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Controllable magnetoresistance effect in a δ-doped and magnetically-confined semiconductor heterostructure
Vacuum 169 (2019) 108891
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Short communication
Controllable magnetoresistance effect in a δ-doped and magneticallyconfined semiconductor heterostructure
T
Dong-Hui Liang, Mao-Wang Lu∗, Xin-Hong Huang, Meng-Rou Huang, Zeng-Lin Cao College of Science, Guilin University of Technology, Guilin, 541004, People's Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Magnetically confined semiconductor heterostructure δ-doping Magnetoresistance effect Magnetoresistance ratio Controllable magnetoresistance device
A magnetoresistance device was proposed by patterning two asymmetric ferromagnetic stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure. To control its performance, a tunable δ-potential is introduced inside this device by wire-like doping. Adopting transfer matrix method and Landauer-Büttiker conductance theory, Schrödinger equation is solved analytically and δ-doping dependent transmission, conductance and magnetoresistance ratio are calculated. An obvious magnetoresistance effect still stays in this device, due to existence of transmission difference between parallel and antiparallel configurations independent of δ-doping. However, its magnetoresistance ratio can be tuned by changing weight or position of δ-doping. These interesting results not only propose an alternative scheme to manipulate magnetoresistance device, but also a structurallycontrollable magnetoresistance device can be obtained for magnetic information storage.
1. Introduction Magnetoresistance (MR) effect [1] has attracted much attention in recent years, due to its scientific significance and practical application in magnetic information storage such as ultrasensitive magnetic field sensor, random access memorie, and read head [2–5]. In general, MR effect is observed in a sandwich-like structure, where adjacent ferromagnetic (FM) layers are separated by a thin nonmagnetic layer. Because of the spin-dependent carriers’ scattering, conductance of such a heterogeneous structure strikingly drops, when magnetizations of neighboring magnetic layers are switched from parallel (P) alignment to antiparallel (AP) one under an external magnetic field. Degree of MR effect is hence characterized by so-called magnetoresistance ratio (MRR), which can be usually defined as MRR = (GP GAP )/GAP or (GP GAP )/(GP + GAP ) with and GAP standing for conductance of P and AP configurations, respectively. For a MR device, a high MRR at a relatively low switching magnetic field is desired in practical application. An alternative scheme to achieve a large MRR is to exploit the so-called magnetically confined semiconductor heterostructure (MCSH) [6], which is often constructed on a semiconductor heterostructure by restricting motion of two-dimensional electron gas (2DEG) with an inhomogeneous magnetic field on nanometer scale. A MCSH is actually a type of hybrid semiconductor nanostructure consisting of magnetic materials and semiconductors, where the former provides an inhomogeneous magnetic field locally influencing motion of electrons in
∗
the latter [7]. Because of small size, low dimensionality and quantum magnetic confinement, there are some of strange quantum effects in the MCSH, one of which, e.g., is interesting MR effect. MCSH-based MR effect can be used to develop new-typed MR devices for magnetic information storage. In 2002, Zhai et al. [8] first studied MR effect by considering a simplified MCSH comprising of δ-function magnetic barriers (MBs), which can be fabricated by patterning two FM stripes in parallel configuration on top of a semiconductor heterostructure. In physics mechanism, they found that differing from metal-based MR effect this kind of MR effect makes no use of spin degree of freedom, originating from transmission discrepancy between P and AP configuration. On the other hand, it possesses a very high MRR (~106% ) at a relatively low switching magnetic field, which may be an ideal candidate for magnetic information storage. However, their results obtain from ideal δ-MBs may deviate from practical cases. Subsequently, Lu ea al [9]. took realistic magnetic profiles instead of simplified MCSH to explore MR effect in MCSH, and supported Zhai et al.’s findings. Since then, MR effect in other MCSHs were reported in sequence and corresponding MR devices were put forwards successfully; see partial references [10–19]. Edified by the atomic layer doping technique [20] such as molecular beam epitaxy (MBE) and metal-organic chemical-vapor deposition (MOCVD), in 2015 Kong et al. [21] proposed to manipulate MCSHbased MR devices via δ-doping. They found that MRR is closely related to position or weight of the δ-doping in a δ-function-shaped MB MCSH.
Corresponding author. E-mail address: [email protected] (M.-W. Lu).
https://doi.org/10.1016/j.vacuum.2019.108891 Received 23 June 2019; Received in revised form 19 August 2019; Accepted 23 August 2019 Available online 29 August 2019 0042-207X/ © 2019 Elsevier Ltd. All rights reserved.
Vacuum 169 (2019) 108891
D.-H. Liang, et al. 2
H=
(P + eA ) + U (x ) + V (x 2m
eg Z BZ (x ), 2m 0 2
x 0) +
(1)
where m , m 0 , and g are effective mass, free electron mass, and effective Landé factor, respectively, p is momentum vector, Z = ±1
corresponds to spin-up/down electron, and A = [0, Ay (x ), 0] with Ay (x ) = B1 (x d) (L x ) + ( B2 B1) (x )(d x ) is magnetic vector potential given in the Landau gauge. By introducing cyclotron /eB0 with B0 being some typical frequency and magnetic length B = magnetic field, all relevant quantities are written as dimensionless x B and Ay (x ) B0 B Ay (x ) . form, e.g., x For an electron in 2DEG, motion in y -direction is conserved, thus its wave function can be written as (x , y ) = e iky y (x ) , where k y is longitudinal wave vector. The wave function (x ) satisfies reduced onedimensional (1D) Schrödinger equation n (x )
+ 2[E
(2)
Ueff (x )] (x) = 0
where the effective potential experienced by electrons in the MR device can be written as
Ueff (x ) = [k y + Ay (x )]2 /2 + V (x
x 0) +
g z Bz (x ) 4m 0
(3)
The last term in Ueff represents the Zeeman coupling between electronic spin Z and magnetic field BZ (x ) . Comparing to other items, absolute value of such a term is much smaller for GaAs materials. Therefore, spin-dependent term plays a minor role in determining electronic transport properties and thus can be omitted reasonably. Clearly, this effective potential depends on longitudinal wave vector k y , magnetic configuration BZ (x ) , and δ-doping V (x x 0 ) . By employing transfer matrix method (TMM) [30], we can exactly solve Eq. (2). In incident and outgoing regions, wave functions can be exp( ikl x ) , assumed as , and x<0 l (x ) = exp(ikl x ) + exp(ikr x ) , x > L , where kl = kr = 2E k y 2 , and / is rer (x ) = flection/transmission amplitude. In device region of 0 < x < L , wave function can be expressed as
Fig. 1. (a) MCSH-based MR device, (b) P and (c) AP alignments, where two asymmetric FM stripes are patterned on top and bottom of GaAs/AlxGa1-xAs heterostructure and a δ-doping V (x x 0) is embedded inside.
Later, Lu et al. [22] extended to realistic MB-type MCSH and found a sizeable MR effect even if a δ-doping is included inside. They also confirmed that MRR can be tuned by the δ-doping. Subsequently, several groups explored modulation of MR effect in other MCSH-based MR devices; cf. relevant Refs. [23–27]. By pattering two asymmetric FM stripes on top and bottom of the semiconductor heterostructure, Wang et al. [19] proposed a new MR device, and studied influences of magnetic field, external electric barrier (EB) and temperature on MR effect. To control this MR device, in the present work we introduce a δ-doping into this device with the help of atomic layer doping technique. Adopting theoretical analysis combined with numerical calculations, we theoretically explore impact of the δ-doping on MR effect and a controllable MR device is put forward for magnetic information storage. Our considered MR device [19] is shown schematically in Fig. 1(a), which can be experimentally realized [28] by depositing two asymmetric nanosized FM stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure, respectively. Fig. 1(b) and (c) represent the P and AP alignments of this device, respectively, where a δ-doping with tunable weight V and changeable position x 0 is contained in this device. It should be noted that, such a δ-doping is produced from the “wire”-like doping not the “layer”-doping [29]. For a very small distance between FM stripes and 2DEG, magnetic field distribution generated by magnetized FM stripes can be approximated as, where B1 and B2 are magnetic field strength, L or d is the width of FM stripe, and represents magnetization configurations ( ± 1 or P and AP). When a negative voltage ( Vg ) is applied on the upper metallic FM stripe, a rectangular electric barrier (EB) U (x ) = U0 (x ) (d x ) is induced within 2DEG, where U0 is EB height and (x ) is the Heaviside step function. In the framework of single particle and effective mass approximation, the motion of an electron in the 2DEG assumed to be located at xy -plane can be described by the Hamiltonian,
in (x ) =
C1 exp(ik1 x ) + D1 exp( ik1 x ), 0 < x < d C2exp(ik2 x ) + D2 exp( ik2 x ), d < x < x 0 , C3 exp(ik3 x ) + D3 exp( ik3 x ), x 0 < x < L
(4)
where k1 = 2(E U ) (k y + B2 B1 k2 = k3 = 2E (k y B1) 2 , and Ci and Di (i = 1,2,3) are constants. According to continuity of wave function at boundaries, we can get
)2 ,
exp(ikl x 0 ) ikl exp(ikl x 0 ) =M
exp( ikl x 0 ) ikl exp( ikl x 0 )
exp(ikr xL +) ikr exp(ikr xL +)
1
exp( ikr xL +) ikr exp( ikr xL +)
0
,
(5)
where m11 m12 M= m m = M1 × M2 × M3 21 22 = ×
cos k1 d k1 sin k1 d
cos k2 (x 0 d) + 2V sin k2 (x 0 d )/ k2 sin k1 d/ k1 × cos k1 d k2 sin k2 (x 0 d ) 2V cos k2 (x 0 d )
cos k3 (L x 0) k3 sin k3 (L x 0)
sin k3 (L cos k3 (L
sin k2 (x 0 d )/ k2 cos k2 (x 0 d )
x 0)/ k3 . x 0)
(6) Therefore, we can obtain transmission coefficient for an electron with incident energy E and longitudinal wave vector k y , 2
T (E , k y ) = 1
m11 kr m22 / kl + i (kr m12 + m21/ kl ) . m11 + kr m22 / kl + i (kr m12 m21/ kl )
(7)
At zero temperature, ballistic conductance can be calculated from well-known Landauer-Büttiker theory [31]. 2
Vacuum 169 (2019) 108891
D.-H. Liang, et al.
Fig. 3. Conductance varies with the Fermi energy, where δ-doping is the same as in Fig. 2. As a function of the Fermi energy, magnetoresistance ratio is plotted in the inset.
can be understood from the fact that a non-symmetric multiple barriers/wells structure in effective potential Ueff for the AP configuration makes electronic transmission incomplete. More importantly, a great difference of transmission between P and AP configurations exists in the device. In contrast to the P configuration, TAP spectrum moves to the high-energy region and its value is greatly suppressed, which is because the effective potential Ueff has a strong dependence on the magnetic configuration of the MR device. Fig. 3 shows the Fermi energy () dependence of conductance (dashed line) and GAP (dotted line) for electrons tunneling through P and AP configurations, where both structural parameters and δ-doping are the same as in Fig. 2. From this figure, we can clearly see a distinct difference of conductance between P and AP configurations, which is because conductance is obtained by integrating transmission coefficient over all incident angles. For AP alignment, conductance is strongly suppressed, because of greatly reduced TAP compared to the P configuration. In fact, it is such a discrepancy of conductance between P and AP configurations that gives rise to an obvious MR effect in our considered MCSH. The magnetoresistance ratio (MRR ) versus the Fermi energy () is shown in the inset. A large MRR , especially in the lowenergy region, can be observed evidently from this inset, namely, a considerable MR effect still exists in the MCSH device embedded a δdoping. This is because for a MCSH MR effect originates from the difference of transmission or conductance between P and AP alignments, which is independent of whether or not a δ-doping is embedded in the MCSG. However, the δ-doping will impact on the degree of MR effect (i.e.MRR ), as the effective potential Ueff experienced by electrons in the MCSH depends on the δ-doping [see Eq. (3)]. In the following, we explore how a δ-doping influences degree of MR effect for our considered MCSH device as shown in Fig. 1. Quantitative results of both weight and position of the δ-doping effects on MRR for the MCSH device will be presented detailedly. To begin with, we fix the position of the δ-doping to discuss its weight impact on MR effect. Calculated results are shown in Fig. 4, where the δ-doping is assumed to be located in x 0 = 0.5. Fig. 4(a) gives MRR versus the Fermi energy for three δ-doping weights: V = 0.0 (solid curve), 1.0 (dashed curve) and 2.0 (dotted curve). This figure clearly shows the apparent modulation of the weight of the δ-doping to the magnetoresistance ratio MRR . The MRR curve moves upwards as the weight V increases in energy ranges of 5 EF 9.5 and 14 EF 23, while it moves down in other energy ranges. In addition, with the increase of the Fermi energy the MRR gradually reduces. Influence of the weight of δ-doping on the MR effect can be seen more clearly from
Fig. 2. Transmission coefficients of electrons tunneling P and AP configurations versus incident energy for given longitudinal wave vector and δ-doping. /2
G (EF ) = G0
T (EF , /2
2EF sin )cos d ,
(8)
where G0 = 2e 2m vF L y / h2 , is incident angle relative to x -direction, vF is the Fermi velocity, and L y is length of the device in y -direction. In numerical calculations, GaAs is taken as 2DEG material, which m = 0.067m 0 and g = 0.44 and leads to B = 81.3nm and E0 = 0.17meV for B0 = 0.1T . Partial structural parameters are set as B1 = 4.0 , B2 = 3.0 , U = 5.0 , L = 2.0 and d = 0.4 . Previous investigation [19] reported that our considered MCSH possesses a considerable MR effect due to the significant difference of conductance between P and AP configuration. Now, we wonder whether it still has a MR effect when a δ-doping is embedded inside. In order to answer this question, as a function of incident energy Fig. 2 plots transmission coefficients for electrons tunneling through (a) P and (b) AP configurations, where a δ-doping is 2.0 (x 0.5) and k y = 2.0 (solid curve), 0.0 (dashed curve) and 2.0 (dotted curve). A significant anisotropy with longitudinal wave vector is clearly shown in this figure, which is due to an essentially two-dimensional process for electrons across a MCSH. For the P alignment, several resonance peaks with unity value appear in low-energy region at the case of k y 0 . However, when the system turns to AP alignment, these peaks almost disappear, which 3
Vacuum 169 (2019) 108891
D.-H. Liang, et al.
Fig. 5. Magnetoresistance ratio as a function of the Fermi energy or position of δ-doping, where weight of δ-doping is set to be V = 5.0 .
Fig. 4. Magnetoresistance ratio varies with the Fermi energy and weight of δdoping, respectively, where position of δ-doping is fixed at x 0 = 0.5 .
x 0 on MRR is associated closely to the Fermi energy —MRR gradually decreases with increasing the Fermi energy. Therefore, we also can control the MCSH-based MR device by tuning position of -doping and achieve a position-manipulable MR device. In summary, we have theoretically studied how to control a MCSHbased MR device by δ-doping. This MCSH can be realized experimentally by patterning two asymmetric FM stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure; while δ-doping can be embedded inside by atomic layer doping technique. An obvious MR effect still exists in the MCSH device even if a δ-doping is comprised inside, because significant difference in transmission or conductance between P and AP configurations is independent of existence of the doping. However, the δ-doping can modify magnetoresistance ratio of the MCSH device by adjusting weight or position of δ-doping. Therefore, these interesting results obtained in this work not only propose an alternative way to manipulate MCSH-based MR devices, but also a structurally-controllable MR device can be achieved for magnetic information storage.
Fig. 4 (b), where variation of MRR with V is presented directly for three Fermi energies: EF = 10.0 (solid line), 15.0 (dashed line) and 20.0 (dotted line). These three curves shown that MRR is altered greatly by V , i.e., δ-doping's weight has an evident effect on MR effect. Moreover, this modulation is related to Fermi energy , e.g., for EF = 10.0 the V has larger effect than those for EF = 15.0 and 20.0 . Clearly, this modulation results from V -dependence of effective potential Ueff of electrons in the MCSH device. On the other hand, it means that the MR device shown in Fig. 1 can be manipulated by adjusting weight of δ-doping, giving rise to a controllable MR device for magnetic information storage. Eq. (3) shows that effective potential (Ueff ) felt by electrons in the MCSH device depends on not only weight (V ) but also position ( x 0 ) of the δ-doping. Therefore, the position of the δ-doping also generates an impact on the MR device, as confirmed by Fig. 5, where weight of the δdoping is taken as V = 2.0 . In Fig. 5 (a), we plot MRR versus the Fermi energy for three different δ-doping positions: x 0 = 0.5 (solid curve), 1.0 (dashed Curve) and 1.5 (dotted curve). Indeed, from this figure we can see that MRR is closely related to x 0 of δ-doping: MRR changes complicatedly with for different positions of δ-doping. Fig. 5 (b) directly gives MRR as a function of position of the δ-doping for three different Fermi energies: EF = 6.0 (solid curve), 9.0 (dashed curve) and 12.0 (dotted curve). It can be clearly seen that MRR varies drastically with position ( x 0 ) of the -doping. In other words, position of δ-doping has a large impact on the MR effect in the MCSH device. Besides, influence of
Funding This work was supported by the Nation Natural Science Foundation of China (Grant No. 11864009).
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Vacuum 169 (2019) 108891
D.-H. Liang, et al.
References
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5
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Short communication
Controllable magnetoresistance effect in a δ-doped and magneticallyconfined semiconductor heterostructure
T
Dong-Hui Liang, Mao-Wang Lu∗, Xin-Hong Huang, Meng-Rou Huang, Zeng-Lin Cao College of Science, Guilin University of Technology, Guilin, 541004, People's Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Magnetically confined semiconductor heterostructure δ-doping Magnetoresistance effect Magnetoresistance ratio Controllable magnetoresistance device
A magnetoresistance device was proposed by patterning two asymmetric ferromagnetic stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure. To control its performance, a tunable δ-potential is introduced inside this device by wire-like doping. Adopting transfer matrix method and Landauer-Büttiker conductance theory, Schrödinger equation is solved analytically and δ-doping dependent transmission, conductance and magnetoresistance ratio are calculated. An obvious magnetoresistance effect still stays in this device, due to existence of transmission difference between parallel and antiparallel configurations independent of δ-doping. However, its magnetoresistance ratio can be tuned by changing weight or position of δ-doping. These interesting results not only propose an alternative scheme to manipulate magnetoresistance device, but also a structurallycontrollable magnetoresistance device can be obtained for magnetic information storage.
1. Introduction Magnetoresistance (MR) effect [1] has attracted much attention in recent years, due to its scientific significance and practical application in magnetic information storage such as ultrasensitive magnetic field sensor, random access memorie, and read head [2–5]. In general, MR effect is observed in a sandwich-like structure, where adjacent ferromagnetic (FM) layers are separated by a thin nonmagnetic layer. Because of the spin-dependent carriers’ scattering, conductance of such a heterogeneous structure strikingly drops, when magnetizations of neighboring magnetic layers are switched from parallel (P) alignment to antiparallel (AP) one under an external magnetic field. Degree of MR effect is hence characterized by so-called magnetoresistance ratio (MRR), which can be usually defined as MRR = (GP GAP )/GAP or (GP GAP )/(GP + GAP ) with and GAP standing for conductance of P and AP configurations, respectively. For a MR device, a high MRR at a relatively low switching magnetic field is desired in practical application. An alternative scheme to achieve a large MRR is to exploit the so-called magnetically confined semiconductor heterostructure (MCSH) [6], which is often constructed on a semiconductor heterostructure by restricting motion of two-dimensional electron gas (2DEG) with an inhomogeneous magnetic field on nanometer scale. A MCSH is actually a type of hybrid semiconductor nanostructure consisting of magnetic materials and semiconductors, where the former provides an inhomogeneous magnetic field locally influencing motion of electrons in
∗
the latter [7]. Because of small size, low dimensionality and quantum magnetic confinement, there are some of strange quantum effects in the MCSH, one of which, e.g., is interesting MR effect. MCSH-based MR effect can be used to develop new-typed MR devices for magnetic information storage. In 2002, Zhai et al. [8] first studied MR effect by considering a simplified MCSH comprising of δ-function magnetic barriers (MBs), which can be fabricated by patterning two FM stripes in parallel configuration on top of a semiconductor heterostructure. In physics mechanism, they found that differing from metal-based MR effect this kind of MR effect makes no use of spin degree of freedom, originating from transmission discrepancy between P and AP configuration. On the other hand, it possesses a very high MRR (~106% ) at a relatively low switching magnetic field, which may be an ideal candidate for magnetic information storage. However, their results obtain from ideal δ-MBs may deviate from practical cases. Subsequently, Lu ea al [9]. took realistic magnetic profiles instead of simplified MCSH to explore MR effect in MCSH, and supported Zhai et al.’s findings. Since then, MR effect in other MCSHs were reported in sequence and corresponding MR devices were put forwards successfully; see partial references [10–19]. Edified by the atomic layer doping technique [20] such as molecular beam epitaxy (MBE) and metal-organic chemical-vapor deposition (MOCVD), in 2015 Kong et al. [21] proposed to manipulate MCSHbased MR devices via δ-doping. They found that MRR is closely related to position or weight of the δ-doping in a δ-function-shaped MB MCSH.
Corresponding author. E-mail address: [email protected] (M.-W. Lu).
https://doi.org/10.1016/j.vacuum.2019.108891 Received 23 June 2019; Received in revised form 19 August 2019; Accepted 23 August 2019 Available online 29 August 2019 0042-207X/ © 2019 Elsevier Ltd. All rights reserved.
Vacuum 169 (2019) 108891
D.-H. Liang, et al. 2
H=
(P + eA ) + U (x ) + V (x 2m
eg Z BZ (x ), 2m 0 2
x 0) +
(1)
where m , m 0 , and g are effective mass, free electron mass, and effective Landé factor, respectively, p is momentum vector, Z = ±1
corresponds to spin-up/down electron, and A = [0, Ay (x ), 0] with Ay (x ) = B1 (x d) (L x ) + ( B2 B1) (x )(d x ) is magnetic vector potential given in the Landau gauge. By introducing cyclotron /eB0 with B0 being some typical frequency and magnetic length B = magnetic field, all relevant quantities are written as dimensionless x B and Ay (x ) B0 B Ay (x ) . form, e.g., x For an electron in 2DEG, motion in y -direction is conserved, thus its wave function can be written as (x , y ) = e iky y (x ) , where k y is longitudinal wave vector. The wave function (x ) satisfies reduced onedimensional (1D) Schrödinger equation n (x )
+ 2[E
(2)
Ueff (x )] (x) = 0
where the effective potential experienced by electrons in the MR device can be written as
Ueff (x ) = [k y + Ay (x )]2 /2 + V (x
x 0) +
g z Bz (x ) 4m 0
(3)
The last term in Ueff represents the Zeeman coupling between electronic spin Z and magnetic field BZ (x ) . Comparing to other items, absolute value of such a term is much smaller for GaAs materials. Therefore, spin-dependent term plays a minor role in determining electronic transport properties and thus can be omitted reasonably. Clearly, this effective potential depends on longitudinal wave vector k y , magnetic configuration BZ (x ) , and δ-doping V (x x 0 ) . By employing transfer matrix method (TMM) [30], we can exactly solve Eq. (2). In incident and outgoing regions, wave functions can be exp( ikl x ) , assumed as , and x<0 l (x ) = exp(ikl x ) + exp(ikr x ) , x > L , where kl = kr = 2E k y 2 , and / is rer (x ) = flection/transmission amplitude. In device region of 0 < x < L , wave function can be expressed as
Fig. 1. (a) MCSH-based MR device, (b) P and (c) AP alignments, where two asymmetric FM stripes are patterned on top and bottom of GaAs/AlxGa1-xAs heterostructure and a δ-doping V (x x 0) is embedded inside.
Later, Lu et al. [22] extended to realistic MB-type MCSH and found a sizeable MR effect even if a δ-doping is included inside. They also confirmed that MRR can be tuned by the δ-doping. Subsequently, several groups explored modulation of MR effect in other MCSH-based MR devices; cf. relevant Refs. [23–27]. By pattering two asymmetric FM stripes on top and bottom of the semiconductor heterostructure, Wang et al. [19] proposed a new MR device, and studied influences of magnetic field, external electric barrier (EB) and temperature on MR effect. To control this MR device, in the present work we introduce a δ-doping into this device with the help of atomic layer doping technique. Adopting theoretical analysis combined with numerical calculations, we theoretically explore impact of the δ-doping on MR effect and a controllable MR device is put forward for magnetic information storage. Our considered MR device [19] is shown schematically in Fig. 1(a), which can be experimentally realized [28] by depositing two asymmetric nanosized FM stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure, respectively. Fig. 1(b) and (c) represent the P and AP alignments of this device, respectively, where a δ-doping with tunable weight V and changeable position x 0 is contained in this device. It should be noted that, such a δ-doping is produced from the “wire”-like doping not the “layer”-doping [29]. For a very small distance between FM stripes and 2DEG, magnetic field distribution generated by magnetized FM stripes can be approximated as, where B1 and B2 are magnetic field strength, L or d is the width of FM stripe, and represents magnetization configurations ( ± 1 or P and AP). When a negative voltage ( Vg ) is applied on the upper metallic FM stripe, a rectangular electric barrier (EB) U (x ) = U0 (x ) (d x ) is induced within 2DEG, where U0 is EB height and (x ) is the Heaviside step function. In the framework of single particle and effective mass approximation, the motion of an electron in the 2DEG assumed to be located at xy -plane can be described by the Hamiltonian,
in (x ) =
C1 exp(ik1 x ) + D1 exp( ik1 x ), 0 < x < d C2exp(ik2 x ) + D2 exp( ik2 x ), d < x < x 0 , C3 exp(ik3 x ) + D3 exp( ik3 x ), x 0 < x < L
(4)
where k1 = 2(E U ) (k y + B2 B1 k2 = k3 = 2E (k y B1) 2 , and Ci and Di (i = 1,2,3) are constants. According to continuity of wave function at boundaries, we can get
)2 ,
exp(ikl x 0 ) ikl exp(ikl x 0 ) =M
exp( ikl x 0 ) ikl exp( ikl x 0 )
exp(ikr xL +) ikr exp(ikr xL +)
1
exp( ikr xL +) ikr exp( ikr xL +)
0
,
(5)
where m11 m12 M= m m = M1 × M2 × M3 21 22 = ×
cos k1 d k1 sin k1 d
cos k2 (x 0 d) + 2V sin k2 (x 0 d )/ k2 sin k1 d/ k1 × cos k1 d k2 sin k2 (x 0 d ) 2V cos k2 (x 0 d )
cos k3 (L x 0) k3 sin k3 (L x 0)
sin k3 (L cos k3 (L
sin k2 (x 0 d )/ k2 cos k2 (x 0 d )
x 0)/ k3 . x 0)
(6) Therefore, we can obtain transmission coefficient for an electron with incident energy E and longitudinal wave vector k y , 2
T (E , k y ) = 1
m11 kr m22 / kl + i (kr m12 + m21/ kl ) . m11 + kr m22 / kl + i (kr m12 m21/ kl )
(7)
At zero temperature, ballistic conductance can be calculated from well-known Landauer-Büttiker theory [31]. 2
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Fig. 3. Conductance varies with the Fermi energy, where δ-doping is the same as in Fig. 2. As a function of the Fermi energy, magnetoresistance ratio is plotted in the inset.
can be understood from the fact that a non-symmetric multiple barriers/wells structure in effective potential Ueff for the AP configuration makes electronic transmission incomplete. More importantly, a great difference of transmission between P and AP configurations exists in the device. In contrast to the P configuration, TAP spectrum moves to the high-energy region and its value is greatly suppressed, which is because the effective potential Ueff has a strong dependence on the magnetic configuration of the MR device. Fig. 3 shows the Fermi energy () dependence of conductance (dashed line) and GAP (dotted line) for electrons tunneling through P and AP configurations, where both structural parameters and δ-doping are the same as in Fig. 2. From this figure, we can clearly see a distinct difference of conductance between P and AP configurations, which is because conductance is obtained by integrating transmission coefficient over all incident angles. For AP alignment, conductance is strongly suppressed, because of greatly reduced TAP compared to the P configuration. In fact, it is such a discrepancy of conductance between P and AP configurations that gives rise to an obvious MR effect in our considered MCSH. The magnetoresistance ratio (MRR ) versus the Fermi energy () is shown in the inset. A large MRR , especially in the lowenergy region, can be observed evidently from this inset, namely, a considerable MR effect still exists in the MCSH device embedded a δdoping. This is because for a MCSH MR effect originates from the difference of transmission or conductance between P and AP alignments, which is independent of whether or not a δ-doping is embedded in the MCSG. However, the δ-doping will impact on the degree of MR effect (i.e.MRR ), as the effective potential Ueff experienced by electrons in the MCSH depends on the δ-doping [see Eq. (3)]. In the following, we explore how a δ-doping influences degree of MR effect for our considered MCSH device as shown in Fig. 1. Quantitative results of both weight and position of the δ-doping effects on MRR for the MCSH device will be presented detailedly. To begin with, we fix the position of the δ-doping to discuss its weight impact on MR effect. Calculated results are shown in Fig. 4, where the δ-doping is assumed to be located in x 0 = 0.5. Fig. 4(a) gives MRR versus the Fermi energy for three δ-doping weights: V = 0.0 (solid curve), 1.0 (dashed curve) and 2.0 (dotted curve). This figure clearly shows the apparent modulation of the weight of the δ-doping to the magnetoresistance ratio MRR . The MRR curve moves upwards as the weight V increases in energy ranges of 5 EF 9.5 and 14 EF 23, while it moves down in other energy ranges. In addition, with the increase of the Fermi energy the MRR gradually reduces. Influence of the weight of δ-doping on the MR effect can be seen more clearly from
Fig. 2. Transmission coefficients of electrons tunneling P and AP configurations versus incident energy for given longitudinal wave vector and δ-doping. /2
G (EF ) = G0
T (EF , /2
2EF sin )cos d ,
(8)
where G0 = 2e 2m vF L y / h2 , is incident angle relative to x -direction, vF is the Fermi velocity, and L y is length of the device in y -direction. In numerical calculations, GaAs is taken as 2DEG material, which m = 0.067m 0 and g = 0.44 and leads to B = 81.3nm and E0 = 0.17meV for B0 = 0.1T . Partial structural parameters are set as B1 = 4.0 , B2 = 3.0 , U = 5.0 , L = 2.0 and d = 0.4 . Previous investigation [19] reported that our considered MCSH possesses a considerable MR effect due to the significant difference of conductance between P and AP configuration. Now, we wonder whether it still has a MR effect when a δ-doping is embedded inside. In order to answer this question, as a function of incident energy Fig. 2 plots transmission coefficients for electrons tunneling through (a) P and (b) AP configurations, where a δ-doping is 2.0 (x 0.5) and k y = 2.0 (solid curve), 0.0 (dashed curve) and 2.0 (dotted curve). A significant anisotropy with longitudinal wave vector is clearly shown in this figure, which is due to an essentially two-dimensional process for electrons across a MCSH. For the P alignment, several resonance peaks with unity value appear in low-energy region at the case of k y 0 . However, when the system turns to AP alignment, these peaks almost disappear, which 3
Vacuum 169 (2019) 108891
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Fig. 5. Magnetoresistance ratio as a function of the Fermi energy or position of δ-doping, where weight of δ-doping is set to be V = 5.0 .
Fig. 4. Magnetoresistance ratio varies with the Fermi energy and weight of δdoping, respectively, where position of δ-doping is fixed at x 0 = 0.5 .
x 0 on MRR is associated closely to the Fermi energy —MRR gradually decreases with increasing the Fermi energy. Therefore, we also can control the MCSH-based MR device by tuning position of -doping and achieve a position-manipulable MR device. In summary, we have theoretically studied how to control a MCSHbased MR device by δ-doping. This MCSH can be realized experimentally by patterning two asymmetric FM stripes on top and bottom of the GaAs/AlxGa1-xAs heterostructure; while δ-doping can be embedded inside by atomic layer doping technique. An obvious MR effect still exists in the MCSH device even if a δ-doping is comprised inside, because significant difference in transmission or conductance between P and AP configurations is independent of existence of the doping. However, the δ-doping can modify magnetoresistance ratio of the MCSH device by adjusting weight or position of δ-doping. Therefore, these interesting results obtained in this work not only propose an alternative way to manipulate MCSH-based MR devices, but also a structurally-controllable MR device can be achieved for magnetic information storage.
Fig. 4 (b), where variation of MRR with V is presented directly for three Fermi energies: EF = 10.0 (solid line), 15.0 (dashed line) and 20.0 (dotted line). These three curves shown that MRR is altered greatly by V , i.e., δ-doping's weight has an evident effect on MR effect. Moreover, this modulation is related to Fermi energy , e.g., for EF = 10.0 the V has larger effect than those for EF = 15.0 and 20.0 . Clearly, this modulation results from V -dependence of effective potential Ueff of electrons in the MCSH device. On the other hand, it means that the MR device shown in Fig. 1 can be manipulated by adjusting weight of δ-doping, giving rise to a controllable MR device for magnetic information storage. Eq. (3) shows that effective potential (Ueff ) felt by electrons in the MCSH device depends on not only weight (V ) but also position ( x 0 ) of the δ-doping. Therefore, the position of the δ-doping also generates an impact on the MR device, as confirmed by Fig. 5, where weight of the δdoping is taken as V = 2.0 . In Fig. 5 (a), we plot MRR versus the Fermi energy for three different δ-doping positions: x 0 = 0.5 (solid curve), 1.0 (dashed Curve) and 1.5 (dotted curve). Indeed, from this figure we can see that MRR is closely related to x 0 of δ-doping: MRR changes complicatedly with for different positions of δ-doping. Fig. 5 (b) directly gives MRR as a function of position of the δ-doping for three different Fermi energies: EF = 6.0 (solid curve), 9.0 (dashed curve) and 12.0 (dotted curve). It can be clearly seen that MRR varies drastically with position ( x 0 ) of the -doping. In other words, position of δ-doping has a large impact on the MR effect in the MCSH device. Besides, influence of
Funding This work was supported by the Nation Natural Science Foundation of China (Grant No. 11864009).
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