choose physical parameters for p-type ZnO/Mgx Zn1−x O/Liy Zn1−y O with x = 0.2 and y = 0.05. The parameters for Zn1−x Mgx O are obtained from linear combinations of the parameters for ZnO and MgO [9]. However, many of the material parameters for MgO are not well known, so we assumed the parameters to be equal to those of ZnO as a first approximation in the case of a lack of published data because the Mg composition
S.H. Park et al. / Solid State Communications 143 (2007) 395–398
Fig. 1. HDQW system having a structure of MgZnO/ZnO/MgZnO/LiZnO/Mg ZnO with upper ZnO and lower LiZnO wells and MgZnO barriers.
in the ZnMgO barrier is relatively small (x = 0.2). Also, the parameters for Zn1−y Liy O are assumed to be equal to those of ZnO because of a small composition of Li. We used 65/35 as the ratio between the conduction- and the valence-band offsets (1E c /1E v ) in the ZnO/ZnMgO heterostructure. Fig. 2 shows the potential profile and the wave functions (HH1) at zone centre (kk ) for dipole down (left) and dipole up (right) obtained by a self-consistent calculation (solid lines). For comparison, we plotted those (dashed lines) obtained by the flat-band model. Here, the depolarizing field is Fd = 15 meV/nm and the electrostatic potential at the interface due to screening charges is assumed to be V1 = 0.02 eV. The selfconsistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . In the case of a low bias voltage Fg , the wave functions are shown to be located on the right side well. On the other hand, in the case of a high bias voltage Fg = 2 meV/nm, the wave functions of the self-consistent solution for both dipole down (left) and dipole up (right) cases are located in the right side while those of the flat-band model are located on the left side. This is because of the band-bending effect due to the free-carrier screening, which makes the the bottom in the potential well to exist on the right side. Fig. 3 shows the dependence of the ratio of ferromagnetic transition temperatures Tc /Tco on bias voltage Fg applied across the Mn-doped HDQW for dipole down (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. The self-consistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . For comparison, we plotted those (dashed lines) obtained by the flat-band model. The selfconsistent solution shows that, in the case of the dipole right, the change 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 electrostatic potential due to screening charges. On the other hand, the results of the flat-band model shows that the change in Curie temperature occurs at lower bias voltage than that of the self-
397
Fig. 2. Potential profile and the wave functions (HH1) at zone centre (kk ) for dipole down (left) and dipole up (right) obtained by a self-consistent calculation (solid lines). For comparison, we plotted those (dashed lines) obtained by the flat-band model.
Fig. 3. Dependence of the ratio of ferromagnetic transition temperatures Tc /Tco on bias voltage Fg applied across the Mn-doped HDQW for dipole down (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 selfconsistent treatment to obtain more exact result at higher carrier density. Fig. 4 shows the dependence of the ratio of ferromagnetic transition temperatures Tc /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 voltage Fg = −10 meV/nm across the sample. The self-consistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . For comparison, we plotted those (dashed lines) obtained by the flat-band model. In the case of the flat-band model, we know that the ratio of ferromagnetic transition temperatures Tc /Tco has a constant value and is independent of Fd in an investigated range of the Fd value. This is because, with the applied voltage Fg = −10 meV/nm, the wave functions for the flat-band model are located in the left side for both dipole down (left) and dipole up (right) cases. However, Tc /Tco of the self-consistent calculation shows a
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S.H. Park et al. / Solid State Communications 143 (2007) 395–398
change in Curie temperature for the dipole right and the dipole left occurs near Fg = 17 and 10 meV/nm, respectively. This effect is caused by asymmetry of electrostatic potential due to screening charges. On the other hand, the results of the 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 of the 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. Acknowledgements Fig. 4. Dependence of the ratio of ferromagnetic transition temperatures Tc /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 the case of the dipole down (left), the transition of Tc /Tco from 1.0 to 0.0 is observed near Fd = 10 meV/nm. On the other hand, its transition occurs near Fd = −3 meV/nm for the case of the dipole up (right). This means that, for a given the Fd value, smaller applied voltage Fg is needed to obtain the transiton to Tc /Tco = 1.0 for the dipole down case, as shown in Fig. 3. In summary, the ferromagnetic transition temperature as a function of the electric field for a HDQW system is investigated by using a self-consistent method. The band structures and wave functions are obtained by solving the Schr¨odinger equation for electrons and the 3 × 3 Hamiltonian for holes. These results are also compared with those obtained from the flat-band model. The self-consistent solution shows that, the
This work was supported by the Korea Science and Engineering Foundation through the Quantum-functional Semiconductor Research Centre at Dongguk University. Partially, this work was supported by the second stage of BK21. References [1] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, D. Ferrand, Science 287 (2000) 1019. [2] M. Joseph, H. Tabata, T. Kawai, Appl. Phys. Lett. 74 (1999) 2534. [3] N. Kim, H. Kim, J.W. Kim, S.J. Lee, T.W. Kang, Phys. Rev. B 73 (2006) 033318. [4] S.L. Chuang, C.S. Chang, Phys. Rev. B 54 (1996) 2491. [5] S.H. Park, S.L. Chuang, J. Appl. Phys. 87 (2000) 353. [6] S.L. Chuang, Physics of Optoelectronic Devices, Wiley, New York, 1995 (Chapter 4). [7] T. Dietl, A. Haury, Y.M. D’aubigne, Phys. Rev. B 55 (1996) R3347. [8] B. Lee, T. Jungwirth, A.H. Macdonald, Phys. Rev. B 61 (2000) 15606. [9] S.H. Park, K.J. Kim, S.N. Yi, D. Ahn, S.J. Lee, J. Korean Phys. Soc. 47 (2005) 448.
Self-consistent solutions of the Curie temperature in ferromagnetic/ferroelectric hybrid double quantum wells S.H. Park a,∗ , S.W. Ryu a , J.J. Kim a , W.P. Hong a , H.M. Kim a , N. Kim b , J.W. Kim b , T.W. Kang b , S.J. Lee b , S.N. Yi c a Department of Electronics Engineering, Catholic University of Daegu, Hayang, Kyeongsan, Kyeongbuk 712-702, South Korea b 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. Akai Available 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 10 meV/nm, respectively. This effect is caused by asymmetry of electrostatic potential due to screening charges. On the other hand, the results of the 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 of the 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 been generally known as one of the most promising candidates for spintronic device materials, many experimental and theoretical studies have been performed on various DMS materials and structures. Because the spintronic devices should ultimately be operated at room temperature, much effort has been focused on increasing the ferromagnetic transition temperature Tc of DMS above room temperature. Among many materials, ZnMnO is considered to have Tc above 300 K with 5% Mn per unit cell and 3×1020 /cm3 holes according to a theoretical prediction [1]. Recently, observation of the ferroelectric properties was reported in Li-doped ZnO bulk samples. The reason for the ferroelectric property is attributed to the following: when the ∗ Corresponding author.
E-mail addresses: [email protected] (S.H. Park), [email protected] (S.J. Lee). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.06.016
˚ is smaller than the host size of the dopant Li atom (0.6 A) ˚ then the Li atoms can occupy off-centre Zn atom (0.74 A), positions, thus locally inducing electric dipoles, thereby leading to ferroelectric behaviour [2]. On the theoretical side, a ferromagnetic/ferroelectric hybrid double quantum well (HDQW) system using ZnMnO and ZnLiO layers has been studied as a new spintronic device [3]. Here, the ferromagnetic transition temperature of the HDQW system was investigated as a function of the applied electric field considering spontaneous electric polarization in the ZnLiO well and the flat-band model was used assuming a regime of low carrier density of 1 × 1011 /cm2 in the wells. On the other hand, there has been little work on the carrier density dependence of the ferromagnetic transition temperature in a HDQW system. In particular, in the case of the high carrier density, the self-consistent treatment is expected to be very important to obtain exact solutions for the wave functions. In this research, we investigate the ferromagnetic transition temperature 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 structures and wave functions are obtained by solving the Schr¨odinger equation for electrons and the 3×3 Hamiltonian for holes [4,5]. These results are also compared with those obtained from the flat-band model. The HDQW system has a structure of MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO and lower LiZnO wells and MgZnO barriers. We assume that the electron and hole are simultaneously injected into the wells. The ZnO well contains an additional Mn-doped p-type layer at the middle of the well or at the upper edge of the well. The lower well LiZnO is the p-type ferroelectric well with spontaneous polarization. The dimension of the structure are chosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and the capping layer is 20 nm. The confinement potential is V0 = 263 meV with 20% of Mg per unit cell in ZnMgO barriers. 2. Theory The self-consistent band structures and wave functions are obtained by iteratively solving the Schr¨odinger equation for electrons, the block-diagonalized Hamiltonian for holes, and Poisson’s equation [4,5]. The total potential profiles for the electrons and the holes are Vc (z) = Vcw (z) − |e|φ(z),
(1)
Vv (z) = Vvw (z) − |e|φ(z),
where Vcw (z) and Vvw (z) are the square-well potentials for the conduction band and the valence band, respectively, and φ(z) is the screening potential induced by the charged carriers and satisfies the Poisson equation d d ε(z) φ(z) = −|e|[ p(z) − n(z)], (2) dz dz where ε(z) is the dielectric constant. The electron and the hole concentrations, p(z) and n(z), are related to the wave functions of the n-th conduction subband and the m-th valence subband by n(z) =
kT m e X π h¯ 2
| f n (z)|2 ln(1 + e[E f c −E cn (0)]/kT )
(3)
n
and p(z) =
X XZ σ =U,L m
×
kk X σ (ν) |gmkk (z)|2 2π v 1 , −E (k )]/kT
dkk
1 + e[E f v
vm
k
(4)
where m e is the effective mass of electrons, h¯ is Planck’s constant divided by 2π , n and m are the quantized subband indices for the conduction and the valence bands, E f c and E f v are the quasi-Fermi levels of the electrons and the holes, respectively, E cn (0) is the quantized energy level of the electrons, E vm (kk ) is the energy for the m-th subband in the valence band, σ denotes the upper (U) and the lower (L) blocks of the Hamiltonian, kk is the in-plane wave vector, ν refers to σ (ν) the new bases for the Hamiltonian, and f n (z) and gmkk (z) are
the envelope functions in the conduction and the valence bands, respectively. The potential φ(z) is obtained by integration [6]: Z z E(z 0 )dz 0 , (5) φ(z) = − −L/2
where Z
z
E(z) = −L/2
1 ρ(z 0 )dz 0 . ε(z)
(6)
The procedures for the SC calculations consist of the following steps: (i) Start with the potential profiles Vc and Vv with φ (0) (z) = 0 in Eq. (1); (ii) solve the Schr¨odinger equation (for electrons) and the block-diagonalized Hamiltonian (for holes) with the potential profiles φ (n−1) (z) in step (i) to obtain band structures and wave functions; (iii) for a given carrier density, obtain the Fermi-energies by using the band structures and 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, the band structures and the wave functions obtained with φ (n−1) (z) are solutions. An adjustable parameter w (0 < w < 1) is typically set to 0.5 at low carrier densities. With increasing carrier densities, a smaller value of w is needed for rapid convergence. We write Tc in the form [7,8] Ts =
Z 2 S(S + 1)J pd m∗ dz|g(z)|4 c(z), 12k B π h¯ 2
(7)
where c(z) is a magnetic ion distribution function, J pd is the exchange integral of carrier–spin exchange interaction, and S is a Mn ion spin. We calculate the change in the fourth power of growth direction envelope functions, |g(z)|4 , of carriers at the lowest energy subband in the HDQW as a function of the applied electric fields. 3. Results and discussion Fig. 1 shows the HDQW system having a structure of MgZnO/ZnO/MgZnO/LiZnO/MgZnO with upper ZnO and lower LiZnO wells and MgZnO barriers. We assume that the electron and the hole are simultaneously injected into the wells. The ZnO well contains an additional Mn δ-doped p-type layer of a 0.5 monolayer at the distance of 2 nm toward the well from the interface between the ZnO well and the MgZnO barrier. The lower well LiZnO is the p-type ferroelectric well with spontaneous polarization. The dimension of the structure are chosen: W1 = 10 nm, W2 = 10 nm, B = 5 nm, and the capping layer is 7 nm. In numerical calculations, we choose physical parameters for p-type ZnO/Mgx Zn1−x O/Liy Zn1−y O with x = 0.2 and y = 0.05. The parameters for Zn1−x Mgx O are obtained from linear combinations of the parameters for ZnO and MgO [9]. However, many of the material parameters for MgO are not well known, so we assumed the parameters to be equal to those of ZnO as a first approximation in the case of a lack of published data because the Mg composition
S.H. Park et al. / Solid State Communications 143 (2007) 395–398
Fig. 1. HDQW system having a structure of MgZnO/ZnO/MgZnO/LiZnO/Mg ZnO with upper ZnO and lower LiZnO wells and MgZnO barriers.
in the ZnMgO barrier is relatively small (x = 0.2). Also, the parameters for Zn1−y Liy O are assumed to be equal to those of ZnO because of a small composition of Li. We used 65/35 as the ratio between the conduction- and the valence-band offsets (1E c /1E v ) in the ZnO/ZnMgO heterostructure. Fig. 2 shows the potential profile and the wave functions (HH1) at zone centre (kk ) for dipole down (left) and dipole up (right) obtained by a self-consistent calculation (solid lines). For comparison, we plotted those (dashed lines) obtained by the flat-band model. Here, the depolarizing field is Fd = 15 meV/nm and the electrostatic potential at the interface due to screening charges is assumed to be V1 = 0.02 eV. The selfconsistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . In the case of a low bias voltage Fg , the wave functions are shown to be located on the right side well. On the other hand, in the case of a high bias voltage Fg = 2 meV/nm, the wave functions of the self-consistent solution for both dipole down (left) and dipole up (right) cases are located in the right side while those of the flat-band model are located on the left side. This is because of the band-bending effect due to the free-carrier screening, which makes the the bottom in the potential well to exist on the right side. Fig. 3 shows the dependence of the ratio of ferromagnetic transition temperatures Tc /Tco on bias voltage Fg applied across the Mn-doped HDQW for dipole down (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. The self-consistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . For comparison, we plotted those (dashed lines) obtained by the flat-band model. The selfconsistent solution shows that, in the case of the dipole right, the change 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 electrostatic potential due to screening charges. On the other hand, the results of the flat-band model shows that the change in Curie temperature occurs at lower bias voltage than that of the self-
397
Fig. 2. Potential profile and the wave functions (HH1) at zone centre (kk ) for dipole down (left) and dipole up (right) obtained by a self-consistent calculation (solid lines). For comparison, we plotted those (dashed lines) obtained by the flat-band model.
Fig. 3. Dependence of the ratio of ferromagnetic transition temperatures Tc /Tco on bias voltage Fg applied across the Mn-doped HDQW for dipole down (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 selfconsistent treatment to obtain more exact result at higher carrier density. Fig. 4 shows the dependence of the ratio of ferromagnetic transition temperatures Tc /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 voltage Fg = −10 meV/nm across the sample. The self-consistent solutions are obtained at the sheet carrier density of N2D = 1 × 1012 /cm2 . For comparison, we plotted those (dashed lines) obtained by the flat-band model. In the case of the flat-band model, we know that the ratio of ferromagnetic transition temperatures Tc /Tco has a constant value and is independent of Fd in an investigated range of the Fd value. This is because, with the applied voltage Fg = −10 meV/nm, the wave functions for the flat-band model are located in the left side for both dipole down (left) and dipole up (right) cases. However, Tc /Tco of the self-consistent calculation shows a
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S.H. Park et al. / Solid State Communications 143 (2007) 395–398
change in Curie temperature for the dipole right and the dipole left occurs near Fg = 17 and 10 meV/nm, respectively. This effect is caused by asymmetry of electrostatic potential due to screening charges. On the other hand, the results of the 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 of the 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. Acknowledgements Fig. 4. Dependence of the ratio of ferromagnetic transition temperatures Tc /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 the case of the dipole down (left), the transition of Tc /Tco from 1.0 to 0.0 is observed near Fd = 10 meV/nm. On the other hand, its transition occurs near Fd = −3 meV/nm for the case of the dipole up (right). This means that, for a given the Fd value, smaller applied voltage Fg is needed to obtain the transiton to Tc /Tco = 1.0 for the dipole down case, as shown in Fig. 3. In summary, the ferromagnetic transition temperature as a function of the electric field for a HDQW system is investigated by using a self-consistent method. The band structures and wave functions are obtained by solving the Schr¨odinger equation for electrons and the 3 × 3 Hamiltonian for holes. These results are also compared with those obtained from the flat-band model. The self-consistent solution shows that, the
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