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Magnetoresistance effect in a hybrid ferromagnetic/semiconductor nanostructure Mao-Wang Lu, Guo-Jian Yang ∗ Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China Received 3 September 2006; received in revised form 15 October 2006; accepted 3 November 2006 by F. Peeters Available online 20 November 2006

Abstract We propose a Magnetoresistance device in a magnetically modulated two-dimensional electron gas, which can be realized experimentally by the deposition of two parallel ferromagnetic strips on the top and bottom of a semiconductor heterostructure. It is shown that there exists a significant transmission difference for electrons through the parallel and antiparallel magnetization configurations of such a device, which leads to a considerable magnetoresistance effect. It is also shown that the magnetoresistance ratio of the device depends greatly on the magnetic strength difference in the two delta barriers of the system. c 2006 Elsevier Ltd. All rights reserved.

PACS: 73.40.-c; 75.75.+a; 72.20.-i; 73.23.-b Keywords: A. Magnetic nanostructures; D. Magnetoresistance effect; D. MR ratio

Large magnetoresistance (MR) effects, especially the socalled giant magnetoresistance (GMR) effects [1], have attracted a great amount of experimental and theoretical attention in MR systems in recent years. At present, the MR effect has given rise to a lot of significant and practical applications in magnetic information storage [2], including ultrasensitive magnetic field sensors, read heads, random access memories, and so on. Generally, the structures where MR is observed consist of ferromagnetic layers separated by thin nonmagnetic layers. In such heterogeneous systems, the MR is characterized by a striking drop in electric resistance when an external magnetic field switches the magnetization of the adjacent magnetic layers from an antiparallel (AP) alignment to a parallel (P) one. For a specific MR device, one hopes, from the viewpoint of practical applications, that the system possesses a high MR ratio under a relatively low saturation magnetic field. To obtain a large MR ratio, an attractive alternative is to use magnetic or superconducting microstructures on the surface of heterostructures containing a two-dimensional electron ∗ Corresponding author.

E-mail address: [email protected] (G.-J. Yang). c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.11.003

gas (2DEG). Microstructured ferromagnets or superconductors provide an inhomogeneous magnetic field which locally influences the motion of the electrons in the semiconductor. Nogaret et al. [3] demonstrated a MR effect in hybrid ferromagnetic and/or semiconductor devices at low temperatures, and a MR ratio of up to 103 % at 4 K has been observed recently [4]. It was also reported that MR oscillations, due to the internal Landau band structure of a 2DEG system, can be observed in a periodic magnetic field [5]. More recently, a spin-independent MR effect has attracted considerable attention in a magnetically modulated 2DEG system [6–9]. Using δ-function magnetic barriers in a twodimensional electron gas (2DEG), a MR device was proposed by Zhai et al. [6] It was found that, although the average magnetic field of the structure is zero, this kind of system possesses a very high MR ratio, and the MR effect makes no use of the spin degree of freedom, which is distinct from conventional MR devices. The MR effect in a realistic case, where the exact magnetic profiles instead of the ideal δfunction magnetic barriers were employed, was also studied [7]. Yang et al. [8] investigated the MR effect of 2DEG systems subjected to a periodically modulated magnetic field, and found that the MR ratio of such periodically modulated systems

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ignored. The magnetization directions of the FM strips are assumed to be parallel in Fig. 1(a), and the resulting magnetic field is presented schematically in Fig. 1(b). For simplicity, the magnetic barriers can be approximated [12] as a delta function in order to demonstrate the principle of operation of this device, as shown in Fig. 1(c) and (d). Here, Fig. 1(c) and (d) correspond to the parallel (P) and antiparallel (AP) configurations of the two FM layers, respectively. The magnetic field can be expressed as Bz (x) = [B1 δ(x + L2 ) − χ B2 δ(x − L 2 )], where B1 and B2 are the magnetic strengths of two δ-function barriers, L is their separation, and χ represents the magnetization configuration (±1 or P/AP). In the single effective-mass approximation, the Hamiltonian describing such a system is H=

[ p y + (e/c)A y (x)]2 px2 eg ∗ σz h¯ + + Bz (x), ∗ ∗ 2m e 2m e 2m e 2c

(1)

where m ∗e is the effective mass, m e is the free electron mass, ( px , p y ) is the electron momentum, g ∗ is the effective Land´e factor of the electron in the 2DEG, σz = +1/−1 for spinup/down electrons, and the magnetic vector potential of the device can be written as AE = [0, A y (x), 0] in the Landau gauge, i.e., 0, x < −L/2 A y (x) = B1 , −L/2 < x < +L/2 (2) B1 + χ B2 , x > L/2 Fig. 1. (a) A schematic illustration of the MR device, where two magnetic strips are placed on the top and bottom of a 2DEG. (b) Magnetic field profile induced in the 2DEG. Magentic-field models (c) and (d) exploited in this work correspond to the P and AP alignments, respectively.

showed a strong dependence on the space between the magnetic potentials and the number of periods. In the present work, we propose an alternative way to realize the MR effect. The proposed MR-device is a 2DEG that is usually formed in a modulation-doped semiconductor heterostructure, subject to modulation by a perpendicular magnetic field. This system can be experimentally realized [10] by depositing two ferromagnetic (FM) strips on the top and bottom of a semiconductor heterostructure, as schematically depicted in Fig. 1(a). The in-plane magnetization of the FM layers creates an out-of-plane fringe magnetic field at both ends. This fringe field constitutes a nonhomogeneous magnetic barrier for electron transport within the 2DEG. The two FM strips are assumed to be asymmetric in length, with a distance L between their right edges. The two FM layers are also different in distance relative to the 2DEG: the distance of the upper FM layer is smaller than that of the FM layer at the bottom, which will result in the magnetic barriers produced by the two FM strips having non-identical strengths (because the magnetic strength increases with the decreasing distance of the FM strips relative to the 2DEG) [11]. Making use of modern nanotechnology, such a system can be deliberately designed to fall short of the left-hand edges of the FM layers, so that the effects of the fringe field there can be

which results in Bz (x) = dA y (x)/dx. For convenience, we introduce two characteristic quantities, the cyclotron frequency √ ωc = eB0 /cm ∗ and the magnetic length l B = h¯ c/eB0 . Thus, all the relevant quantities can be expressed as dimensionless forms in terms of ωc and l B : the magnetic field Bz (x) → B0 Bz (x), the magnetic vector potential A(x) → B0l B A(x), the coordinate x → l B x, and the energy E → h¯ ωc E (=E 0 E). In our calculation, we take B0 = 0.1 T, which leads to the units l B = 81.3 nm and E 0 = 0.17 meV for the GaAs system with m ∗e = 0.067m e and g ∗ = 0.44. Because of the translational invariance of the system along the y direction, the total electronic wave-function can be written as Φ(x, y) = eik y y ψ(x), where k y is the wave-vector component in the y direction. Accordingly, the wave function ψ(x) satisfies the following reduced one-dimensional (1D) Schr¨odinger equation 2 m ∗e g ∗ σz d 2 − [k y + A y (x)] + 2 E − Bz (x) ψ(x) = 0. 4m e dx 2 (3) It is useful to introduce the effective potential Ueff (x, k y ) = 2 k y + A y (x) /2+m ∗e g ∗ σz Bz (x)/(4m e ). Clearly, this effective potential of the system depends strongly not only on the magnetic configuration Bz (x), but also on the longitudinal wave vector k y . The k y -dependence renders the motion of electrons an essentially two-dimensional (2D) process, as would be expected from the classical analogy. From the dependence of the Ueff on the magnetic profile Bz (x), one can easily see that for the device presented in Fig. 1, when the P

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alignment (Fig. 1(c)) turns to the inverse (Fig. 1(d)), Ueff varies substantially. It is the dependence on the magnetic configuration of Ueff that results in the MR effect in the involved systems. The reduced 1D Schr¨odinger Equation (3) can be solved exactly in each region [13], where the wave function is expressed by a linear combination of the plane waves. In the left and right regions of the device, the wave functions can be written as ψleft (x, y) = exp(ik y y)[exp(ikl x) + γ exp(−ikl x)], x < −L/2 and ψright (x, y) = τ exp(ik y y) exp(ik q r x), x > L/2, where kl = q

2E − [A yl (x) + k y ]2 , kr = 2E − [A yr (x) + k y ]2 , and γ /τ is the reflection/transmission amplitude. In the intermediate region, −L/2 < x < L/2, the wave function can be assumed as ψq in (x, y) = exp(ik y y)[A exp(ikx) + B exp(−ikx)], where

k = 2E − [B1 + k y ]2 , as well and A and B are two unknown constants determined by the use of the boundary conditions. Thus, one can readily obtain the transmission coefficient as T (E, k y ) = kkrl |τ |2 . Furthermore, we can calculate the ballistic conductance at zero temperature from the well-known Landauer–B¨uttiker formula [14] Z π/2 p (4) G(E F ) = G 0 T E F , 2E F sin θ cos θdθ, −π/2

with θ the incident angle relative to the x direction. The conductance is presented in units of G 0 = 2e2 m ∗e v F L y / h 2 , where v F is the Fermi velocity and L y is the longitudinal length of the system. For a MR device, the MR ratio usually has two kinds of definitions [6,9], i.e., MRR = (G P − G AP )/G AP or (G P − G AP )/G P and MMRR = (G P − G AP )/(G P + G AP ), where G P and G AP are the conductance for the parallel and antiparallel alignments, respectively. Obviously, the MR ratios as calculated by the different definitions are distinct for some cases. In this work, we adopt the MMRR definition to study the MR effect. Although the delta function Bz (x) is locally infinite, the effect of the polarization g ∗ m ∗e Bz (x)/m e on the MR effect extends to the whole infinite space. But, the Zeeman coupling term on the MR depends on the quantity g ∗ Bm ∗e /4m e , which equals 0.0369 for B = 5 and for GaAs. Comparing it to the other terms in Ueff , the absolute value of such a Zeeman term is much smaller. Therefore, the spin-dependent term plays a minor role in determining the transport properties [6], and will be ignored in the subsequent discussion [9]. First of all, in order to display the transmissions for the electrons tunneling through the P configuration (Fig. 1(c)) and the AP one (Fig. 1(d)), we have calculated the corresponding transmission coefficients, TP and TAP . Fig. 2 shows these transmission coefficients versus the incident energy E for electrons with a different wave vector k y = 0 (solid curve), 1 (dashed curve), and −1 (dotted curve), where Fig. 2(a) and (b) correspond to the P and AP configurations, respectively. The structural parameters are chosen to be B1 = 1.0, B2 = 1.2, and L = 3.0 for both the configurations. From this

Fig. 2. (a) and (b) are the transmission coefficients for the electrons tunneling through the P and AP configurations, respectively, where the structural parameters are chosen to be B1 = 1.0, B2 = 1.2, L = 3.0 and the wave vector components of the electrons are taken to be k y = 0 (solid curve), 1 (dashed curve), and −1 (dotted curve).

figure, we can see that, for the P and AP configurations, the transmission spectrum exhibits clear longitudinal-wave-vector dependent tunneling features, as confirmed previously [11], due to the essentially 2D process for the electrons passing through the magnetic-barrier structures. Moreover, one can obviously see that in the low-energy region, there are several incomplete resonant peaks for the P configuration. However, when the system switches from the P configuration to the AP configuration, one can see from Fig. 2(b) that the electron transmission is greatly altered, because of the variation of the Ueff induced by the structure. The transmission curves shift towards the high energy region and are greatly suppressed in contrast to the P configuration. Especially, the low-energy resonant peaks for the AP configuration almost disappear, because the change of the effective potential Ueff makes the electron transmission through AP configuration incomplete. The configuration-dependent transmission features demonstrated above should be reflected in the measurable quantity, the conductance G, which is obtained by integrating the transmission of the electrons over the incident angle, as in Eq. (4). Indeed, our calculated results also confirm this discrepancy in the conductances G P and G AP . In Fig. 3 we present the conductances G P (solid curve) and G AP (dashed curve) for the P and AP alignments versus the Fermi energy E F , where the structural parameters are the same as in Fig. 2 and the conductance is in units of G 0 . The large suppression of the conductance G AP is clearly seen, due to the great reduction of the transmission coefficient TAP , in contrast to the P alignment. It is this large suppression of the conductance of the AP alignment that results in an evident MR effect in the device under consideration. The inset of Fig. 3 shows the magnetoresistance ratio MMRR as a function of the Fermi energy E F for our considered system. A

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Fig. 3. The conductances G P (solid curve) and G AP (dashed curve) of the P and AP alignments as a function of the Fermi energy E F , where the conductances are in units of G 0 = 2e2 m ∗e v F L y / h 2 and the structural parameters are the same as in Fig. 2. Its inset gives the corresponding magnetoresistance ratio MMRR.

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as the discrepancy (B2 − B1 ) increases. This feature implies that one can manipulate and tune the MR effect by changing the difference (B2 − B1 ) in the device. In summary, we propose an MR device based on the combination of nonhomogeneous magnetic fields and a 2DEG, which can be experimentally realized by the deposition, on the top and bottom of a conventional GaAs heterostructure, of two parallel ferromagnetic strips. We have theoretically investigated the MR effect in this system. Our calculations show that, since there exists an evident tunneling difference in the P and AP configurations (especially the transmission suppression for the AP alignment), this device shows a considerable MR effect. We have also exhibited that the MR ratio is greatly influenced by the magnetic strength difference (B2 − B1 ) in the two δ-barriers of the device; thus a much larger MR ratio can be achieved by suitably adjusting the difference (B2 − B1 ). References

Fig. 4. The magnetoresistance ratio MMRR versus the Fermi energy E F , where the device parameters are the same as in Fig. 2 but B2 = 1 (solid curve), 2 (dashed curve), 3 (dotted curve), and 5 (dash-dotted curve).

considerable MR effect can be evidently seen, especially in the low-energy region. The MR effect changes its degree when the Fermi energy varies. In particular, it is striking that the MR ratio MMRR can be up to 100% at certain low-E F , and the MMRR reduces with increasing Fermi energy E F . Finally, we examine the influence of the magnetic strength difference between two δ-barriers on the MR effect for the system, as shown in Fig. 1. Fig. 4 shows the MR ratio MMRR versus the Fermi energy E F for the system, where the structural parameters are the same as in Fig. 3, but B2 = 1.0 (solid curve), 2.0 (dashed curve), 3.0 (dotted curve), and 5.0 (dash-dotted curve). The MMRR shows a strong dependence on the magnetic strength difference (B2 − B1 ). The MMRR curve shifts to the high Fermi energy region and its value is significantly enhanced

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