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ferromagnet junctions
Author’s Accepted Manuscript Conductance spectra of asymmetric ferromagnet/ferromagnet/ferromagnet junctions K. Pasanai
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S0304-8853(16)30614-X http://dx.doi.org/10.1016/j.jmmm.2016.08.050 MAGMA61735
To appear in: Journal of Magnetism and Magnetic Materials Received date: 10 May 2016 Revised date: 15 August 2016 Accepted date: 16 August 2016 Cite this article as: K. Pasanai, Conductance spectra of asymmetric ferromagnet/ferromagnet/ferromagnet junctions, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.08.050 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 galley proof before it is published in its final citable 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.
Conductance spectra of asymmetric ferromagnet/ferromagnet/ferromagnet junctions K. Pasanai Theoretical Condensed Matter Physics Research Unit, Department of Physics, Faculty of Science, Maha-sarakham University, Khamriang Sub-District, Kantarawichai District, Maha-Sarakham 44150, Thailand
Abstract A theory of tunneling spectroscopy of ferromagnet/ferromagnet/ferromagnet junctions was studied. We applied a delta-functional approximation for the interface scattering properties under a one-dimensional system of a free electron approach. The reflection and transmission probabilities were calculated in the ballistic regime, and the conductance spectra were then calculated using the Landauer formulation. The magnetization directions were set to be either parallel (P) or anti-parallel (AP) alignments, for comparison. We found that the conductance spectra was suppressed when increasing the interfacial scattering at the interfaces. Moreover, the electron could exhibit direct transmission when the thickness was rather thin. Thus, there was no oscillation in this case. However, in the case of a thick layer the conductance spectra oscillated, and this oscillation was most prominent when the middle layer thickness increased. In the case of direct transmission, the conductance spectra of P and AP systems were definitely suppressed with increased exchange energy of the middle ferromagnet. This also refers to an increase in the magnetoresistance of the junction. In the case of oscillatory behavior, the positions of the resonance peaks were changed as the exchange energy was changed. Keywords: tunneling conductance spectra, magnetic junction, magnetism effect, ferromagnetic double junction 2015 MSC: code, 2015 ✩
Email address: [email protected] (K. Pasanai)
Preprint submitted to Journal of Magnetism and Magnetic Materials
August 18, 2016
1. Introduction In the past decades, an exciting field called spin-electronics, or spintronics in short, has emerged. It uses electron spin, in addition to charge, to control the motion of an electron across a magnetic tunnel junction, for example, a Fe/Cr/Fe multilayer [1, 2, 3, 4]. Among the many applications of spintronics devices are the read heads of hard disks, which have greatly contributed to the fast rise in the density of stored information and led to the extension of hard disk technology to consumer electronics. These devices are constructed from layers of alternating ferromagnetic and non-magnetic materials, which exhibit giant magnetoresistance (GMR) [5, 6]. Basically, the resistance of the layers strongly depends on the magnetization directions of the magnetic layers and the number of material layers that compose the junctions. Due to the fact that the magnetic junctions are composed of various novel materials and interfaces between them, in theory these characterize and specify the physics of the junctions. In this work, the interesting parameters consist of the exchange energy of a ferromagnet, the thickness of the middle layer, and the potential strength at the interfaces. For the first parameter, it also refers to the magnetization value of the magnetic material, and can be particularly improved by, for example, the doping concentration for a suitable condition [7]. This parameter directly affects the conductance and resistance of the junctions. For the second parameter, it can cause the direct transmission and oscillation behavior of the conductance spectra [8]. However, there is no present work that explains the relationship between this parameter and the exchange energy of the system. For the last parameter, it can control the transport properties of the junctions. Moreover, the inserted insulating sheet that has a non spin-flip scattering potential can enhance the efficiency of the junctions [9, 10, 11]. Furthermore, there is another kind of potential scattering, called spin-flip scattering, that can change the spin direction of the electrons. In particular, this kind of scattering can be experimentally produced by embedding magnetic impurities at the interface [12, 13, 14, 15, 16]. Surprisingly, when these two kinds of scattering occur at the interfaces, the tunneling conductance spectra of the junction increases [17, 18]. In some cases, the spin-flip-scattering can cause the resonance peak of the magnetic tunnel junction to split [19]. These indicate how exchange energy and interface properties are crucial. 2
Thus, in this work we are interested in the theoretical study of the tunneling phenomena of FM/FM/FM junctions. The approach we used was elastic scattering at the interface between the materials [20]. In the calculation process, we developed a model Hamiltonian of the junctions, in which the dispersion relations of the materials were assumed to be one band parabolic approximations, as discussed in Sec.2. For the two band coupling model, s-band and d-band coupling of the ferromagnet, the coupling between bands could cause a kink in the spectra and density of states, but the effect of the parameters under one-band and two-band models on the conductance should be the same [21, 22]. Other than that, the interface between materials was explained by the delta potential model, and not the insulating sheet with finite width model. This is because the finite width model just gives rise to an oscillation behavior, and both of the models give similar results when the interface layer was rather thin [8]. The main area of interest was to focus on the effect of the exchange energy of the ferromagnet, the thickness of the middle ferromagnetic layer, and the interface properties on the conductance spectra, as shown in Sec. 3. Finally, we finish with the conclusion as in Sec. 4. 2. Model and formalism In this work, the energy dispersions of FM/FM/FM junctions are illustrated in Fig.1. The magnetization directions of the system were set to be either P or AP alignments, for comparison. The two interfaces between the materials were explained by the delta potential, in which there is no width but infinite height. The Hamiltonian of the junctions was described by a 2 × 2 matrix, as follows [17]: 1 H(x) = p p + V (x), (1) 2m(x) where p is the momentum operator in a one-dimensional system. The position dependence of the electron effective mass m(x) is given by [23]: m(x) = m∗1 Θ (−x) + m∗2 [Θ (x) − Θ (x − L)] + m∗3 Θ (x − L) ,
(2)
where m∗1 , m∗2 , and m∗3 are respectively the electron effective mass in the first, second, and third regions. Here, Θ(±x) is the Heaviside step function. The 3
Figure 1: Schematic illustration of energy dispersion of FM/FM/FM junctions of P and AP alignments. In reality, the structures of the FM/FM/FM double barrier potentials can be generated using Co, CoCr, CoFeB, Fe, and NiFe as the ferromagnets, and Al2 O3 or MgO as the potential barriers.
4
potential energy consists of: ˆ ·σ ˆ − Eex1 ) Θ (−x) + (Eex2 m ˆ ·σ ˆ − Eex2 ) [Θ (x) − Θ (x − L)] V (x) = (Eex1 m + (Eex3 m ˆ ·σ ˆ − Eex3 ) Θ (x − L) + U[δ(x) + δ(x − L)], (3) where Eex1,2,3 refers to the exchange energy in the F M1,2,3 . In this case, we choose a special energy range, and this will cause the spectra to occur in the positive region only (see Fig.1 for illustration). δ(x) is the Dirac delta function, m ˆ is the unit vector of the magnetization in a ferromagnetic material, and σ ˆ is the vector of the Pauli spin matrices. When the external magnetic field and spin-orbit coupling were not considered, m ˆ was set to point along the +z direction. Thus, spin-up and spin-down directions point along +z and −z , respectively. The scattering was modeled by at the interfaces u0 uf the delta potential, and written as: U = , where the diagonal and uf u0 off-diagonal elements refer to the strength of the non-spin-flip and spin-flip scattering, respectively. By diagonalizing the above Hamiltonian, one can obtain the energy dispersion relations for spin-up (−) and spin-down (+) bands in each region as follows: 2 k 2 (q 2 )(p2 ) Ek(q)(p) = ∓ Eex1(2)(3) + Eex1(2)(3) , (4) 2m1∗ (2∗ )(3∗ ) where k, q, p are the magnitudes of the wave vectors k, q, p of the quasiparticles. The spin-up and spin-down eigenstates in each region are respectively: 1 ↑ eik↑ (q↑ )(p↑ )·r , Ψ1(2)(3) (r) = (5) 0 0 ↓ eik↓ (q↓ )(p↓ )·r . Ψ1(2)(3) (r) = (6) 1 In the tunneling scenario, the electron transmission and reflection amplitudes of the junctions can be obtained by considering the linear combination of the wave functions of the electron incident, reflection, or transmission. To obtain these amplitudes, we applied the appropriate matching conditions of the junctions for the two interfaces (x = 0 and x = L) [24]. For ΨI (x = 0) = ΨII (x = 0) ≡ Ψ0 and x = 10,dΨone can obtain: 1 dΨII ∗ I − m∗ dx x=0 = 2kF Z1 Ψ0 , where the unitless parameter Z1∗ repm∗2 dx x=0 1 resents the barrier strength of the non-spin-flip and spin-flip scattering at 5
1
1
0.8
0.8
kFL = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.4
kFL = 0 =1 =2 =3 =4
0.6
GT
GT
0.6
0.4
0.2
0.2
Zf = 0 0
0
2
4
eV
6
Zf = 0 8
10
0
0
2
4
eV
6
8
10
Figure 2: (color online) Plot of total conductance, GT , as function of applied voltage eV for different middle ferromagnetic thicknesses. Parameters are for Z1 = Z2 = 1, Zf 1 = Zf 2 = 0, Eex1 = Eex2 = Eex3 = 0.5Eex , and m∗1 = m∗2 = m∗3 = me where me is the electron effective mass.
Z1 Zf 1 . For x = L, the first interface. It is defined by: ≡ = Zf 1 Z1 the matching conditions become: Ψ∗II (x = L) = ∗ΨIII (x = L) ≡∗ ΨL and 1 dΨIII 1 dΨII − = 2kF Z2 ΨL , where Z2 is similar to Z1 . Then, m∗3 dx m∗2 dx x=L x=L one can determine the reflection and transmission probabilities as explained in Ref. [17]. Eventually, the conductance spectra in the one-dimensional system at zero temperature can be determined using the following equation [25]: Z1∗
Gσ (eV ) ≡
m∗1 U 2 k F
e2 1 djσx (eV ) = Tk (eV ), L dV 2π
(7)
where Tk (eV ) is the total probability of the junctions. 3. Results and discussion The main area of interest in this work was to focus on the effect of the exchange energy on the conductance spectra. However, it was found that the effects of the middle layer thickness and the interfacial scattering were also interesting under some conditions, as shown in Fig. 2. In this figure, we set the interfacial non spin-flip scattering, Z1 and Z2 , to be present at the interfaces, but the spin-flip scattering was ignored, and the middle layer thickness was varied. When the middle layer thickness was rather thin, as shown in the first inset, the conductance spectra did not behave like an oscillation. This meant that the electron could transfer to the right material without generating an interference effect, and it was called direct transmission. When the 6
1
1
0.8
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
GT
GT
0.8
0.4
Zf1 = Zf2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
0.4
0.2
0.2
Zf = 0 0
0
2
4
eV
6
Z=0 8
0
10
0
1
2
4
0.8
GT
GT
0.6
0.4
8
10
Zf1 = Zf2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
0.4
0.2
0.2
Zf = 0 0
6
1
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.8
0
eV
2
4
eV
6
Z=0 8
10
0
0
2
4
eV
6
8
10
Figure 3: (color online) Plot of total conductance, GT , as function of applied voltage eV with varying (left column) non-spin-flip scattering Z and (right column) spin-flip scattering Zf . Parameters are for Eex1 = Eex2 = Eex3 = 0.5Eex , m∗1 = m∗2 = m∗3 = me , and (first row) kF L = 0.2 and (second row) kF L = 5.
middle layer thickness increased, the conductance spectra started to oscillate. In particular, the oscillatory behavior was dominant when the middle layer was thicker. The minima peak was not smaller than the curve of the conductance spectra in the case of kF L = 0. By considering the direct or oscillation transmissions, the practical application of this plot could be that the thickness of the middle layer can be approximated in terms of thinness or thickness. We moved to consider the effect of the potential strength at the interfaces on the conductance spectra. In the first row of Fig 3, the middle ferromagnetic layer was set to be thin and the potential strengths at the two interfaces were set to be symmetric. It was found that the conductance spectra in the case of Zf = 0 was suppressed with an increase in Z, as expected. When Zf was varied (Z = 0), the conductance in the region of eV > 2Eex was still suppressed with Zf , but in the region of eV < 2Eex , the conductance increased with Zf . Moreover, the result in the case of a thick layer was similar, as shown in the second row of the figure, unless the conductance spectra showed an oscillation behavior and resonance peak. Even if the result in this case did not result in much excitement, it could be used to verify the model 7
1
1
P
AP
0.8
0.8
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
0.4
0.6
GT
GT
0.6
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
1
eV
Zf1 = Zf2 = 0 6
8
0
10
0
2
P
6
0.8
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
8
10
0.4
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
0.6
GT
GT
0.6
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
eV
AP
0.8
0
4
1
2
4
eV
Zf1 = Zf2 = 0 6
8
10
0
0
2
4
eV
6
8
10
Figure 4: (color online) Plot of total conductance, GT , as function of applied voltage eV with varying small value of exchange energy of middle ferromagnet Eex2 for P and AP alignments. Eex refers to exchange energy of first and third ferromagnets of FM/FM/FM junctions. Parameters are for (first row) kF L = 0.2, (second row) kF L = 5, and Eex1 = Eex3 = Eex .
of the system. The main area of interest was to consider the effect of the exchange energy on the conductance spectra as shown in Figs. 4 - 6 for P and AP alignments, for comparison. It was found that in the direct transmission case the conductance spectra at the energy that was larger than the bottom of the minority spin band was slightly suppressed with increases in Eex2 . In the case of the oscillation behavior, the position of the resonance peak was modified by the change of Eex2 as shown in the second row of Fig. 4. The reason for this was the energy of the electron depends on the exchange energy, as shown in Eq. 4. However, the amplitude of oscillation in this case was almost unaffected by changing the small value of Eex2 . In the case of a change in the large value of Eex2 as depicted in Fig. 5, in the case of direct transmission, a decrease in the conductance spectra at an energy that was larger than the bottom of the minority band was shown distinctly. In the oscillatory behavior case, the period of oscillation was definitely affected by a change in the large value of Eex2 8
1
1
P
AP
0.8
0.8
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
0.6
GT
GT
0.6
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
1
eV
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
Zf1 = Zf2 = 0 6
8
0
10
0
2
4
1
P
eV
6
8
10
AP
0.6
0.6
GT
0.8
GT
0.8
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
eV
6
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
8
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 10
0
0
2
4
eV
6
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
8
10
Figure 5: (color online) These plots are the same as Fig. 4 except Eex2 is larger.
When we pay attention to Fig. 4, it can be seen that the interfacial scattering and the exchange energy could similarly suppress the conductance spectra in the case of direct transmission. To deeply consider this effect, we plotted the conductance spectra as a function of Eex2 with various values of Z1 and Z2 , as shown in Fig. 6. We found that the conductance for P and AP alignments were definitely suppressed when Eex2 increased. This referred to the increase of the resistance with increasing Eex2 . This was an interesting result to improve the conductance or resistance of the junctions by Eex2 because this parameter could be experimentally modified by, for example, doping under a suitable condition, as mentioned previously. 4. Conclusions We have presented a theory of tunneling spectroscopy across the magnetic junctions between ferromagnet/ferromagnet/ferromagnet, where the exchange energy of the three ferromagnets were distinguishable. The calculations were done within the delta functional approximation for the interface scattering of a free electron approach. The reflection and transmission probabilities were first calculated in the ballistic regime, and then the conductance spectra was calculated using the Landauer formulation. In the junctions, 9
1
1
P
0.9
0.8
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6
0.7
GT
GT
0.8
0.6
0.5
Zf1 = Zf2 = 0 0
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6
0.7 0.6
0.5 0.4
AP
0.9
5
Zf1 = Zf2 = 0 10
Eex2
15
0.4
0
5
Eex2
10
15
Figure 6: (color online) Plot of total conductance, GT , as function of exchange energy Eex2 with varying potential strength Z1 = Z2 for P and AP alignments. Parameters are for Eex1 = Eex3 = Eex , kF L = 0.2, Zf 1 = Zf 2 = 0, and eV = 4Eex1 .
the magnetization directions were either parallel or anti-parallel alignments, for comparison. The main area of interest was to focus on the effect of the exchange energy on the transport properties of the junction. It was found that the conductance spectra was suppressed when increasing the interfacial scattering at the interfaces, as expected. For the effect of the middle ferromagnetic thickness, the electron experienced direct transmission through the junctions when the thickness was thin. Thus, there was no oscillation in this case. However, in the case of a thick layer, the conductance spectra gave rise to an oscillation due to the interference effect of the traveling waves in that layer. This oscillation was dominant with increases in the middle layer thickness. For the effect of the exchange energy of the middle ferromagnetic layer, in the case of direct transmission, the conductance spectra of the parallel and anti-parallel systems were definitely suppressed with increases in this exchange energy. In the case of oscillatory behavior, the positions of the resonance peaks were changed when changing the exchange energy. In particular, the exchange energy parameter plays a similar role as the potential strength, but it seems to be much more efficiency because this parameter can be adjusted in a real experiment. Acknowledgments We would like to acknowledge financial support from the National Research Council of Thailand (NRCT) via Maha-Sarakram University for the main funding and Theoretical Condensed Matter Physics Research Unit (TCMPRU). Especially, we thank J. Dodgson for critical reading of the manuscript. 10
References [1] G. Prinz, Phys. Today 48 (1959) 58–60. ˇ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76 (2004) 323–410. [2] I. Zuti´ [3] A. Fert, Rev. Mod. Phys. 80 (2008) 1517–1530. [4] P. A. Gr¨ unberg, Rev. Mod. Phys. 80 (2008) 1531–1540. [5] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472–2475. [6] G. Binasch, P. Gr¨ unberg, F. Saurenbach, W. Zinn, Phys. Rev. B 39 (1989) 4828–4830. [7] D. Y. Petrovykh, H. Altmann, K. N.and Hochst, M. Laubscher, S. Maat, G. J. Mankey, Appl. Phys. Lett. 72 (2007) 3459–3461. [8] K. Pasanai, Journal of Magnetism and Magnetic Materials 401 (2016) 463–471. [9] A. T. Filip, B. H. Hoving, F. J. Jedema, B. J. van Wees, B. Dutta, S. Borghs, Phys. Rev. B 62 (2000) 9996–9999. [10] E. I. Rashba, Phys. Rev. B 62 (2000) R16267–R16270. [11] A. Fert, H. Jaffr`es, Phys. Rev. B 64 (2001) 184420. [12] F. Guinea, Phys. Rev. B 58 (1998) 9212–9216. [13] P. Lyu, D. Y. Xing, J. Dong, Phys. Rev. B 58 (1998) 54–57. [14] J. Inoue, S. Maekawa, Journal of Magnetism and Magnetic Materials 198199 (1999) 167 – 169. [15] R. Jansen, J. S. Moodera, Phys. Rev. B 61 (2000) 9047–9050. [16] A. Vedyayev, D. Bagrets, A. Bagrets, B. Dieny, Phys. Rev. B 63 (2001) 064429. [17] K. Pasanai, P. Pairor, Phys. Rev. B 84 (2011) 224432. 11
[18] B. Srisongmuang, Physica E: Low-dimensional Systems and Nanostructures 54 (2013) 167–170. [19] K. Pasanai, Chiang Mai J. Sci. 43 (2016) 1280–1287. [20] G. E. Blonder, M. Tinkham, T. M. Klapwijk, Phys. Rev. B 25 (1982) 4515–4532. [21] K. Pasanai, Journal of Magnetism and Magnetic Materials 357 (2014) 35–40. [22] K. Pasanai, P. Pairor, Journal of Magnetism and Magnetic Materials 328 (2013) 35–40. [23] U. Z¨ ulicke, C. Schroll, Phys. Rev. Lett. 88 (2001) 029701. [24] E. L. Wolf, Principle of electron tunneling spectroscopy (Oxford University Press: New York) (1989). [25] K. Pasanai, Physica E: Low-dimensional Systems and Nanostructures 77 (2016) 13–22.
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www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)30614-X http://dx.doi.org/10.1016/j.jmmm.2016.08.050 MAGMA61735
To appear in: Journal of Magnetism and Magnetic Materials Received date: 10 May 2016 Revised date: 15 August 2016 Accepted date: 16 August 2016 Cite this article as: K. Pasanai, Conductance spectra of asymmetric ferromagnet/ferromagnet/ferromagnet junctions, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.08.050 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 galley proof before it is published in its final citable 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.
Conductance spectra of asymmetric ferromagnet/ferromagnet/ferromagnet junctions K. Pasanai Theoretical Condensed Matter Physics Research Unit, Department of Physics, Faculty of Science, Maha-sarakham University, Khamriang Sub-District, Kantarawichai District, Maha-Sarakham 44150, Thailand
Abstract A theory of tunneling spectroscopy of ferromagnet/ferromagnet/ferromagnet junctions was studied. We applied a delta-functional approximation for the interface scattering properties under a one-dimensional system of a free electron approach. The reflection and transmission probabilities were calculated in the ballistic regime, and the conductance spectra were then calculated using the Landauer formulation. The magnetization directions were set to be either parallel (P) or anti-parallel (AP) alignments, for comparison. We found that the conductance spectra was suppressed when increasing the interfacial scattering at the interfaces. Moreover, the electron could exhibit direct transmission when the thickness was rather thin. Thus, there was no oscillation in this case. However, in the case of a thick layer the conductance spectra oscillated, and this oscillation was most prominent when the middle layer thickness increased. In the case of direct transmission, the conductance spectra of P and AP systems were definitely suppressed with increased exchange energy of the middle ferromagnet. This also refers to an increase in the magnetoresistance of the junction. In the case of oscillatory behavior, the positions of the resonance peaks were changed as the exchange energy was changed. Keywords: tunneling conductance spectra, magnetic junction, magnetism effect, ferromagnetic double junction 2015 MSC: code, 2015 ✩
Email address: [email protected] (K. Pasanai)
Preprint submitted to Journal of Magnetism and Magnetic Materials
August 18, 2016
1. Introduction In the past decades, an exciting field called spin-electronics, or spintronics in short, has emerged. It uses electron spin, in addition to charge, to control the motion of an electron across a magnetic tunnel junction, for example, a Fe/Cr/Fe multilayer [1, 2, 3, 4]. Among the many applications of spintronics devices are the read heads of hard disks, which have greatly contributed to the fast rise in the density of stored information and led to the extension of hard disk technology to consumer electronics. These devices are constructed from layers of alternating ferromagnetic and non-magnetic materials, which exhibit giant magnetoresistance (GMR) [5, 6]. Basically, the resistance of the layers strongly depends on the magnetization directions of the magnetic layers and the number of material layers that compose the junctions. Due to the fact that the magnetic junctions are composed of various novel materials and interfaces between them, in theory these characterize and specify the physics of the junctions. In this work, the interesting parameters consist of the exchange energy of a ferromagnet, the thickness of the middle layer, and the potential strength at the interfaces. For the first parameter, it also refers to the magnetization value of the magnetic material, and can be particularly improved by, for example, the doping concentration for a suitable condition [7]. This parameter directly affects the conductance and resistance of the junctions. For the second parameter, it can cause the direct transmission and oscillation behavior of the conductance spectra [8]. However, there is no present work that explains the relationship between this parameter and the exchange energy of the system. For the last parameter, it can control the transport properties of the junctions. Moreover, the inserted insulating sheet that has a non spin-flip scattering potential can enhance the efficiency of the junctions [9, 10, 11]. Furthermore, there is another kind of potential scattering, called spin-flip scattering, that can change the spin direction of the electrons. In particular, this kind of scattering can be experimentally produced by embedding magnetic impurities at the interface [12, 13, 14, 15, 16]. Surprisingly, when these two kinds of scattering occur at the interfaces, the tunneling conductance spectra of the junction increases [17, 18]. In some cases, the spin-flip-scattering can cause the resonance peak of the magnetic tunnel junction to split [19]. These indicate how exchange energy and interface properties are crucial. 2
Thus, in this work we are interested in the theoretical study of the tunneling phenomena of FM/FM/FM junctions. The approach we used was elastic scattering at the interface between the materials [20]. In the calculation process, we developed a model Hamiltonian of the junctions, in which the dispersion relations of the materials were assumed to be one band parabolic approximations, as discussed in Sec.2. For the two band coupling model, s-band and d-band coupling of the ferromagnet, the coupling between bands could cause a kink in the spectra and density of states, but the effect of the parameters under one-band and two-band models on the conductance should be the same [21, 22]. Other than that, the interface between materials was explained by the delta potential model, and not the insulating sheet with finite width model. This is because the finite width model just gives rise to an oscillation behavior, and both of the models give similar results when the interface layer was rather thin [8]. The main area of interest was to focus on the effect of the exchange energy of the ferromagnet, the thickness of the middle ferromagnetic layer, and the interface properties on the conductance spectra, as shown in Sec. 3. Finally, we finish with the conclusion as in Sec. 4. 2. Model and formalism In this work, the energy dispersions of FM/FM/FM junctions are illustrated in Fig.1. The magnetization directions of the system were set to be either P or AP alignments, for comparison. The two interfaces between the materials were explained by the delta potential, in which there is no width but infinite height. The Hamiltonian of the junctions was described by a 2 × 2 matrix, as follows [17]: 1 H(x) = p p + V (x), (1) 2m(x) where p is the momentum operator in a one-dimensional system. The position dependence of the electron effective mass m(x) is given by [23]: m(x) = m∗1 Θ (−x) + m∗2 [Θ (x) − Θ (x − L)] + m∗3 Θ (x − L) ,
(2)
where m∗1 , m∗2 , and m∗3 are respectively the electron effective mass in the first, second, and third regions. Here, Θ(±x) is the Heaviside step function. The 3
Figure 1: Schematic illustration of energy dispersion of FM/FM/FM junctions of P and AP alignments. In reality, the structures of the FM/FM/FM double barrier potentials can be generated using Co, CoCr, CoFeB, Fe, and NiFe as the ferromagnets, and Al2 O3 or MgO as the potential barriers.
4
potential energy consists of: ˆ ·σ ˆ − Eex1 ) Θ (−x) + (Eex2 m ˆ ·σ ˆ − Eex2 ) [Θ (x) − Θ (x − L)] V (x) = (Eex1 m + (Eex3 m ˆ ·σ ˆ − Eex3 ) Θ (x − L) + U[δ(x) + δ(x − L)], (3) where Eex1,2,3 refers to the exchange energy in the F M1,2,3 . In this case, we choose a special energy range, and this will cause the spectra to occur in the positive region only (see Fig.1 for illustration). δ(x) is the Dirac delta function, m ˆ is the unit vector of the magnetization in a ferromagnetic material, and σ ˆ is the vector of the Pauli spin matrices. When the external magnetic field and spin-orbit coupling were not considered, m ˆ was set to point along the +z direction. Thus, spin-up and spin-down directions point along +z and −z , respectively. The scattering was modeled by at the interfaces u0 uf the delta potential, and written as: U = , where the diagonal and uf u0 off-diagonal elements refer to the strength of the non-spin-flip and spin-flip scattering, respectively. By diagonalizing the above Hamiltonian, one can obtain the energy dispersion relations for spin-up (−) and spin-down (+) bands in each region as follows: 2 k 2 (q 2 )(p2 ) Ek(q)(p) = ∓ Eex1(2)(3) + Eex1(2)(3) , (4) 2m1∗ (2∗ )(3∗ ) where k, q, p are the magnitudes of the wave vectors k, q, p of the quasiparticles. The spin-up and spin-down eigenstates in each region are respectively: 1 ↑ eik↑ (q↑ )(p↑ )·r , Ψ1(2)(3) (r) = (5) 0 0 ↓ eik↓ (q↓ )(p↓ )·r . Ψ1(2)(3) (r) = (6) 1 In the tunneling scenario, the electron transmission and reflection amplitudes of the junctions can be obtained by considering the linear combination of the wave functions of the electron incident, reflection, or transmission. To obtain these amplitudes, we applied the appropriate matching conditions of the junctions for the two interfaces (x = 0 and x = L) [24]. For ΨI (x = 0) = ΨII (x = 0) ≡ Ψ0 and x = 10,dΨone can obtain: 1 dΨII ∗ I − m∗ dx x=0 = 2kF Z1 Ψ0 , where the unitless parameter Z1∗ repm∗2 dx x=0 1 resents the barrier strength of the non-spin-flip and spin-flip scattering at 5
1
1
0.8
0.8
kFL = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.4
kFL = 0 =1 =2 =3 =4
0.6
GT
GT
0.6
0.4
0.2
0.2
Zf = 0 0
0
2
4
eV
6
Zf = 0 8
10
0
0
2
4
eV
6
8
10
Figure 2: (color online) Plot of total conductance, GT , as function of applied voltage eV for different middle ferromagnetic thicknesses. Parameters are for Z1 = Z2 = 1, Zf 1 = Zf 2 = 0, Eex1 = Eex2 = Eex3 = 0.5Eex , and m∗1 = m∗2 = m∗3 = me where me is the electron effective mass.
Z1 Zf 1 . For x = L, the first interface. It is defined by: ≡ = Zf 1 Z1 the matching conditions become: Ψ∗II (x = L) = ∗ΨIII (x = L) ≡∗ ΨL and 1 dΨIII 1 dΨII − = 2kF Z2 ΨL , where Z2 is similar to Z1 . Then, m∗3 dx m∗2 dx x=L x=L one can determine the reflection and transmission probabilities as explained in Ref. [17]. Eventually, the conductance spectra in the one-dimensional system at zero temperature can be determined using the following equation [25]: Z1∗
Gσ (eV ) ≡
m∗1 U 2 k F
e2 1 djσx (eV ) = Tk (eV ), L dV 2π
(7)
where Tk (eV ) is the total probability of the junctions. 3. Results and discussion The main area of interest in this work was to focus on the effect of the exchange energy on the conductance spectra. However, it was found that the effects of the middle layer thickness and the interfacial scattering were also interesting under some conditions, as shown in Fig. 2. In this figure, we set the interfacial non spin-flip scattering, Z1 and Z2 , to be present at the interfaces, but the spin-flip scattering was ignored, and the middle layer thickness was varied. When the middle layer thickness was rather thin, as shown in the first inset, the conductance spectra did not behave like an oscillation. This meant that the electron could transfer to the right material without generating an interference effect, and it was called direct transmission. When the 6
1
1
0.8
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
GT
GT
0.8
0.4
Zf1 = Zf2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
0.4
0.2
0.2
Zf = 0 0
0
2
4
eV
6
Z=0 8
0
10
0
1
2
4
0.8
GT
GT
0.6
0.4
8
10
Zf1 = Zf2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.6
0.4
0.2
0.2
Zf = 0 0
6
1
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6 = 0.8
0.8
0
eV
2
4
eV
6
Z=0 8
10
0
0
2
4
eV
6
8
10
Figure 3: (color online) Plot of total conductance, GT , as function of applied voltage eV with varying (left column) non-spin-flip scattering Z and (right column) spin-flip scattering Zf . Parameters are for Eex1 = Eex2 = Eex3 = 0.5Eex , m∗1 = m∗2 = m∗3 = me , and (first row) kF L = 0.2 and (second row) kF L = 5.
middle layer thickness increased, the conductance spectra started to oscillate. In particular, the oscillatory behavior was dominant when the middle layer was thicker. The minima peak was not smaller than the curve of the conductance spectra in the case of kF L = 0. By considering the direct or oscillation transmissions, the practical application of this plot could be that the thickness of the middle layer can be approximated in terms of thinness or thickness. We moved to consider the effect of the potential strength at the interfaces on the conductance spectra. In the first row of Fig 3, the middle ferromagnetic layer was set to be thin and the potential strengths at the two interfaces were set to be symmetric. It was found that the conductance spectra in the case of Zf = 0 was suppressed with an increase in Z, as expected. When Zf was varied (Z = 0), the conductance in the region of eV > 2Eex was still suppressed with Zf , but in the region of eV < 2Eex , the conductance increased with Zf . Moreover, the result in the case of a thick layer was similar, as shown in the second row of the figure, unless the conductance spectra showed an oscillation behavior and resonance peak. Even if the result in this case did not result in much excitement, it could be used to verify the model 7
1
1
P
AP
0.8
0.8
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
0.4
0.6
GT
GT
0.6
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
1
eV
Zf1 = Zf2 = 0 6
8
0
10
0
2
P
6
0.8
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
8
10
0.4
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = 0.2Eex = 0.4Eex = 0.6Eex = 0.8Eex
0.6
GT
GT
0.6
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
eV
AP
0.8
0
4
1
2
4
eV
Zf1 = Zf2 = 0 6
8
10
0
0
2
4
eV
6
8
10
Figure 4: (color online) Plot of total conductance, GT , as function of applied voltage eV with varying small value of exchange energy of middle ferromagnet Eex2 for P and AP alignments. Eex refers to exchange energy of first and third ferromagnets of FM/FM/FM junctions. Parameters are for (first row) kF L = 0.2, (second row) kF L = 5, and Eex1 = Eex3 = Eex .
of the system. The main area of interest was to consider the effect of the exchange energy on the conductance spectra as shown in Figs. 4 - 6 for P and AP alignments, for comparison. It was found that in the direct transmission case the conductance spectra at the energy that was larger than the bottom of the minority spin band was slightly suppressed with increases in Eex2 . In the case of the oscillation behavior, the position of the resonance peak was modified by the change of Eex2 as shown in the second row of Fig. 4. The reason for this was the energy of the electron depends on the exchange energy, as shown in Eq. 4. However, the amplitude of oscillation in this case was almost unaffected by changing the small value of Eex2 . In the case of a change in the large value of Eex2 as depicted in Fig. 5, in the case of direct transmission, a decrease in the conductance spectra at an energy that was larger than the bottom of the minority band was shown distinctly. In the oscillatory behavior case, the period of oscillation was definitely affected by a change in the large value of Eex2 8
1
1
P
AP
0.8
0.8
0.4
Z1 = Z2 = 0.6
0.2
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
0.6
GT
GT
0.6
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
1
eV
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
Zf1 = Zf2 = 0 6
8
0
10
0
2
4
1
P
eV
6
8
10
AP
0.6
0.6
GT
0.8
GT
0.8
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 0
0
2
4
eV
6
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
8
0.4
Z1 = Z2 = 0.6
0.2
Zf1 = Zf2 = 0 10
0
0
2
4
eV
6
Eex2 = 0 = Eex = 2Eex = 4Eex = 6Eex
8
10
Figure 5: (color online) These plots are the same as Fig. 4 except Eex2 is larger.
When we pay attention to Fig. 4, it can be seen that the interfacial scattering and the exchange energy could similarly suppress the conductance spectra in the case of direct transmission. To deeply consider this effect, we plotted the conductance spectra as a function of Eex2 with various values of Z1 and Z2 , as shown in Fig. 6. We found that the conductance for P and AP alignments were definitely suppressed when Eex2 increased. This referred to the increase of the resistance with increasing Eex2 . This was an interesting result to improve the conductance or resistance of the junctions by Eex2 because this parameter could be experimentally modified by, for example, doping under a suitable condition, as mentioned previously. 4. Conclusions We have presented a theory of tunneling spectroscopy across the magnetic junctions between ferromagnet/ferromagnet/ferromagnet, where the exchange energy of the three ferromagnets were distinguishable. The calculations were done within the delta functional approximation for the interface scattering of a free electron approach. The reflection and transmission probabilities were first calculated in the ballistic regime, and then the conductance spectra was calculated using the Landauer formulation. In the junctions, 9
1
1
P
0.9
0.8
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6
0.7
GT
GT
0.8
0.6
0.5
Zf1 = Zf2 = 0 0
Z1 = Z2 = 0 = 0.2 = 0.4 = 0.6
0.7 0.6
0.5 0.4
AP
0.9
5
Zf1 = Zf2 = 0 10
Eex2
15
0.4
0
5
Eex2
10
15
Figure 6: (color online) Plot of total conductance, GT , as function of exchange energy Eex2 with varying potential strength Z1 = Z2 for P and AP alignments. Parameters are for Eex1 = Eex3 = Eex , kF L = 0.2, Zf 1 = Zf 2 = 0, and eV = 4Eex1 .
the magnetization directions were either parallel or anti-parallel alignments, for comparison. The main area of interest was to focus on the effect of the exchange energy on the transport properties of the junction. It was found that the conductance spectra was suppressed when increasing the interfacial scattering at the interfaces, as expected. For the effect of the middle ferromagnetic thickness, the electron experienced direct transmission through the junctions when the thickness was thin. Thus, there was no oscillation in this case. However, in the case of a thick layer, the conductance spectra gave rise to an oscillation due to the interference effect of the traveling waves in that layer. This oscillation was dominant with increases in the middle layer thickness. For the effect of the exchange energy of the middle ferromagnetic layer, in the case of direct transmission, the conductance spectra of the parallel and anti-parallel systems were definitely suppressed with increases in this exchange energy. In the case of oscillatory behavior, the positions of the resonance peaks were changed when changing the exchange energy. In particular, the exchange energy parameter plays a similar role as the potential strength, but it seems to be much more efficiency because this parameter can be adjusted in a real experiment. Acknowledgments We would like to acknowledge financial support from the National Research Council of Thailand (NRCT) via Maha-Sarakram University for the main funding and Theoretical Condensed Matter Physics Research Unit (TCMPRU). Especially, we thank J. Dodgson for critical reading of the manuscript. 10
References [1] G. Prinz, Phys. Today 48 (1959) 58–60. ˇ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76 (2004) 323–410. [2] I. Zuti´ [3] A. Fert, Rev. Mod. Phys. 80 (2008) 1517–1530. [4] P. A. Gr¨ unberg, Rev. Mod. Phys. 80 (2008) 1531–1540. [5] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472–2475. [6] G. Binasch, P. Gr¨ unberg, F. Saurenbach, W. Zinn, Phys. Rev. B 39 (1989) 4828–4830. [7] D. Y. Petrovykh, H. Altmann, K. N.and Hochst, M. Laubscher, S. Maat, G. J. Mankey, Appl. Phys. Lett. 72 (2007) 3459–3461. [8] K. Pasanai, Journal of Magnetism and Magnetic Materials 401 (2016) 463–471. [9] A. T. Filip, B. H. Hoving, F. J. Jedema, B. J. van Wees, B. Dutta, S. Borghs, Phys. Rev. B 62 (2000) 9996–9999. [10] E. I. Rashba, Phys. Rev. B 62 (2000) R16267–R16270. [11] A. Fert, H. Jaffr`es, Phys. Rev. B 64 (2001) 184420. [12] F. Guinea, Phys. Rev. B 58 (1998) 9212–9216. [13] P. Lyu, D. Y. Xing, J. Dong, Phys. Rev. B 58 (1998) 54–57. [14] J. Inoue, S. Maekawa, Journal of Magnetism and Magnetic Materials 198199 (1999) 167 – 169. [15] R. Jansen, J. S. Moodera, Phys. Rev. B 61 (2000) 9047–9050. [16] A. Vedyayev, D. Bagrets, A. Bagrets, B. Dieny, Phys. Rev. B 63 (2001) 064429. [17] K. Pasanai, P. Pairor, Phys. Rev. B 84 (2011) 224432. 11
[18] B. Srisongmuang, Physica E: Low-dimensional Systems and Nanostructures 54 (2013) 167–170. [19] K. Pasanai, Chiang Mai J. Sci. 43 (2016) 1280–1287. [20] G. E. Blonder, M. Tinkham, T. M. Klapwijk, Phys. Rev. B 25 (1982) 4515–4532. [21] K. Pasanai, Journal of Magnetism and Magnetic Materials 357 (2014) 35–40. [22] K. Pasanai, P. Pairor, Journal of Magnetism and Magnetic Materials 328 (2013) 35–40. [23] U. Z¨ ulicke, C. Schroll, Phys. Rev. Lett. 88 (2001) 029701. [24] E. L. Wolf, Principle of electron tunneling spectroscopy (Oxford University Press: New York) (1989). [25] K. Pasanai, Physica E: Low-dimensional Systems and Nanostructures 77 (2016) 13–22.
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