Tunneling currents along the c-axis of the majority and minority spin electrons have been studied for a magnetic semiconductor (MS)/insulator (I)/superconductor (S) tunneling junction consisting of a Ga_{1−x}Mn_{
x}As MS with x = 1/32, a nonmagnetic I with a realistic dimension, and a HgBa2Ca2Cu3O8.4 (Hg-1223) high-Tc S. The normalized charge and spin currents, QT,Cμ′Vex and QT,Sμ′Vex, and the flows of the majority ↑ and minority ↓ spin electrons, QT,↑μ′Vex and QT,↓μ′Vex, have been calculated at a fixed external voltage Vex, as a function of the magnetic moment μ′ (≡μ/μB) per a Mn atom which is deduced from the band structure calculations. It is found that the tunneling due to the minority spin electron dominates when μ′<2.4, but such a phenomenon is not found for μ′>2.4. We have pointed out that the present MS/I/S tunneling junction seems to work as a switching device in which the ↑ and ↓ spin flows can be easily controlled by the external magnetic field.

1. Introduction

Spintronics, in which both the charge and spin of an electron should be controlled, is one of the most attractive subjects in solid state physics and technology. Therefore, if one can make a device in which the spin flow can be easily controlled, then such a device may play an essential role in the field of the spintronics. By using ferromagnetic materials such as 3d-transition metal compounds, spin-polarized electrons can be easily injected into the other materials including superconductors. In the nonsuperconducting states, a phenomenon such as the tunneling magnetoresistance (TMR) has been clearly observed in the magnetic tunneling junction (MTJ) consisting of two ferromagnetic (F) electrodes separated by an insulating (I) barrier, that is, F/I/F-junction. Parkin et al. [1] and Yuasa et al. [2] have measured very large TMR values for Fe/MgO/Fe junctions, and Belashchenko et al. [3] have theoretically studied the electronic structure and spin-dependent tunneling in epitaxial Fe/MgO/Fe(001) tunnel junctions and found that interface resonant states in Fe/MgO/Fe(001) tunnel junctions contribute to the conductance in the antiparallel configuration and are responsible for the decrease of TMR at a small barrier thickness, which explains the experimental results of Yuasa et al. [2].

Many studies have been done for the superconductor (S)/insulator (I)/superconductor (S) tunneling junctions, that is, Josephson junction, from the experimental and theoretical points of view. Barone and Paterno [4] presented to us a guide principle to study the Josephson effect. We have also studied the current- (I-) voltage (V) characteristics observed in the BSCCO intrinsic Josephson junctions from both the experimentally [5–8] and theoretically [9–11]. If a junction is made from F and S layers, the further interesting phenomena could be observed. For such junctions, there are two valuable review articles, one is by Golubov et al. [12] and the other is by Buzdin [13]. The one of the interesting phenomena found in a junction consisting of the ferromagnetic F and superconducting S layers could be an occurrence of the S/F/Sπ-junction [14–21]. It has already been known that the π-junction is caused to the damped oscillatory behavior of the Cooper pair (CP) wave function in the ferromagnetic layer.

Very recently, we have theoretically studied the c-axis charge and spin currents in F/I/S tunneling junction [22], in which Hg-1223 copper-oxides high-Tc superconductor HgBa2Ca2Cu3O8.4 and a ferromagnetic Fe metal have been selected as the S and F layers. Our recent study [22] has showed that an interesting result such that the minority spin current exceeds the majority one is surely found in the junction consisting of the nonmagnetic insulating layer; however, more clear and remarkable result is found in the junction including the magnetic insulating layer. Magnetic insulator (MI) can be made by doping the magnetic impurities into the nonmagnetic insulator, but it may not be so easy to make a tunneling device such as F/MI/S junction whose magnetizations are in antiparallel configuration. In the present paper, therefore, we further study the magnetic semiconductor (MS)/insulator (I)/superconductor (S) tunneling junction. As S, the Hg-1223 copper-oxides high-Tc superconductor is selected again, and a Ga1-xMnxAs with x=1/32 is selected as MS. For the ferromagnetic III-V semiconductors, there is an excellent article written by Ohno [23], in which he presented the properties of III-V-based ferromagnetic semiconductors (In,Mn)As and (Ga,Mn)As. Some of the interesting results obtained for the Ga1-xMnxAs MS are that (1) no ferromagnetism is observed below x=0.005 and (2) the relation between x and the ferromagnetic transition temperature Tc(F) is found as Tc(F)≃2000x±10 K up to x=0.05. The x in the present study is fixed to 1/32=0.03125, so that the Tc(F) of the present Ga0.96875Mn0.03125As MS is calculated as 62.5±10 K. It is well known that the Ga1-xMnxAs MS shows some phases such as magnetic semiconductor, half-metal, and ferromagnetic metal due to the change of the magnetization, that is, the change of the external magnetic field. Therefore, it is expected that an interesting phenomenon could be observed in the current- (I-) voltage (V) characteristics of the present MS/I/S tunneling junction. This is a motivation of the use of MS.

The transition temperature Tc of Hg-based copper-oxides superconductors is fairly higher than the liquid nitrogen temperature TLN (=77 K), so that the Hg-1223 high-Tc superconductor with δ=0.4, that is, HgBa2Ca2Cu3O8.4 whose Tc is 135 K, has been selected as a superconducting layer S. As already stated, the ferromagnetic transition temperature Tc(F) of Ga1-xMnxAs MS with x=1/32 is calculated as about 60±10 K. Therefore, it is certain that the superconductivity of Hg-1223 high-Tc superconductor is fairly well kept at the temperature region below 70 K, since the Tc of the Hg-1223 high-Tc superconductor is 135 K. This is the reason why we have selected the Hg-1223 high-Tc superconductor as S layer.

The transport problem in the MS/S/MS tunneling junction has already been studied by Tao and Hu [24] and Shokri and Negarestani [25]. Here it should be noted that they have selected s-symmetry low-Tc superconductor as the S and adopted the Blonder-Tinkham-Klapwijk (BTK) model [26], which is based on the effective mass approximation. In the present paper, we consider the c-axis tunneling of the majority and minority spin electrons in Ga0.96875Mn0.03125 As MS/insulator I/HgBa2Ca2Cu3O8.4 high-Tc superconductor S tunneling junction within the framework of the tunneling Hamiltonian model. In the present junction, there are facts that (1) the electron states in the vicinity of the Fermi level EF mainly come from 3d orbitals of Mn and Cu atoms, (2) the density of states (DOS) that originated from the 3d orbital shows a pointed structure meaning the localized nature, on the contrary to the DOS from s and p orbitals which show a broadened structure, that is, the extended nature, therefore, (3) the effective mass approximation, which is valid for the extended nature, may not be so good for the present system in which the electron states near the EF are fairly well localized, and (4) the I layer is not a delta-functional but in a real dimensional size, whose barrier strength is large enough, so it must be noted that (5) the BTK model reaches the tunneling Hamiltonian model since the probability of Andreev reflection decreases with increasing the barrier strength of the I layer. The above are just a reason why we have adopted the tunneling Hamiltonian model based on the electrons with the Bloch states which are decided from the band structure calculations. It must be noted here that we do not set here a realistic size such as a width of the insulating layer. We think that it may be enough to state that the insulating layer works well as a tunneling barrier so that the tunneling Hamiltonian model is valid.

2. Theoretical

Tunneling current iT,σ(V) as a function of an applied voltage V of a ferromagnet- (F-) insulator- (I-) superconductor (S) tunneling junction is given by [22](1)iT,σ(V)=2πeℏT~2∑μS∑LSκσ(F)(μS,LS,V)≡iT,σ(F)(V),(2)κσ(F)(μS,LS,V)=ησ∑k2ΩSΘσ(F)(ξk2(S),Δk2,eV)λLS(μS)(k2)2,where ΩS is the first Brillouin zone of S. The λLS(μS)(k) is the coefficient in the expansion by the Bloch orbitals χLS(μS)(k,r) of the total wave funtion Ψk(r) of S such that(3)Ψk(r)=∑μS∑LSλLS(μS)(k)χLS(μS)(k,r),where μS and LS are the site to be considered and the quantum state of atomic orbital of S, respectively.

As already stated in our previous paper [22], the ησ is the tunneling probability of a σ-spin electron in the F/I/S tunneling junction defined by(4)ησ=Tσ2T↑2+T↓2≡Tσ2T~2,so that the value of ησ strongly depends on the magnetic nature of an insulating layer I. It is clear that when the I shows no magnetic nature, the tunneling probabilities of majority and minority spin electrons should be equal; that is, η↑=η↓=1/2, and when the I shows magnetic nature, those should differ from each other; that is, η↑≠η↓. In the present study, only the nonmagnetic I layer is considered, so that the tunneling probabilities η↑ and η↓ of the majority and minority spin electrons are equal to each other; that is, only the case of T↑2=T↓2 is considered here. As a tunneling process, coherent, incoherent, and WKB cases can be considered. In the present paper, the incoherent tunneling is mainly studied. The reason is described later.

In the incoherent tunneling case, the Θσ(F)(ξk2(S),Δk2,eV) in (2), which is written as Θσ(F)(ξk2(S),Δk2,eV)Inc, is given by [22](5)ΘσFξk2S,Δk2,eVInc=fEk2-eV-fEk2DσFEk2-eV+fEk2-fEk2+eVDσF-Ek2-eV,where f is a Fermi-Dirac distribution function and Dσ(F)(x) is the TDOS of the ferromagnetic layer, that is, Ga1-xMnxAs with x=1/32MS layer, for σ spin state as a function of x. For the spin symbol σ used in our studies, it is noted that ↑ and ↓ mean the majority and minority spin electrons, respectively. The Ek is a quasiparticle excitation energy defined by ξk2+Δk2, where the ξk is one electron energy relative to the Fermi level EF and the Δk is a superconducting energy gap given by Δ(T)cos2θk.

The one electron energy ξk is calculated on the basis of the band theory using a universal tight-binding parameters (UTBP) method proposed by Harrison [27]. The energies of the atomic orbitals used in the band structure calculations have been calculated by using the spin-polarized self-consistent-field (SP-SCF) atomic structure calculations based on the Herman and Skillman prescription [28] using the Schwarz exchange correlation parameters [29]. The calculation procedure of the present band structure calculation is the same as that of our previous calculation [22]. Present band structure calculation for the Ga1-xMnxAs MS with x=1/32 has been done using a unit cell consisting of 8 cubes such as a 2×2×2-structure by a primitive cube. The unit cell includes 32 cations (=Ga or Mn) and 32 anions (=As); therefore, the condition x=1/32 used in the present study means that the one of the 32 Ga atoms is replaced by Mn atom.

3. Results and Discussion3.1. Density of States

The densities of states (DOSs) of Ga1-xMnxAs magnetic semiconductor MS with x=1/32 have been calculated as a function of the d-electron configuration of Mn atom. The electron configuration used in the spin-polarized self-consistent-field (SP-SCF) atomic structure calculation for the Mn atom is 3d↑x3d↓y4s↑14s↓14p↑04p↓0 with x+y=5, that for Ga atom is 4s↑14s↓14p↑0.54p↓0.5, and that for As atom is 4s↑14s↓14p↑1.54p↓1.5. The DOSs calculated by setting (x,y) to (2.5,2.5), (2.75,2.25), (3,2), (3.25,1.75), (3.5,1.5), (3.75,1.25), (4,1), (4.25,0.75), (4.5,0.5), (4.75,0.25), and (5,0) are shown in Figures 1(a), 1(b), 1(c), 1(d), 1(e), 1(f), 1(g), 1(h), 1(i), 1(j), and 1(k), respectively. Resultant magnetic moment μ/μB calculated per Mn atom is 0.246, 0.480, 0.980, 1.972, 3.340, 3.514, 3.684, 3.849, 4.021, and 4.128 for (b), (c), (d), (e), (f), (g), (h), (i), (j), and (k), respectively. Calculated DOSs clearly show that (a) is a nonmagnetic semiconductor, (b) is a ferromagnetic semiconductor, (c) is a ferromagnetic zero-gap semiconductor, (d) and (e) are ferromagnetic metals, (f) and (g) are half-metals, and (h), (i), (j), and (k) are ferromagnetic metals. The phase change mentioned above is closely related to the energy position of the e2-band of the minority spin electron denoted as e2↓-band. Actually, we can see that the e2↓-band of (d) locates below the Fermi level EF, that of (e) is very close to the EF, and that of (f) locates above the EF. The energy shift of the e2↓-band makes the rapid change of the magnetization of the Ga1-xMnxAs MS. Such a rapid change in the magnetic moment is really observed between (d) and (f).

Densities of states (DOSs) calculated for Ga1-xMnxAs magnetic semiconductor MS with x=1/32. The electron configuration used in the spin-polarized self-consistent-field (SP-SCF) atomic structure calculation for Mn atom is 3d↑x3d↓y4s↑14s↓14p↑04p↓0 with x+y=5, that for Ga atom is 4s↑14s↓14p↑0.54p↓0.5, and that for As atom is 4s↑14s↓14p↑1.54p↓1.5. (a), (b), (c), (d), (e), (f), (g), (h), (i), (j), and (k) are the results calculated for (x,y)=(2.5,2.5), (2.75,2.25), (3,2), (3.25,1.75), (3.5,1.5), (3.75,1.25), (4,1), (4.25,0.75), (4.5,0.5), (4.75,0.25), and (5,0), respectively. Magnetic moment μ/μB calculated per Mn atom is 0.246, 0.480, 0.980, 1.972, 3.340, 3.514, 3.684, 3.849, 4.021, and 4.128 for (b), (c), (d), (e), (f), (g), (h), (i), (j), and (k), respectively. Present results show that (a) is a nonmagnetic semiconductor, (b) is a ferromagnetic semiconductor, (c) is a ferromagnetic zero-gap semiconductor, (d) and (e) are ferromagnetic metals, (f) and (g) are half-metals, and (h), (i), (j), and (k) are ferromagnetic metals.

Finally, it is noted that the DOS of Hg-1223 high-Tc superconductor with δ=0.4, that is, HgBa2Ca2Cu3O8.4 with Tc=135 K, has already been given in our recent paper [22].

3.2. Spin Flow

First of all, we did calculations for two cases in which the sample temperature Tsamp has been set to 5 and 60 K. The BCS curve gives the values of 75 and 73.3 meV as the amplitudes Δ(T) of superconducting gap at Tsamp=5 and 60 K, respectively. The difference between these two values is small, so we have found that there is no significant difference between the current- (I-) voltage (V) characteristics calculated for these two temperatures, as is expected. In the following, therefore, the Tsamp is set to 5 K. Here note that the magnetic field dependence of the magnetization of Ga1-xMnxAs MS with x=0.035 has already been measured at 5 K by Ohno [23].

The calculations for the coherent and WKB cases need a very large CPU time as compared with the incoherent one [22]. Therefore, first, the I-V characteristics for a given 3d electron configuration, that is, a resultant magnetic moment, have been calculated for the coherent, incoherent, and WKB cases. As a result, we have found that the results calculated for three cases are fairly similar to each other. In the following, therefore, only the incoherent tunneling case is considered because of the CPU times in the numerical calculations.

The normalized charge and spin currents, iT(+)(V)Nor and iT(-)(V)Nor, calculated for the present MS/I/S tunneling junction are shown in Figure 2. Here note that the iT(+)(V)Nor and iT(-)(V)Nor have already been defined by (11) in our recent paper [22] and that the MS used in (a) to (k) in Figure 2 is the same as that in (a) to (k) in Figure 1. Figure 2 clearly shows that the charge and spin currents are changed due to the change of the magnetization of MS. In order to directly see the currents due to the majority (↑) and minority (↓) spin electrons, we have drawn in Figure 3 the normalized currents calculated for the ↑ and ↓ spin electrons, iT,↑(V)Nor and iT,↓(V)Nor. Here note that the normalized current iT,σ(V)Nor is equal to ∑μS∑LSκσ(F)(μS,LS,V)Nor defined by (12) in our previous paper [22], so that a relation iT(±)(V)Nor=iT,↑(V)Nor±iT,↓(V)Nor is satisfied. (a) to (k) in Figure 3 correspond to those in Figure 2. Figure 3 shows that the tunneling nature changes due to the change of the magnetic moment, that is, the magnetization of Ga1-xMnxAs MS. For example, if the normalized voltage is fixed to 4, then we can see in (b), (c), (d), and (e) an interesting result such that the tunneling current due to the ↓ spin electron is larger than the ↑ one, but such a result is not found in (f), (g), (h), (i), (j), and (k). It is clear that the result is closely related to the electronic structural change of MS which causes the change of the magnetization.

Normalized charge and spin currents iT(+)(V)Nor and iT(-)(V)Nor calculated for MS/I/S tunneling junction, where the MS is Ga1-xMnxAs with x=1/32 and the S is Hg-1223 high-Tc superconductor with δ=0.4. The MS used in (a) to (k) is the same as that in (a) to (k) in Figure 1. The normalized voltage is defined by eV/Δ(T). Note that Δ(5)=75 meV.

Normalized currents iT,↑(V)Nor and iT,↓(V)Nor calculated for the majority (↑) and minority (↓) spin electrons. (a) to (k) correspond to those in Figure 2. The normalized voltage is defined by eV/Δ(T), where Δ(5)=75 meV.

Experimentally, it may be possible to observe the external magnetic field dependence of the tunneling current at a fixed external voltage Vex. In order to reproduce such an experimental situation, we have calculated the magnetic moment dependence of charge and spin currents. The normalized charge current QT,C(F)(Vex) for the present purpose is defined by(6)QT,CFVex≡∑σiT,σFVex∑σiT,σNMVex=∑σ∑μS∑LSκσ(F)(μS,LS,Vex)∑σ∑μS∑LSκσ(NM)(μS,LS,Vex)≡QT,C(μ′)(Vex).Here NM means the nonmagnetic phase, so that the magnetic moment μ/μB (≡μ′) per Mn atom is 0. F means the ferromagnetic phase; therefore, in the following, the symbol F is replaced by the symbol μ′. Using (6), we can calculate the normalized charge current QT,C(μ′)(Vex) as a function of μ′. The raw values of ∑μS∑LSκ↑(F)(μS,LS,Vex) and ∑μS∑LSκ↓(F)(μS,LS,Vex) have already been calculated numerically, so that the ratio of those raw values is easily given. As a result, we can get the normalized spin current QT,S(μ′)(Vex) as a function of μ′. The calculated normalized charge and spin currents, QT,C(μ′)(Vex) and QT,S(μ′)(Vex), are shown in Figure 4(a) as a function of the calculated μ′. By using the above QT,C(μ′)(Vex) and QT,S(μ′)(Vex), we can easily get the flows QT,↑(μ′)(Vex) and QT,↓(μ′)(Vex) of the majority (↑) and minority (↓) spin electrons. Those are shown in Figure 4(b) as a function of μ′. Figure 4 shows that the nature of the spin flow is changed at the μ′ with the value around 2.4. Namely, the tunneling due to the minority spin electron dominantly occurs when μ′<2.4, but for the case of μ′>2.4, such a tunneling phenomenon is not found. It is certain that such a change is closely related to the variation of the e2↓-band of the Ga1-xMnxAs MS. The value of the magnetization M can be easily controlled by the external magnetic field Bext. The M-Bext curve at 5 K of Ga1-xMnxAs MS with x=0.035 has already been drawn in Figure 3 in Ohno’s paper [23], which clearly shows that the Bext=0.02 T is a large external magnetic field enough to get the saturation of the magnetization. Here note that we have checked that the magnetic induction with the value of 0.02 T has no considerable effect on the present superconductor.

(a) Normalized charge and spin currents, QT,C(μ′)(Vex) and QT,S(μ′)(Vex), defined by (6) and (b) flows of the majority (↑) and minority (↓) spin electrons, QT,↑(μ′)(Vex) and QT,↓(μ′)(Vex). Those have been calculated as a function of μ′, where μ′ is the magnetic moment μ/μB calculated for Mn atom. The Vex is the normalized voltage applied to the MS/I/S tunneling junction, which has been set to 4, that is, 300(=4×75) mV in real voltage. Note that relations QT,↑(μ′)(Vex)=QT,C(μ′)(Vex)+QT,S(μ′)(Vex)/2 and QT,↓(μ′)(Vex)=QT,C(μ′)(Vex)-QT,S(μ′)(Vex)/2 are held.

3.3. Effect of Nonequilibrium

We are now considering the superconductors consisting of Cooper pairs (CPs) with a spin-singlet state. In the junctions involving the ferromagnetic materials and the superconductors, therefore, it is easily supposed that the unbalance in the numbers of the ↑ and ↓ spin electrons makes a decrease in the number of the CPs. This is just a nonequilibrium effect. The decrease in the number of CPs makes a decrease in the amplitude Δ(T) of the superconducting gap. Therefore, in order to take into account the influence of such a nonequilibrium effect, we have introduced a parameter ζ with a range of 0<ζ≤1, by which the Δ(T) is reduced to ζΔ(T). It is clear that the case of ζ=1 means no consideration for the nonequilibrium effect.

Figure 1(e) shows that the Fermi level just locates on the e2↓-band with a pointed shape, so that a sizable unbalance in the numbers of the ↑ and ↓ spin electrons could be found in this case. Nevertheless, the nonequilibrium effect should not be so large; therefore, as an attempt we have calculated the I-V characteristics by setting ζ to 0.8. The normalized currents calculated for ζ=1 and 0.8 are shown in Figures 5(a) and 5(b), respectively. Apart from the reliability of the value of 0.8, the calculated results have showed that there is no significant difference between them. The above result tells us that a considerable nonequilibrium effect could not be found in the I-V characteristics of the MS/I/S tunneling junction studied here. This means that the present MS/I/S tunneling junction stably works as a device to switch the ↑ and ↓ spin flows by varying the Bext within the range of Bext<0.02 T.

Normalized currents iT,↑(V)Nor and iT,↓(V)Nor calculated for the majority (↑) and minority (↓) spin electrons. (a) is the same as Figure 3(e); that is, the magnetic moment μ/μB per Mn atom is 1.972 and ζ is 1. (b) is the same as (a) but the ζ has been set to 0.8; that is, the normalized currents shown in (b) include the nonequilibrium effect.

4. Summary

The c-axis tunneling of the majority and minority spin electrons has been studied for the MS/I/S tunneling junction consisting of Ga1-xMnxAs magnetic semiconductor MS with x=1/32, an insulator I with a realistic dimension, and HgBa2Ca2Cu3O8.4 (Hg-1223) high-Tc superconductor S. We have deduced the magnetic moment μ′ (≡μ/μB) per Mn atom from the band structure calculations for the Ga1-xMnxAs MS and calculated the normalized charge and spin tunneling currents, QT,C(μ′)(Vex) and QT,S(μ′)(Vex), and the flows of the majority (↑) and minority (↓) spin electrons, QT,↑(μ′)(Vex) and QT,↓(μ′)(Vex), as a function of μ′ at a given external voltage Vex. We have found that the tunneling due to the minority spin electron dominantly occurs when μ′<2.4, but such a phenomenon is not found in the case of μ′>2.4. We have pointed out that the present MS/I/S tunneling junction seems to work as a switching device in which the ↑ and ↓ spin flows can be easily controlled by varying slightly the external magnetic field.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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