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Effects of ferromagnetic and Schottky metal stripes on electronic transport in a GMR device
Accepted Manuscript Effects of ferromagnetic and schottky metal stripes on electronic transport in a GMR device
X.W. Zhang, H.L. He, Y.L. Liu PII:
S0921-4526(18)30735-X
DOI:
10.1016/j.physb.2018.11.033
Reference:
PHYSB 311168
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
19 October 2018
Accepted Date:
17 November 2018
Please cite this article as: X.W. Zhang, H.L. He, Y.L. Liu, Effects of ferromagnetic and schottky metal stripes on electronic transport in a GMR device, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.11.033
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Effects of ferromagnetic and schottky metal stripes on electronic transport in a GMR device X. W. Zhang*, H. L. He, Y. L. Liu
State Key Laboratory of Environment-Friendly Energy Materials, Southwest University of Science and Technology, Mianyang 621010, China.
Abstract: Based on transfer matrix method, the giant magnetoresistance (GMR) effect of 2-dimensional electron gas (2DEG) under magnetic and electrical potential is investigated. Such system can be obtained easily by depositing two ferromagnetic (FM) stripes and one schottky metal (SM) on the surface of semiconductor heterostructure. Our study shows a great difference in electrons tunneling through parallel (P) and antiparallel (AP) magnetization of FM stripes. The corresponding magnetoresistance ratio (MRR) depends strongly on the magnetic intensity of magnetic barrier (MB) and the height (U(x)) of electric barrier (EB). Furthermore, we also study the effect of SM stripe width (ds) on MRR for a given value of U(x). It indicates that the MRR of the device is sensitive to the ds. These results may offer an alternative to get a tunable GMR device by adjusting U(x) and width of the SM stripe. Keywords: GMR; 2-dimensional electron system; transfer matrix method; inhomogeneous magnetic field; Conductance
1
ACCEPTED MANUSCRIPT
1. Introduction In the modulation of magnetic and electric fields, the electronic transport properties on 2DEG has extensively studied during the past few decades[1-15]. In experimental, depositing and then magnetizing two FM stripes on the top of 2DEG, the conductance would be greatly suppressed. More importantly, assembling the parallel or antiparallel magnetization in these stripes, the conductance of the system shows a great difference, which is also called as GMR effect[7-10]. Because of its attractive actual application, such as magnetic information storage, ultrasensitive magnetic field sensors, random access memories, many studies on this effect continue to spring up experimentally and theoretically. More recently, a SM stripe is placed on the surface of 2DEG to provide additional electrical potential[1, 5, 13-15]. These studies show that the magnetoresistance effect would be significantly modified. In this work, we consider a system showed in Fig. 1(a) which can be easily obtained in experiment by modern micromachining technology. The electrical potential can be provided by applying bias voltage on the SM stripe. Our goal here is to adjust the GMR via changing the bias voltage and the width of SM (ds). We must mention that the magnetic field is considered usually as an ideal function in most previous studies, such as δ-function[1, model
[18].
7, 16],
sine-function[17], and Kronig–Penney
From our calculation for the magnetic field profiles as showed in
Fig.1(b,c), these simulation functions deviate from the actual situation to some extent. Therefore, we here use the calculated magnetic field profiles directly for the numerical investigation on MRR. To evaluate the transport properties, we employ the 2
ACCEPTED MANUSCRIPT transfer matrix technique which is very powerful and reliable in computational physics.
2. Theoretical model Now for the presented model showed in Fig.1(a), the single-electron Hamiltonian can be described readily as follow
H
2 1 2 1 2 U ( x) , p e A U ( x ) p p e A ( x ) x y * * 2m 2m
(1)
where m* and e is the effective mass and the charge of an electron in 2DEG, respectively. The U(x) represents for the electric potential induced by the applied voltage on SM stripe. To determinate the magnetic field, we assume the FM stripes are much longer then the width of 2DEG. Now the vertical component of stray magnetic field at the plane of 2DEG can be readily calculated by the fundamental equations of magnetostatics [19-23] as follow 2 x d / 2 d 2 x d / 2 d z0 d z0 d B( x) B0 (2) 2 2 2 2 2 2 2 2 x d / 2 z x d / 2 z x d / 2 z x d / 2 z 0 0 0 0
with B0
Mh . Here M is the x-component of the magnetization in the FM stripes d
Using the Landau gauge, the vector potential is written as A 0, Ay , 0 in this work, x
in which can be deduced straightforward from the formula Ay x = B x dx [24]. For 0
clarity, the computed magnetic profiles for the parallel (P) and antiparallel (AP) configurations of two FM stripes are depicted in Fig. 1(b) and (c) respectively. The electric potential is represented as U(x) as well. According to our assumption (the system is very large in y-direction), the eigenfunction of H determined by Eq.(1) can be reduced to its one-dimensional 3
ACCEPTED MANUSCRIPT version[16] 2 2 d 2 2 e - * 2 + * k y Ay ( x ) U ( x ) ( x ) E ( x ) , 2m dx 2m
(3)
where ky is the wave vector in y-direction. The two-dimensional wave function is written as x, y e
x . The effective potential binding the electrons now is
ik y y
viewed as 2
2 e Veff k Ay ( x ) U ( x ) . * y 2m
(4)
Very clearly, this effective potential is strongly dependent not only on the wave vector ky and the vector potential Ay(x), but also on the EB height U(x). In Fig.3, we visualize 3-dimensional plots of the effective potential Veff as P and AP alignments with the wave vector from k y 10 m 1 to k y 10 m 1 , respectively. For evaluating the electronic transport properties of the system, the transfermatrix technique is applied. The detailed basic idea of transfer-matrix technique can be found in references 19 and 25. Here we only display the main formulae. The total transfer matrix of our system can be written as follow 1 k j ik j D 1 e N N 2 k j 1 M M j j 0 j 0 1 k j ik j D 1 2 k e j 1
k j ik j D 1 1 e 2 k j 1 . k j ik j D 1 1 e 2 k j 1
(5)
Sub-matrix Mj connects the wave functions in the neighbor segments j and j+1. For an electron with given energy, its transmission coefficient is calculated by the expression 2
2
T ( E , k y , U ) t 1 R 1 M 21 M 22 .
(6)
The ballistic conductance, which is an observable quantity, can be reached at zero 4
ACCEPTED MANUSCRIPT temperature from the well-known Landauer-Büttiker formula[26] G G0
2
2
T EF , 2 EF sin , U cos d ,
(7)
which is an integral over a half plane of Fermi surface and G0 e 2 mvF l 2 with l and vF is the length of the structure in y-direction and the Fermi velocity. 3. Results and discussion In Fig.2, we show 3-dimensional transmission coefficients in the plane of (ky, E) for P (upper row) and AP (bottom row) configurations with different EB height U(x). The structure parameters can be referred from the caption of Fig.1. Apparently, these two groups of transmission exhibit significant difference. The AP alignment has symmetrical transmission spectra about ky=0, while P alignment hasn’t. If comparing carefully their corresponding effective potentials, one can find that this is a reasonable result. As can be seen from Fig.(3), the distribution of effective potential is symmetrical about the center in AP configuration and this symmetry is independent on the EB height. It means that incident electrons with positive or negative ky can feel the same effective potential. This immediately causes the symmetrical transmission spectra. However the situation becomes quite different in P configuration. The magnetic barriers at the region of positive ky are much larger than those with negative ky, which can be clearly seen in Fig.3(a). Undoubtedly, an incident electron with positive ky is more difficult to tunnel the device. The computed transmission spectra showed in the upper row of Fig.2 also confirm this analysis. Only some resonant peaks can be noticed at low energy region. On the contrary, the electrons with negative ky possess 5
ACCEPTED MANUSCRIPT wider transmission peak at low energy region. Additionally, with increasing of electric barrier, the transmission suppression is more and more significant. For A configuration (see the upper low of Fig.2), the second and the third resonant peaks get close to each other and shift to the high energy region, while for AP configuration (see the bottom low of Fig.2), the resonant peaks with unity value around ky=0 get very narrow at high EB value. According to the Eq.(7), one can imagine that the observable quantity, conductance, will be reduced greatly because it is an integral over half plane of Fermi surface. To confirm this point, we calculate the conductance G by using Eq.(7), which is showed in Fig.4. For highlighting the effect of electric barrier, we fix the magnetization M of FM stripes and change the electric barrier U from 0 to 3 meV. As expected, the conductance has been suppressed dramatically with increasing of U(x) for both configurations. At low energy region, only some transport peaks can be spotted. But an obvious difference in conductance between P and AP configurations can be also clearly observed. Concretely, the conductance suppression is more significantly as AP configuration and almost no conductance can be seen when the EF is lower than 4meV except some sharp transport peaks. These sharp peaks originate from the narrow transmission peaks around ky=0 (see Fig.2(d)-(f)). For P configuration, the transport peaks at the low energy range are wider and are shrinking with electric barrier growing. But interestingly, the locations of these transport peaks remain roughly unchanged. This feature may bring us to get a considerable conductance at the given Fermi energy without worrying about the change of electric 6
ACCEPTED MANUSCRIPT barrier’s height. The great difference in conductance between A and AP configurations will give us an opportunity to make a GMR device in the corresponding energy range. We next consider the MRR of our system to further understand such conductance difference. In this work, the MRR is defined as MRR = (GP−GAP)/GAP×100%, with GP and GAP the conductance of P and AP configurations respectively. The calculated 3-dimensional plot of lg(MRR) in the (EF, U) plane is shown in Fig.5. Obviously, one can get very high value of MRR at the range of EF<1.5meV where MRR is up to 106 (see inset(a)). This will make it easier for us to prepare a high MRR device. However, low Fermi energy requires low electron concentration and hence reduces the conductance of our system. Fig.4 also shows the same results. When we move EF to higher energy region (1.5meV
ACCEPTED MANUSCRIPT EB, the MRR peak moves to high energy region. The inset(b) also shows this result clearly. That means, for a given GMR device like this, in which the electron concentration is fixed and magnetic stripes can’t be re-deposited, we can adjust the value of MRR by changing the EB height U(x). The results above offer an alternative to get a tunable GMR device which can be controlled by adjusting the voltage on SM stripe and Fermi energy EF. The above investigation tells us the effect of the EB height on MRR. In the following, we discuss how the width of SM stripe affects the transport properties of the system. We choose U=1.0 meV as applied electric potential and then let the width of SM stripe ds equals to 50, 100, and 150nm respectively. The computed results are showed in Fig.6. Clearly, as ds increases, the conductance curves shift towards high energy for both A and AP configurations, as shown in Fig.6(a) and (b). This is a very reasonable result because the electric barrier gets wider so that the electronic transmission coefficient gets smaller. Correspondingly, the MRR peaks move evidently to high energy region. Meanwhile, the amplitude of these peaks only decrease slightly (see Fig.6(c)). This fact lets us have opportunity to get some new systems which possess both high electron concentration (hence high conductance) and high MRR. However, the peak width of MRR as ds=150nm is relatively narrow. It requires us to prepare the sample very carefully so that the Fermi energy just sits in the corresponding range.
4. Summary In summary, we investigated the transport properties and GMR effect of 2DEG 8
ACCEPTED MANUSCRIPT under a magnetic and electrical potential. Our system exhibits very different transmission suppression between P and AP configurations at low energy range. It induces the significant GMR effect because the system shows a very high MRR (up to 106). When EB increases, the MRR peaks shift towards higher energy region, while the value of MRR decreases quickly. When the width of SM stripe increases and the EB remains unchanged, the peak width of MRR becomes narrower and the amplitude only decreases slightly. That means the structure parameters and the electric barrier must be synergistically considered. We hope these results offer an alternative to get a tunable GMR device which can be controlled by adjusting EB height U(x) and Fermi energy EF.
Acknowledgements. This work is supported by the National Natural Science Foundation of China under Grant No. 11304255.
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ACCEPTED MANUSCRIPT
References [1] Y. L. Zhou, M. W. Lu, X. L. Cao, X. H. Huang, M. R. Huang, D. H. Liang, Appl. Phys. A 124 (2018) 705. [2] L. H. Shen, W. Y. Ma, G. X. Liu, J. Magn. Magn. Mater. 40 (2016) 231-235. [3] J. Han, J. Hu, Y. Ouyang, S. X. Wang, J. He, IEEE T Ind. Electron. 62 (2015) 516-524. [4] T. Seki, H. Yako, T. Yamamoto, T. Kubota, Y. Sakuraba, M. Ueda, J. Phys. D: Appl. Phys. 48 (2015) 164010. [5] M. Zhao, J. D. Lu, S. J. Peng, H. Y. Liu, Int. J. Mod. Phys. B 32 (2018) 1850186. [6] M. W. Lu, S. Y. Chen, X. H. Huang, G. L. Zhang, IEEE J. Electron Devi. 6 (2018) 227-232. [7] G. Papp, F. M. Peeters, J. Appl. Phys 100 (2006) 043707. [8] N. Overend, A. Nogaret, B. L. Gallagher, P. C. Main, Appl. Phys. Lett. 72 (1998) 1724-1726. [9] F. Zhai, Y. Guo, B. L. Gu, Phys. Rev. B 66 (2002) 125305. [10] Y. H. Kong, M. W. Lu, W. H. Tang, C. S. Li, G. L. Zhang, Solid State Commun. 142 (2007) 143-147. [11] P. Hoodbhoy, J. Phys: Condens. Matter. 18 (2018) 185303. [12] M. W. Lu, G. J. Yang, Phys. Lett. A 362 (2007) 489-493. [13] Y. H. Kong, S. Y. Chen, G. L. Zhang, J. Nanomater. 10 (2013) 257-264. [14] Y. Wang, N. F. Chen, X. W. Zhang, Phys. Lett. A 373 (2009) 1983-1987. [15] Y. H. Kong, S. Y. Chen, A. H. Li, Vacuum 122 (2015) 43-46. 10
ACCEPTED MANUSCRIPT [16] I. S. Ibrahim, F. M. Peeters, Phys. Rev. B 52 (1995) 17321-17334. [17] D. Buchholz, P. S. Drouvelis, P. Schmelcher, Phys. Rev. B 73 (2006) 235346. [18] L. Dell’Anna, A. De Martino, Phys. Rev. B, 79 (2009) 045420. [19] X. W. Zhang, S. Y. Mou, B. Dai, J. Appl. Phys. 114 (2013) 023706. [20] A. Nogaret, J. Phys: Condens. Matter. 22 (2010) 253201. [21] H. L. He, X. W. Zhang, Z. P. Wang, B. Dai, Y. Ren, Aip Adv. 6 (2016) 14881495. [22] B. Dai, X. X. Liu, Y. Lei, A. Nogaret, Chin. Phys. Lett. 26 (2009) 037202. [23] A. G. Pogosov, M. V. Budantsev, E. Y. Zhdanov, D. A. Pokhabov, A. K. Bakarov, A. I. Toropov, Appl. Phys. Lett. 100 (2012) 181902. [24] A. Nogaret, J. C. Portal, H. E. Beere, D. A. Ritchie and C. Phillips, J. Phys.: Condens. Matter 21 (2009) 025303. [25] J. Q. You, L. Zhang, P. K. Ghosh, Phys. Rev. B 52 (1995) 17243-17247. [26] A. Matulis, F. M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 15181521.
11
ACCEPTED MANUSCRIPT Fig. 1. z
y x
Fig. 1. (a) A schematic illustration of the GMR device, in which two FM stripes and one SM stripe are placed on the top of the GaAs heterostructure. (b) and (c) correspond to the parallel (P) and antiparallel (AP) configurations alignments of the device, respectively. The parameters are taken as x-=-300nm, x+=300nm, z0 = 10nm, h = 20nm, dF = 100nm, ds = 50nm, a = 200nm and Mh=20T•nm.
12
ACCEPTED MANUSCRIPT Fig. 2.
Fig. 2. The 3-dimensional plots of transmission coefficient in the (E, ky) plane as (a) and (d) U = 0 meV, (b) and (e) U = 1.5 meV, (c) and (f) U = 3 meV, respectively. The structural parameters are the same as in Fig.1.
13
ACCEPTED MANUSCRIPT Fig. 3. 5
4
3
Veff
Veff
4
5
P
(a)
2
(b)
AP
3 2 1
0 10 5 0 -5 -10
0 10 5 0 -5 -10
-300 -200 -100 0
x(nm)
100 200 300
-1 )
-1 )
k y( m
k y( m
1
-300 -200 -100 0
100 200 300
x(nm)
Fig.3. (a) and (b) show the 3-dimensional plots of the effective potential Veff as P and AP alignments with the wave vector from k y 10 m 1 to k y 10 m 1 , respectively. The other structural parameters are the same as in Fig.1.
14
ACCEPTED MANUSCRIPT Fig. 4.
Fig.4. (a) and (b) show the 3-dimensional plots of the conductance G in the (EF, U) plane as P and AP configuration, respectively. The conductance is in unit of
G0 e 2 mvF l 2 . The structural parameters are the same as in Fig.1.
15
ACCEPTED MANUSCRIPT Fig. 5.
Fig. 5. The 3-dimensional plot of log(MRR) in the (EF, U) plane and the inset show the MRR as the function of the EF. The red dashed shows the log(MRR) varies with U at EF=3.5meV. The structural parameters are the same as in Fig.1.
16
ACCEPTED MANUSCRIPT Fig. 6.
G/G0
1
ds=50nm ds=100nm
U=1.0meV 0.4
G/G0
P
0.5 0
Log(MRR)
(a)
(b)
0 6
ds=50nm ds=100nm
AP
0.2
ds=150nm
U=1.0meV
ds=150nm
(c)
ds=50nm ds=100nm
U=1.0meV
4
ds=150nm
2 0
1
2
3
EF/E0
4
5
6
7
Fig.6. (a) and (b) are the conductances GP and GAP at a given EB height U = 1.0 meV as P and AP alignments, respectively. (c) shows the log(MRR) as a function of the Fermi energy EF for three different EB width: ds = 50nm, 100nm and 150nm. The other structural parameters are the same as in Fig.1.
17
X.W. Zhang, H.L. He, Y.L. Liu PII:
S0921-4526(18)30735-X
DOI:
10.1016/j.physb.2018.11.033
Reference:
PHYSB 311168
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
19 October 2018
Accepted Date:
17 November 2018
Please cite this article as: X.W. Zhang, H.L. He, Y.L. Liu, Effects of ferromagnetic and schottky metal stripes on electronic transport in a GMR device, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.11.033
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Effects of ferromagnetic and schottky metal stripes on electronic transport in a GMR device X. W. Zhang*, H. L. He, Y. L. Liu
State Key Laboratory of Environment-Friendly Energy Materials, Southwest University of Science and Technology, Mianyang 621010, China.
Abstract: Based on transfer matrix method, the giant magnetoresistance (GMR) effect of 2-dimensional electron gas (2DEG) under magnetic and electrical potential is investigated. Such system can be obtained easily by depositing two ferromagnetic (FM) stripes and one schottky metal (SM) on the surface of semiconductor heterostructure. Our study shows a great difference in electrons tunneling through parallel (P) and antiparallel (AP) magnetization of FM stripes. The corresponding magnetoresistance ratio (MRR) depends strongly on the magnetic intensity of magnetic barrier (MB) and the height (U(x)) of electric barrier (EB). Furthermore, we also study the effect of SM stripe width (ds) on MRR for a given value of U(x). It indicates that the MRR of the device is sensitive to the ds. These results may offer an alternative to get a tunable GMR device by adjusting U(x) and width of the SM stripe. Keywords: GMR; 2-dimensional electron system; transfer matrix method; inhomogeneous magnetic field; Conductance
1
ACCEPTED MANUSCRIPT
1. Introduction In the modulation of magnetic and electric fields, the electronic transport properties on 2DEG has extensively studied during the past few decades[1-15]. In experimental, depositing and then magnetizing two FM stripes on the top of 2DEG, the conductance would be greatly suppressed. More importantly, assembling the parallel or antiparallel magnetization in these stripes, the conductance of the system shows a great difference, which is also called as GMR effect[7-10]. Because of its attractive actual application, such as magnetic information storage, ultrasensitive magnetic field sensors, random access memories, many studies on this effect continue to spring up experimentally and theoretically. More recently, a SM stripe is placed on the surface of 2DEG to provide additional electrical potential[1, 5, 13-15]. These studies show that the magnetoresistance effect would be significantly modified. In this work, we consider a system showed in Fig. 1(a) which can be easily obtained in experiment by modern micromachining technology. The electrical potential can be provided by applying bias voltage on the SM stripe. Our goal here is to adjust the GMR via changing the bias voltage and the width of SM (ds). We must mention that the magnetic field is considered usually as an ideal function in most previous studies, such as δ-function[1, model
[18].
7, 16],
sine-function[17], and Kronig–Penney
From our calculation for the magnetic field profiles as showed in
Fig.1(b,c), these simulation functions deviate from the actual situation to some extent. Therefore, we here use the calculated magnetic field profiles directly for the numerical investigation on MRR. To evaluate the transport properties, we employ the 2
ACCEPTED MANUSCRIPT transfer matrix technique which is very powerful and reliable in computational physics.
2. Theoretical model Now for the presented model showed in Fig.1(a), the single-electron Hamiltonian can be described readily as follow
H
2 1 2 1 2 U ( x) , p e A U ( x ) p p e A ( x ) x y * * 2m 2m
(1)
where m* and e is the effective mass and the charge of an electron in 2DEG, respectively. The U(x) represents for the electric potential induced by the applied voltage on SM stripe. To determinate the magnetic field, we assume the FM stripes are much longer then the width of 2DEG. Now the vertical component of stray magnetic field at the plane of 2DEG can be readily calculated by the fundamental equations of magnetostatics [19-23] as follow 2 x d / 2 d 2 x d / 2 d z0 d z0 d B( x) B0 (2) 2 2 2 2 2 2 2 2 x d / 2 z x d / 2 z x d / 2 z x d / 2 z 0 0 0 0
with B0
Mh . Here M is the x-component of the magnetization in the FM stripes d
Using the Landau gauge, the vector potential is written as A 0, Ay , 0 in this work, x
in which can be deduced straightforward from the formula Ay x = B x dx [24]. For 0
clarity, the computed magnetic profiles for the parallel (P) and antiparallel (AP) configurations of two FM stripes are depicted in Fig. 1(b) and (c) respectively. The electric potential is represented as U(x) as well. According to our assumption (the system is very large in y-direction), the eigenfunction of H determined by Eq.(1) can be reduced to its one-dimensional 3
ACCEPTED MANUSCRIPT version[16] 2 2 d 2 2 e - * 2 + * k y Ay ( x ) U ( x ) ( x ) E ( x ) , 2m dx 2m
(3)
where ky is the wave vector in y-direction. The two-dimensional wave function is written as x, y e
x . The effective potential binding the electrons now is
ik y y
viewed as 2
2 e Veff k Ay ( x ) U ( x ) . * y 2m
(4)
Very clearly, this effective potential is strongly dependent not only on the wave vector ky and the vector potential Ay(x), but also on the EB height U(x). In Fig.3, we visualize 3-dimensional plots of the effective potential Veff as P and AP alignments with the wave vector from k y 10 m 1 to k y 10 m 1 , respectively. For evaluating the electronic transport properties of the system, the transfermatrix technique is applied. The detailed basic idea of transfer-matrix technique can be found in references 19 and 25. Here we only display the main formulae. The total transfer matrix of our system can be written as follow 1 k j ik j D 1 e N N 2 k j 1 M M j j 0 j 0 1 k j ik j D 1 2 k e j 1
k j ik j D 1 1 e 2 k j 1 . k j ik j D 1 1 e 2 k j 1
(5)
Sub-matrix Mj connects the wave functions in the neighbor segments j and j+1. For an electron with given energy, its transmission coefficient is calculated by the expression 2
2
T ( E , k y , U ) t 1 R 1 M 21 M 22 .
(6)
The ballistic conductance, which is an observable quantity, can be reached at zero 4
ACCEPTED MANUSCRIPT temperature from the well-known Landauer-Büttiker formula[26] G G0
2
2
T EF , 2 EF sin , U cos d ,
(7)
which is an integral over a half plane of Fermi surface and G0 e 2 mvF l 2 with l and vF is the length of the structure in y-direction and the Fermi velocity. 3. Results and discussion In Fig.2, we show 3-dimensional transmission coefficients in the plane of (ky, E) for P (upper row) and AP (bottom row) configurations with different EB height U(x). The structure parameters can be referred from the caption of Fig.1. Apparently, these two groups of transmission exhibit significant difference. The AP alignment has symmetrical transmission spectra about ky=0, while P alignment hasn’t. If comparing carefully their corresponding effective potentials, one can find that this is a reasonable result. As can be seen from Fig.(3), the distribution of effective potential is symmetrical about the center in AP configuration and this symmetry is independent on the EB height. It means that incident electrons with positive or negative ky can feel the same effective potential. This immediately causes the symmetrical transmission spectra. However the situation becomes quite different in P configuration. The magnetic barriers at the region of positive ky are much larger than those with negative ky, which can be clearly seen in Fig.3(a). Undoubtedly, an incident electron with positive ky is more difficult to tunnel the device. The computed transmission spectra showed in the upper row of Fig.2 also confirm this analysis. Only some resonant peaks can be noticed at low energy region. On the contrary, the electrons with negative ky possess 5
ACCEPTED MANUSCRIPT wider transmission peak at low energy region. Additionally, with increasing of electric barrier, the transmission suppression is more and more significant. For A configuration (see the upper low of Fig.2), the second and the third resonant peaks get close to each other and shift to the high energy region, while for AP configuration (see the bottom low of Fig.2), the resonant peaks with unity value around ky=0 get very narrow at high EB value. According to the Eq.(7), one can imagine that the observable quantity, conductance, will be reduced greatly because it is an integral over half plane of Fermi surface. To confirm this point, we calculate the conductance G by using Eq.(7), which is showed in Fig.4. For highlighting the effect of electric barrier, we fix the magnetization M of FM stripes and change the electric barrier U from 0 to 3 meV. As expected, the conductance has been suppressed dramatically with increasing of U(x) for both configurations. At low energy region, only some transport peaks can be spotted. But an obvious difference in conductance between P and AP configurations can be also clearly observed. Concretely, the conductance suppression is more significantly as AP configuration and almost no conductance can be seen when the EF is lower than 4meV except some sharp transport peaks. These sharp peaks originate from the narrow transmission peaks around ky=0 (see Fig.2(d)-(f)). For P configuration, the transport peaks at the low energy range are wider and are shrinking with electric barrier growing. But interestingly, the locations of these transport peaks remain roughly unchanged. This feature may bring us to get a considerable conductance at the given Fermi energy without worrying about the change of electric 6
ACCEPTED MANUSCRIPT barrier’s height. The great difference in conductance between A and AP configurations will give us an opportunity to make a GMR device in the corresponding energy range. We next consider the MRR of our system to further understand such conductance difference. In this work, the MRR is defined as MRR = (GP−GAP)/GAP×100%, with GP and GAP the conductance of P and AP configurations respectively. The calculated 3-dimensional plot of lg(MRR) in the (EF, U) plane is shown in Fig.5. Obviously, one can get very high value of MRR at the range of EF<1.5meV where MRR is up to 106 (see inset(a)). This will make it easier for us to prepare a high MRR device. However, low Fermi energy requires low electron concentration and hence reduces the conductance of our system. Fig.4 also shows the same results. When we move EF to higher energy region (1.5meV
ACCEPTED MANUSCRIPT EB, the MRR peak moves to high energy region. The inset(b) also shows this result clearly. That means, for a given GMR device like this, in which the electron concentration is fixed and magnetic stripes can’t be re-deposited, we can adjust the value of MRR by changing the EB height U(x). The results above offer an alternative to get a tunable GMR device which can be controlled by adjusting the voltage on SM stripe and Fermi energy EF. The above investigation tells us the effect of the EB height on MRR. In the following, we discuss how the width of SM stripe affects the transport properties of the system. We choose U=1.0 meV as applied electric potential and then let the width of SM stripe ds equals to 50, 100, and 150nm respectively. The computed results are showed in Fig.6. Clearly, as ds increases, the conductance curves shift towards high energy for both A and AP configurations, as shown in Fig.6(a) and (b). This is a very reasonable result because the electric barrier gets wider so that the electronic transmission coefficient gets smaller. Correspondingly, the MRR peaks move evidently to high energy region. Meanwhile, the amplitude of these peaks only decrease slightly (see Fig.6(c)). This fact lets us have opportunity to get some new systems which possess both high electron concentration (hence high conductance) and high MRR. However, the peak width of MRR as ds=150nm is relatively narrow. It requires us to prepare the sample very carefully so that the Fermi energy just sits in the corresponding range.
4. Summary In summary, we investigated the transport properties and GMR effect of 2DEG 8
ACCEPTED MANUSCRIPT under a magnetic and electrical potential. Our system exhibits very different transmission suppression between P and AP configurations at low energy range. It induces the significant GMR effect because the system shows a very high MRR (up to 106). When EB increases, the MRR peaks shift towards higher energy region, while the value of MRR decreases quickly. When the width of SM stripe increases and the EB remains unchanged, the peak width of MRR becomes narrower and the amplitude only decreases slightly. That means the structure parameters and the electric barrier must be synergistically considered. We hope these results offer an alternative to get a tunable GMR device which can be controlled by adjusting EB height U(x) and Fermi energy EF.
Acknowledgements. This work is supported by the National Natural Science Foundation of China under Grant No. 11304255.
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ACCEPTED MANUSCRIPT Fig. 1. z
y x
Fig. 1. (a) A schematic illustration of the GMR device, in which two FM stripes and one SM stripe are placed on the top of the GaAs heterostructure. (b) and (c) correspond to the parallel (P) and antiparallel (AP) configurations alignments of the device, respectively. The parameters are taken as x-=-300nm, x+=300nm, z0 = 10nm, h = 20nm, dF = 100nm, ds = 50nm, a = 200nm and Mh=20T•nm.
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ACCEPTED MANUSCRIPT Fig. 2.
Fig. 2. The 3-dimensional plots of transmission coefficient in the (E, ky) plane as (a) and (d) U = 0 meV, (b) and (e) U = 1.5 meV, (c) and (f) U = 3 meV, respectively. The structural parameters are the same as in Fig.1.
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ACCEPTED MANUSCRIPT Fig. 3. 5
4
3
Veff
Veff
4
5
P
(a)
2
(b)
AP
3 2 1
0 10 5 0 -5 -10
0 10 5 0 -5 -10
-300 -200 -100 0
x(nm)
100 200 300
-1 )
-1 )
k y( m
k y( m
1
-300 -200 -100 0
100 200 300
x(nm)
Fig.3. (a) and (b) show the 3-dimensional plots of the effective potential Veff as P and AP alignments with the wave vector from k y 10 m 1 to k y 10 m 1 , respectively. The other structural parameters are the same as in Fig.1.
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ACCEPTED MANUSCRIPT Fig. 4.
Fig.4. (a) and (b) show the 3-dimensional plots of the conductance G in the (EF, U) plane as P and AP configuration, respectively. The conductance is in unit of
G0 e 2 mvF l 2 . The structural parameters are the same as in Fig.1.
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ACCEPTED MANUSCRIPT Fig. 5.
Fig. 5. The 3-dimensional plot of log(MRR) in the (EF, U) plane and the inset show the MRR as the function of the EF. The red dashed shows the log(MRR) varies with U at EF=3.5meV. The structural parameters are the same as in Fig.1.
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ACCEPTED MANUSCRIPT Fig. 6.
G/G0
1
ds=50nm ds=100nm
U=1.0meV 0.4
G/G0
P
0.5 0
Log(MRR)
(a)
(b)
0 6
ds=50nm ds=100nm
AP
0.2
ds=150nm
U=1.0meV
ds=150nm
(c)
ds=50nm ds=100nm
U=1.0meV
4
ds=150nm
2 0
1
2
3
EF/E0
4
5
6
7
Fig.6. (a) and (b) are the conductances GP and GAP at a given EB height U = 1.0 meV as P and AP alignments, respectively. (c) shows the log(MRR) as a function of the Fermi energy EF for three different EB width: ds = 50nm, 100nm and 150nm. The other structural parameters are the same as in Fig.1.
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