Charge and spin currents along the c-axis in ferromagnet-insulator-superconductor (F/I/S) tunneling junctions have been studied within the framework of the tunneling Hamiltonian model. As a superconductor S, HgBa_{2}Ca_{2}Cu_{3}O_{8+δ
} (Hg-1223) with δ=0.4 copper-oxide high-Tc superconductor has been selected, and as a ferromagnet F, Fe metal with bcc structure has been selected for simplicity. The electronic structures of above materials have been calculated on the basis of the band theory using the spin-polarized self-consistent-field data for the atomic orbital energies and the universal tight-binding parameters (UTBP) for the interactions. For the η↑ and η↓(=1-η↑) defined in the present paper, which are tunneling probabilities of the majority and the minority spin electrons, it is shown that the condition η↑=η↓ means the standard F/I/S tunneling junction with a nonmagnetic insulating layer, and the condition η↑≠η↓means the F/I/S tunneling junction with a magnetic insulating layer showing a detectable magnetization. We have found that the charge current and the differential conductance nearly remain the same as the change of η↑, but the spin current is largely changed due to the change of η↑. As an experimental method to detect the change of the spin current, the validity of an X-ray magnetic circular dichroism (XMCD) has been pointed out.
1. Introduction
In 1982, Blonder, Tinkham, and Klapwijk (BTK) presented a pioneering paper for an interface at the normal (N) material and superconductor (S), in which they proposed a simple theory for the current-voltage (I-V) curves of normal (N)-superconducting (S) microconstriction contacts which describes the crossover from metallic to tunnel junction behavior [1]. Their model based on the Bogoliubov-de Gennes (BdG) equation, now called as “BTK model," worked well to understand the transmission and reflection of particle at N-S interface. Their results told us that the probability of Andreev reflection decreases with increasing the barrier strength Z at the interface and the BTK model reaches to the tunneling Hamiltonian model due to the increasing Z-value. Based on the BTK model, Kashiwaya et al. theoretically studied the origin of zero-bias conductance peaks (ZBCPs) observed in the YBCO high-Tc superconductors and found that the calculation is in good agreement with the experiment [2]. The ZBCPs are observed at the I-V characteristics of the NS-interface. Within the framework of the BTK model, Kashiwaya et al. further studied the I-V characteristics of N/I/S and F/I/S junctions [3, 4]. Here it is noted that they have adopted the BTK model, so that the potential of the real insulating layer I has been treated as a delta-functional and the tunneling electron has been regarded as a free particle under the effective mass approximation. Similar considerations have also been done by Annunziata et al. [5].
In the present paper, we consider the c-axis electron tunneling in the F/I/S tunneling junction using the d-wave high-Tc superconductors. The insulating layer I with a real dimension is considered here, so that the tunneling Hamiltonian model could be reliable for the present purpose because of a large Z-value. Furthermore, the electron is treated as not free but Bloch type, so that the electron states have been calculated on the basis of the band theory. It is well known that the superconducting energy gaps Δ of d-wave copper-oxide high-Tc superconductors strongly depend on the wavenumber vector k characterizing the CP, that is, Δ=Δk, and that the k-dependence of Δk is given by kx2-ky2. It should be emphasized here that, in the tunneling Hamiltonian model adopted in the present paper, the k-dependence of the Δk is exactly taken into account in all the calculation procedures.
Recently, an interesting experimental paper has been presented by Takeshita et al. [6], in which they have measured the electrical resistivity of a HgBa2Ca2Cu3O8+δ (Hg-1223) polycrystalline sample under the high pressure and found the Tc with 153.0 K at the hydrostatic pressure of 15 GPa. It was already reported and well known that the Hg-based copper-oxides high-Tc superconductors HgBa2Can-1CunO2n+2+δ with n=1,2,3 and 4, which are denoted as Hg-1201, Hg-1212, Hg-1223, and Hg-1234, show higher Tc as compared with the other copper-oxides superconductors such as BSCCO and YBCO. The Tc in K is 97, 127, 135, and 127, respectively [7], for the Hg-1201, -1212, -1223, and -1234, and the superconducting gap at low temperature region Δ(0) in meV is 33, 50, and 75 for the Hg-1201, -1212, and -1223, respectively [8]. It has been reported that the measured values of δ in high-Tc samples are approximately 0.06 in Hg-1201, 0.22 in Hg-1212, and 0.4 in Hg-1223 [9]. The Tc of Hg-based copper-oxides superconductors is fairly higher than the liquid nitrogen temperature TLN(=77 K), so it is surely expected that the superconductivity is well kept even in the case that the Hg-based copper-oxides superconductors have been operated at TLN. From the experimental and the applicational points of view, in the present paper, the Hg-1223 high-Tc superconductor with δ=0.4, that is, HgBa2Ca2Cu3O8.4 whose Tc is 135 K, is selected as the superconducting (S) layer. The sample temperature in the present calculations is fixed to TLN, so that the value of TLN/Tc is 0.57. The value of Δ(0) is 75 meV, so that the BCS curve for the energy gap Δ(τ) as a function of the reduced temperature τ≡T/Tc gives a value of 70 meV at T=TLN.
In the present paper, for simplicity, Fe metal with bcc structure has been adopted as a ferromagnetic (F) layer. It is sure that the electron states in the vicinity of the Fermi level mainly come from 3d orbital of the transition metals and that 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. It may be sure that the effective mass approximation works well for the extended nature but not so for the localized one. This is just a reason why we have adopted the tunneling Hamiltonian model based on the electrons with the Bloch states.
2. Theoretical
The charge and spin currents, IT(+)(V) and IT(-)(V), as a function of an applied voltage V can be evaluated by using (A.8) and (A.9). In the following, therefore, we consider the tunneling current iT,σ(V) given by (A.9); that is,
(1)iT,σ(V)=2πeℏ∑k1∑k2|Tk1k2σ|2g(ξk1σ(F),ξk2(S),Δk2,eV),(2)g(ξk1σ(F),ξk2(S),Δk2,eV)={(f(ξk1σ(F))-f(Ek2))δ(ξk1σ(F)+eV-Ek2)+(f(ξk1σ(F))+f(Ek2)-1)δ(ξk1σ(F)+eV+Ek2)}.
Generally, |Tk1k2σ|2 can be written as |Tσ|2τ(k1,k2). Here |Tσ|2 is a value depending only the spin state σ and the τ(k1,k2) is a dimensionless value as a function of k1 and k2. As a value of τ(k1,k2), we consider here three cases; that is, the first one is the coherent tunneling case denoted by τCoh(k1,k2), the second one is the incoherent case τInc(k1,k2), and the third one is the case based on the WKB approximation τWKB(k1,k2). It is clear that τCoh(k1,k2)=δk1,k2 and τInc(k1,k2)=1. In the WKB treatment, the momentum parallel to the junction plane is conserved in the tunneling processes [11]. If we use a geometrical configuration such that the axis normal to the junction plane is z-axis which is parallel to the c-axis, then τWKB(k1,k2) can be written as q1zq2zδk1x,k2xδk1y,k2y, where q1z≡k1z/k1z(max) and q2z≡k2z/k2z(max). Here it should be noted that two wavenumbers k1z(max) and k2z(max) have been introduced so as to satisfy the condition such that the q1z and q2z must be the dimensionless values. Namely, it must be emphasized that the amplitudes of k1z(max) and k2z(max) are not essential in the evaluation of τWKB(k1,k2) and that the influence of the amplitudes k1z(max) and k2z(max) is included in the value of |Tσ|2. From the above, the iT,σ(V) given by (1) can be rewritten as follows:
(3)iT,σ(V)=2πeℏ∑k2|Tσ|2Θσ(F)(ξk2(S),Δk2,eV).
For the function Θσ(F)(ξk2(S),Δk2,eV), there are three functions Θσ(F)(ξk2(S),Δk2,eV)Coh, Θσ(F)(ξk2(S),Δk2,eV)Inc, and Θσ(F)(ξk2(S),Δk2,eV)WKB corresponding to the coherent, incoherent, and WKB cases. Those are given as follows:
(4)Θσ(F)(ξk2(S),Δk2,eV)Coh=g(ξk2σ(F),ξk2(S),Δk2,eV),(5)Θσ(F)(ξk2(S),Δk2,eV)Inc={{f(Ek2-eV)-f(Ek2)}Dσ(F)(Ek2-eV)+{f(Ek2)-f(Ek2+eV)}Dσ(F)(-Ek2-eV)},(6)Θσ(F)(ξk2(S),Δk2,eV)WKB=∑k1zq1zq2zg(ξk1′σ(F),ξk2(S),Δk2,eV),
where k1′≡(k2x,k2y,k1z) and the Dσ(F)(x) is the total density of states (TDOS) of the ferromagnetic layer F for the σ-spin state as a function of x.
In the present paper, one-electron energies such as ξk1σ(F) and ξk2(S), which are for ferromagnetic F and superconducting S layers, are calculated on the basis of the band theory using a universal tight-binding parameters (UTBP) method proposed by Harrison [12]. 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 [13] using the Schwarz exchange correlation parameters [14]. The calculation procedure of the present band structure calculation is the same as that of our previous band structure calculation [15].
The energy of an electron with a σ-spin state is calculated for the wavenumber k within the 1st Brillouin zone (BZ) Ω and the total wave function Ψkσ(r) is expanded by using Bloch orbitals χL(μ)(kσ,r) as follows:
(7)Ψkσ(r)=∑μ∑LλL(μ)(kσ)χL(μ)(kσ,r),
where μ and L are the site to be considered and the quantum state of atomic orbital, respectively, and λL(μ)(kσ) is the coefficient in the Bloch expansion of the total wave funtion. Therefore, the following relation is found for the superconducting layer S:
(8)〈Ψk2(r)∣Ψk2(r)〉=∑μS∑LS|λLS(μS)(k2)|2=1.
Here it is noted again that the k2σ can be regarded as k2 when the CPs in superconducting layer S are in the spin-singlet state. An index “S" in μS and LS means the superconducting layer S, so the μS and LS are the site and quantum state adopted in the band structure calculation of the S.
Using (8), (3) can be rewritten as follows:
(9)iT,σ(V)=2πeℏT~2∑μS∑LSησ∑k2ΩSΘσ(F)(ξk2(S),Δk2,eV)|λLS(μS)(k2)|2≡2πeℏT~2∑μS∑LSκσ(F)(μS,LS,V),
where ΩS is the first BZ of S. ησ is defined by
(10)ησ=|Tσ|2|T↑|2+|T↓|2≡|Tσ|2T~2.
The ησ defined here is the tunneling probability of a σ-spin electron in the F/I/S tunneling junction, so it is sure 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 that when the I shows magnetic nature, those should differ from each other, that is, η↑≠η↓. As already stated, the present model does not explicitly include the details of the magnetic nature of the I; therefore, the value of η↑(=1-η↓) should be treated as a parameter if we wish to extend the present model to the case in which the I shows the magnetic nature. It is sure that when the magnetization of F, MF, is parallel to that of I, MI, that is, MF∥MI, the current due to the majority spin IMaj is larger than the minority one IMin, and in the case of MF∥-MI, the IMaj is largely decreased as compared with the parallel case MF∥MI. Namely, a relation η↑>η↓ should be satisfied for the parallel case MF∥MI, and a relation η↑<η↓ should be satisfied for the antiparallel case MF∥-MI. In the present paper, therefore, η↑(=1-η↓) is treated as a parameter characterizing the magnetic nature of the insulating layer I.
The κσ(F)(μS,LS,V) defined in (9) shows the partial current term due to the tunneling of a σ spin electron. Only the band structure calculation makes it possible to evaluate the κσ(F)(μS,LS,V). Therefore, from the calculations of κσ(F)(μS,LS,V), we can directly see which atomic orbitals contribute to the F/I/S tunneling phenomena. This is just an advantage of the use of the κσ(F)(μS,LS,V).
In the calculation of the I-V characteristics, if we write the maximum voltage Vmax(>0) as nmaxΔ(T)/e, that is, eVmax=nmaxΔ(T), then the calculation done for nmax≃5 is good enough to see the overall profile of the I-V characteristics, since the I-V curve gradually reaches to the ohmic line due to increasing the voltage V. On the practical point of view, it is convenient to define the normalized current iT(±)(V)Nor. We define it as follows:
(11)iT(±)(V)Nor≡nmaxIT(±)(V)IT(+)(Vmax)=∑μS∑LS(κ↑(F)(μS,LS,V)Nor±κ↓(F)(μS,LS,V)Nor).
Here, κσ(F)(μS,LS,V)Nor is defined by
(12)κσ(F)(μS,LS,V)Nor≡nmaxκσ(F)(μS,LS,V)∑μS∑LS∑σκσ(F)(μS,LS,Vmax)≡nmaxK(F)(Vmax)κσ(F)(μS,LS,V),
using κσ(F)(μS,LS,V) defined in (9). Here note that K(F)(Vmax)>0.
From the above considerations, it is concluded that, in order to understand the nature of the charge and spin transfers in the F/I/S tunneling junctions, it is necessary to calculate the κσ(F)(μS,LS,V)Nor defined by (12), that is, κσ(F)(μS,LS,V) defined in (9), as correctly as possible.
On the above considerations, we have implicitly assumed that the external magnetic field H applying the junctions is not strong; that is, the Zeeman energy μBH is small as compared with the amplitude of the superconducting gap. Therefore, the effect of the magnetic field H can be ignored in the present considerations. Usually, such a treatment has no problem because the value of μBH is very small; for example, that is only 5.79 meV even in the case of the extremely strong magnetic field H=100 T. Exactly speaking, however, the effect of the Zeeman energy must be taken into account. In the present paper, its effect has been taken into account by using the same method done by Tedrow and Meservey [16]. Namely, the quasiparticle excitation energy Ek has been replaced by Ek∓μBH for the majority and the minority spin, respectively. Actually, we did a calculation for the external magnetic induction B with the value of 1 T and found that there is no detectable difference between the calculations for B=0 and 1 T. In the present paper, therefore, the effect of the external magnetic field has not been considered anymore.
3. Results and Discussion3.1. Electronic Structures
The electronic structures of a Hg-1223 high-Tc superconductor with δ=0.4, that is, HgBa2Ca2Cu3O8.4, and the ferromagnetic Fe with bcc structure have been calculated on the basis of the band theory using the UTBP method. In the band structure calculations, the following atomic orbitals have been used: 6s and 6p orbits of Hg, 6s and 6p ones of Ba, 4s and 4p of Ca, 3d, 4s, and 4p of Cu, 2s and 2p of O, and 3d, 4s, and 4p orbits of Fe. In order to get the electronic structure of HgBa2Ca2Cu3O8.4 as correctly as possible, we have calculated it by using a unit cell consisting of 2Hg, 4Ba, 4Ca, 6Cu, and 17O. The unit cell of Hg2Ba4Ca4Cu6O17(= HgBa2Ca2Cu3O8.5) is shown in Figure 1, and the crystallographic data for Hg-1223 with δ=0.44, HgBa2Ca2Cu3O8.44, are tabulated in Table 1. In the present paper, the electronic structure of HgBa2Ca2Cu3O8.4 has been obtained by hole-doping into the electronic structure of HgBa2Ca2Cu3O8.5 we have calculated. The calculated densities of states (DOSs) are shown in Figure 2. Figure 2 clearly shows that the electronic structures near the Fermi level EF are made from Cu 3d and O 2p orbitals. This result allows us to suppose that the electronic structures near the EF come from CuO2 layers in Hg-1223 high-Tc superconductor. Actually, we have clearly found it from the calculated results. This is a common feature found in all the copper-oxide high-Tc superconductors.
Crystallographic data [8] of unit cell HgBa2Ca2Cu3O8.5=Hg2Ba4Ca4Cu6O17 shown in Figure 1. Lattice constants a(=b) and c are 3.8502 and 31.5658 Å, respectively. The position (x,y,z) of an atom in the unit cell is defined as (ua,vb,wc), and the values of u, v, and w are tabulated here.
Atom
u
v
w
Hg1
0
0
0
Hg2
0
0
0.5
Ba1
0.5
0.5
0.08865
Ba2
0.5
0.5
0.4113
Ba3
0.5
0.5
0.5887
Ba4
0.5
0.5
0.9113
Ca1
0.5
0.5
0.1977
Ca2
0.5
0.5
0.3023
Ca3
0.5
0.5
0.6977
Ca4
0.5
0.5
0.8023
Cu1
0
0
0.1489
Cu2
0
0
0.25
Cu3
0
0
0.3511
Cu4
0
0
0.6489
Cu5
0
0
0.75
Cu6
0
0
0.8511
O1
0
0
0.0618
O2
0
0.5
0.1489
O3
0.5
0
0.1489
O4
0
0.5
0.25
O5
0.5
0
0.25
O6
0
0.5
0.3511
O7
0.5
0
0.3511
O8
0
0
0.4382
O9
0.5
0.5
0.5
O10
0
0
0.5618
O11
0
0.5
0.6489
O12
0.5
0
0.6489
O13
0
0.5
0.75
O14
0.5
0
0.75
O15
0
0.5
0.8511
O16
0.5
0
0.8511
O17
0
0
0.9382
Unit cell of Hg2Ba4Ca4Cu6O17(=HgBa2Ca2Cu3O8.5) used in the band structure calculation. Lattice constants a(=b) and c are 3.8502 and 31.5658 Å, respectively.
Density of states (DOS) (states/eV·unit cell) calculated for the high-Tc superconductor HgBa2Ca2Cu3O8.4. Energy (eV) is measured from Fermi level EF. Here note that the electronic structure of HgBa2Ca2Cu3O8.4 is obtained by hole-doping into that of HgBa2Ca2Cu3O8.5. (a) is the total density of states (TDOS). (b), (c), (d), (e), and (f) are sum of partial density of states (PDOS) of respective atoms, namely, those are that of 2 Hg, 4 Ba, 4 Ca, 6 Cu and 17 O atoms, respectively. For example, the DOS of Cu: 3d drawn in (e) has been obtained by adding the Cu: 3d PDOSs of six Cu atoms (Cu1 to Cu6) shown in Figure 1. Here note that the Cu: 3d PDOS of the respective Cu atom is made by adding the five PDOSs for Cu 3d_{{xy}}, 3d_{{yz}}, 3d_{{zx}}, 3d{x2-y2} and 3d{3z2-r2} orbitals. The present figure clearly shows that the states near the EF are made from Cu 3d and O 2p orbitals shown in (e) and (f).
The electron configuration used in the SP-SCF atomic structure calculation for the spin-polarized iron is 3d↑x3d↓y4s↑14s↓14p↑04p↓0, where x+y=6. By using the spin-polarized self-consistent atomic data, the electronic structures of ferromagnetic Fe have been calculated for bcc structure. The calculated results are shown in Figures 3(a) to 3(e). Figure 3(a) is for nonmagnetic phase, that is, (x,y)=(3,3), and Figures 3(b), 3(c), 3(d), and 3(e) are for (x,y)=(3.5,2.5), (4,2), (4.5,1.5), and (5,1). Magnetic moment μ/μB calculated per an atom is 0.639, 2.067, 2.507, and 2.900 for Figures 3(b), 3(c), 3(d), and 3(e). Here note that the measured value of μ is 2.218μB [10].
TDOSs (states/eV·atom) calculated for bcc Fe. The electron configuration used in the SP-SCF atomic structure calculation is 3d↑x3d↓y4s↑14s↓14p↑04p↓0, where x+y=6. (a), (b), (c), (d), and (e) are the results calculated for (x,y)=(3,3), (3.5,2.5), (4,2), (4.5,1.5), and (5,1). The values of DOS Dσ(F)(0) at EF are follows: (D↑(F)(0),D↓(F)(0))=(2.829,2.829), (2.669,1.403), (1.533,1.355), (1.368,0.921), and (1.091,1.743) for (a), (b), (c), (d), and (e), respectively. Here note that a relation D↑(F)(0)>D↓(F)(0) is satisfied except for (e). Magnetic moment μ/μB calculated per an atom is 0.639, 2.067, 2.507, and 2.900 for (b), (c), (d), and (e), respectively. The calculated value should be compared with the experimental one 2.218μB [10].
As a tunneling process, we study here three processes such as coherent, incoherent, and WKB cases. In order to do so, we must evaluate the value of the function Θσ(F)(ξk2(S),Δk2,eV) in (3). The TDOS Dσ(F)(x) in (5) can be calculated separately; therefore, many CPU time is not required to calculate the function Θσ(F)(ξk2(S),Δk2,eV)Inc for the incoherent case, because two band structure calculations for the S and F layers can be done independently. In the calculations of Θσ(F)(ξk2(S),Δk2,eV)Coh and Θσ(F)(ξk2(S),Δk2,eV)WKB, however, many CPU time is needed since two band structure calculations for the S and F are correlated with each other, as can be seen from (4) and (6). In the calculation of Θσ(F)(ξk2(S),Δk2,eV)WKB, moreover, the summation over k1z must be done so that a very large CPU time is needed for the WKB case.
The I-V characteristics calculated for three cases are shown in Figure 4. Figures 4(a), 4(b), and 4(c) on the first line are results calculated for the nonmagnetic Fe, that is, N/I/S junction, Figures 4(d), 4(e), and 4(f) on the second one are for the spin-polarized Fe of the electron configuration 3d↑x3d↓y with (x,y)=(4,2), and Figures 4(g), 4(h), and 4(i) on the third one are for that with (x,y)=(5,1). The curves drawn in Figures 4(a), 4(d), and 4(g), which are on the left column, are results calculated as the coherent tunneling, those in Figures 4(b), 4(e), and 4(h) on the central column are those as the incoherent one, and those in Figures 4(c), 4(f), and 4(i) on the right one are those as the WKB treatment. When the tunneling occurs coherently, Figures 4(a), 4(d), and 4(g) clearly show some regions in which the differential conductance is evaluated as the negative value which is not reasonable. Namely, the present calculation tells us that the coherent tunneling does not occur in the F/I/S and N/I/S junctions considered here. It should be noted here that the accumulation number of the k2 in (3) varies as a function of the applied voltage V. This may be the reason why the negative differential conductance is found in the case of coherent tunneling.
I-V characteristics calculated for the ferromagnet-insulator-superconductor (F/I/S) tunneling junctions. Ferromagnet is an iron with bcc structure and the superconductor is Hg-1223 high-Tc superconductor HgBa2Ca2Cu3O8.4. The normalized charge current iT(+)(V)Nor is defined by (11), and the normalized voltage VN is defined by eV/Δ(T). (a), (b), and (c) on the first line are results calculated for the nonmagnetic phase, (d), (e), and (f) on the second line are for the electron configuration 3d↑x3d↓y with (x,y)=(4,2), and (g), (h), and (i) on the third one are for that with (x,y)=(5,1). (a), (d), and (g) on the left column are results calculated for the case of coherent tunneling, (b), (e), and (h) on the central one are for the case of incoherent tunneling, and (c), (f), and (i) on the right column are for the case of WKB treatment. In (a), (d), and (g) calculated as the coherent tunneling, there are some regions in which the differential conductance is calculated as the negative value which is unreasonable. Moreover, also in (c) for the WKB case, the region of the negative differential conductance is found in the normalized voltage around -3. Note that all the calculations have been done for η↑=η↓=1/2.
In the results calculated as the incoherent and WKB cases, there is no region of the negative differential conductance, except for the region around the normalized voltage -3 in Figure 4(c) for the WKB treatment. Furthermore, it seems that the overall profiles of the I-V characteristics calculated for the incoherent and WKB cases are similar to each other. As already stated, the numerical calculations for the WKB case need a huge CPU time. In the following, therefore, we present only the results calculated for the incoherent case, that is, the case such that |Tk1k2σ|2 is evaluated as |Tσ|2 depending only on the spin valuable σ.
The I-V characteristics calculated for a ferromagnet-insulator-superconductor (F/I/S) tunneling junction are shown in Figure 5. The normalized charge and spin currents, iT(+)(V)Nor and iT(-)(V)Nor defined by (11), are drawn together with the corresponding differential conductance dIN/dVN obtained from the normalized current IN and the normalized voltage VN. Here note that the VN is defined by eV/Δ(T) using the real voltage V and the amplitude Δ(T) of Δk. In Figure 5, (a), (b), and (c) on the first line are results calculated for the N/I/S junction; that is, the iron is in nonmagnetic phase, (d), (e), and (f) on the second line are those for the F/I/S junction calculated by using a ferromagnetic iron with the 3d electron configuration 3d↑x3d↓y of (x,y)=(3.5,2.5), (g), (h), and (i) on the third line are those of (x,y)=(4,2), (j), (k), and (l) on the fourth one are of (x,y)=(4.5,1.5), and (m), (n), and (o) on the bottom line are those of (x,y)=(5,1). The I-V characteristics drawn in (a), (d), (g), (j), and (m) on the left column have been obtained by using (η↑,η↓)=(3/4,1/4), those in (b), (e), (h), (k), and (n) on the middle column are for (η↑,η↓)=(1/2,1/2), and those in (c), (f), (i), (l), and (o) on the right column are for (η↑,η↓)=(1/4,3/4).
(a), (b), and (c) on the first line are results calculated for the N/I/S junction; that is, the iron is in nonmagnetic phase, whose DOS is shown in Figure 3(a). (d) to (o) are for the F/I/S junction. (d), (e), and (f) on the second line are results calculated for a ferromagnetic iron with the electron configuration 3d↑x3d↓y of (x,y)=(3.5,2.5), whose DOS is shown in Figure 3(b). (g), (h), and (i) on the third line are those for (x,y)=(4,2), whose DOS is shown in Figure 3(c), (j), (k), and (l) on the fourth one are for (x,y)=(4.5,1.5), whose DOS is shown in Figure 3(d), and (m), (n), and (o) on the bottom line are for (x,y)=(5,1), whose DOS is shown in 3(e). The I-V characteristics drawn in (a), (d), (g), (j), and (m) on the left column have been obtained by setting a parameter η↑(=1-η↓) to 3/4, those in (b), (e), (h), (k), and (n) on the middle column are for η↑=1/2, and those in (c), (f), (i), (l), and (o) on the right column are for η↑=1/4.
The normalized current iT(±)(V)Nor defined by (11) can be rewritten as follows:
(13)iT(±)(V)Nor=nmaxK(F)(Vmax)∑μS∑LS∑k2|λLS(μS)(k2)|2×(η↑Θ↑(F)(ξk2(S),Δk2,eV)±η↓Θ↓(F)(ξk2(S),Δk2,eV)).
Here, Θσ(F)(ξk(S),Δk,eV)Inc for the incoherent case defined by (5) can be well approximated as follows:
(14)Θσ(F)(ξk(S),Δk,eV)Inc={{f(Ek-eV)-f(Ek)}×Dσ(F)(Ek-eV)forV>0-{f(Ek-e|V|)-f(Ek)}×Dσ(F)(-(Ek-e|V|))forV<0.
Equation (14) tells us that only the quasiparticle excitation energy Ek with the energy region of 0≤Ek≤e|V| is valid for the evaluations of tunneling currents. Therefore, the DOS Dσ(F)(ρ) as a function of the energy ρ is evaluated only in the energy region of |ρ|≤e|V|. From a relation e|V|≤nmaxΔ(T) and our calculation conditions nmax=5 and Δ(TLN)=70 meV, the DOS Dσ(F)(ρ) of ferromagnetic iron is evaluated only in the energy region of |ρ|≤350 meV, whose energy is measured from the EF.
As already stated, the present case corresponds to the case when the insulating layer I shows no magnetic nature, that is, the case of a standard F/I/S junction. From (b), (e), (h), (k), and (n) on the middle column of Figure 5, in which a relation η↑=η↓=1/2 is satisfied, we can see that the I-V curve, the resultant differential conductance, and the spin current are changed due to the change of MF. For the spin current, moreover, an interesting result such that the minority spin current exceeds the majority one is found. This interesting result is clearly found in the positive voltage region of Figure 5(n).
On the positive voltage region, the normalized spin current iT(-)(V)Nor can be represented as
(15)iT(-)(V)Nor=nmaxK(F)(Vmax)∑μS∑LS∑k2|λLS(μS)(k2)|2×{f(Ek2-eV)-f(Ek2)}×12(D↑(F)(Ek2-eV)-D↓(F)(Ek2-eV)).
Equation (15) clearly tells us that the interesting result shown in Figure 5(n) originated from the difference between the DOSs of the majority and minority spin bands of ferromagnetic iron. The calculated results shown in Figure 3 show that the values of D↑(F)(0) and D↓(F)(0) satisfy the condition of D↑(F)(0)>D↓(F)(0) except for the case shown in Figure 3(e). Here note that the results shown in Figures 5(m), 5(n), and 5(o) have been calculated by using the DOS shown in Figure 3(e). We can say that the condition D↑(F)(0)<D↓(F)(0) has led to an interesting result mentioned above.
Present case corresponds to the case such that the insulating layer I shows a magnetic nature such as a detectable magnetization MI. As already stated, a relation η↑>η↓ corresponds to the MF∥MI, and the η↑<η↓ does to the MF∥-MI. In the FIS tunneling junction with the magnetic insulating barrier, it may be expected that a proximity effect could be found between the magnetic barrier and the superconducting layer. In the present study, however, all possible proximity effects have been neglected for simplicity.
The I-V characteristics calculated for the case of (η↑,η↓)=(3/4,1/4) are shown in (a), (d), (g), (j), and (m) on the left column, and those for (η↑,η↓)=(1/4,3/4) are in (c), (f), (i), (l), and (o) on the right column of Figure 5. In the present subsection, we consider the case of η↑≠η↓. Therefore, in the following, only the case of the F/I/S tunneling junction, that is, the case excepting Figures 5(a), 5(b), and 5(c) for the N/I/S junction, is considered.
From (13) and (14), we can see that (I) the condition η↑>η↓ works by increasing the DOS D↑(F)(0) and decreasing the D↓(F)(0) and (II) the condition η↑<η↓ inversely works as mentioned in (I), that is, by decreasing the majority spin band and increasing the minority one. A result such that the minority spin current exceeds the majority one is clearly found in Figures 5(f), 5(l), and 5(o). We can see that the I-V curve and the resultant diffrential conductance nearly remain the same as the change of η↑(=1-η↓), but the spin current is largely changed due to the change of η↑. Such a behavior is clearly found from the comparison of three Figures 5(d), 5(e), and 5(f) on the second line. In the next subsection, therefore, we consider the spin currents found in Figures 5(d), 5(e), and 5(f).
3.2.3. Spin Current
It has already been stated that the κσ(F)(μS,LS,V) defined in (9) shows the partial current term due to the tunneling of a σ spin electron, so that the present calculation is able to show which atomic orbitals contribute to the F/I/S tunneling phenomena. This is just an advantage of our calculation method. It is easy to divide the spin currents drawn in Figures 5(d), 5(e), and 5(f) into the atomic orbital parts, s, px, py, pz, dxy, dyz, dzx, dx2-y2, and d3z2-r2. The calculated result clearly showed that the most important atomic parts are 3d orbitals of Cu, as is expected. The dxy, dyz, dzx, dx2-y2, and d3z2-r2 parts of the spin currents drawn in Figures 5(d), 5(e), and 5(f) are shown in Figures 6(a), 6(b), and 6(c). The calculated results tell us that the most dominant spin current is observed in an electron tunneling between the 3d orbits of ferromagnetic Fe and the Cu 3dx2-y2 orbit of Hg-1223 high-Tc superconductor. Cu 3dx2-y2 and O 2px and 2py atomic orbitals make a pdσ-bonding state which acts as a stage of an electron transfer within a CuO2 superconducting layer. This is a common feature found in all the copper-oxide high-Tc superconductors; therefore, it seems that the change of the spin state of the Cu 3dx2-y2 atomic orbital makes a large effort to the superconductivity. In the following, therefore, we focus our attention on only the Cu 3dx2-y2 atomic orbital and consider how the change of spin should be observed.
The dxy, dyz, dzx, dx2-y2, and d3z2-r2 atomic orbital parts of the spin currents drawn in Figures 5(d), 5(e), and 5(f). As d orbitals, the Cu 3d orbitals of Cu2 atom shown in Figure 1 have been selected. (a), (b), and (c) of the present figure correspond to Figures 5(d), 5(e), and 5(f), respectively. Present figure clearly shows that the most dominant spin current is observed in an electron tunneling between 3d orbits of ferromagnetic iron and Cu 3dx2-y2 orbit of Hg-1223 high-Tc superconductor.
It is well known that the experiment of the X-ray magnetic circular dichroism (XMCD) is a powerful tool to see directly the nature of the spin dependent electron state near the EF. Now, let us consider two absorption processes such as the |2p1/2〉→|3dx2-y2〉 and the |2p3/2〉→|3dx2-y2〉 for the Cu atom in Hg-1223 high-Tc superconductor. The Cu 3dx2-y2 orbit |3dx2-y2〉 is written as
(16)|3dx2-y2〉=R3d(r)12(Y22(r^)+Y2-2(r^)),
where R3d(r) is the radial wave function of Cu 3d atomic orbital and Ylm(r^) is the spherical harmonics. Therefore, the hole state |3dx2-y2(hole)〉 of |3dx2-y2〉, which is the electron state just above the EF, can be written as
(17)|3dx2-y2(hole)〉=(a|α〉+b|β〉)|3dx2-y2〉,
where |α〉≡|↑〉, |β〉≡|↓〉, and a relation |a|2+|b|2=1 is satisfied. Using Clebsch-Gordan coefficients, the |2p1/2〉 and |2p3/2〉 states can be written as follows:
(18)|2p1/2〉=R2p(r)×{-13Y10(r^)|α〉+23Y11(r^)|β〉(=|12,12〉)-23Y1-1(r^)|α〉+13Y10(r^)|β〉(=|12,-12〉),(19)|2p3/2〉=R2p(r)×{Y11(r^)|α〉(=|32,32〉)23Y10(r^)|α〉+13Y11(r^)|β〉(=|32,12〉)13Y1-1(r^)|α〉+23Y10(r^)|β〉(=|32,-12〉)Y1-1(r^)|β〉(=|32,-32〉).
For the circular polarized lights with the helicity h=±1, the Hamiltonian H^′ of the dipole-allowed transition is written as
(20)H^′=eE12(x±iy)=∓eEr4π3Y1±1(r^),
where E is the amplitude of the electric field E. It is clear that (20) becomes a standard Hamiltonian eErcosθ for the dipole-allowed transition when a zero helicity light h=0 is considered, that is, when Y1±1(r^) is replaced by Y10(r^). There are some allowed-absorption processes. In the present paper, only the allowed transition with h=1 is considered for simplicity.
It is easy to calculate the oscillator strength |〈3dx2-y2(hole)|H^′|j,mj〉|2(≡Γ). When h=1, the Γ is evaluated as follows:
(21)Γ=A|(a*〈α|+b*〈β|)〈Y22|Y11|j,mj〉|2,
where A≡(2π/3)e2E2I2p→3d2 and I2p→3d≡∫0∞R3d(r)R2p(r)r3dr. An integral 〈Y22|Y11|j,mj〉 in (21) is calculated as follows:
(22)〈Y22|Y11|j,mj〉={23〈Y22|Y11|Y11〉|β〉for|12,12〉〈Y22|Y11|Y11〉|α〉for|32,32〉13〈Y22|Y11|Y11〉|β〉for|32,12〉.
Here 〈Y22|Y11|Y11〉(≡G) is a Gaunt integral. In the |2p1/2〉→|3dx2-y2〉 and |2p3/2〉→|3dx2-y2〉 XMCD absorption spectra with helicity h=1, the oscillator strength Γ is finally written as follows:
(23)Γ(|12,12〉⟶Y22)=23AG2|b|2(≡LII(1)),Γ(|32,32〉⟶Y22)=AG2|a|2(≡LIII(1)),Γ(|32,12〉⟶Y22)=13AG2|b|2(≡LIII(2)),
where we note that |a|2+|b|2=1. By using (23), the absorption ratio LII(1)/(LIII(1)+LIII(2))≡Λ is calculated as
(24)Λ=2|b|23-2|b|2.
The value of |b|2 is exactly equal to 1/2 when no spin current is observed. In such a situation, therefore, the value of Λ remains 1/2. However, if a sizable spin current is found, then it is sure that the value of |b|2 differs from 1/2. As can be seen from (17), the |b|2 reflects the nature of the minority spin hole. Therefore, if the spin state of electrons injected into the superconductor is largely changed, such as the case of Figures 5(d), 5(e), and 5(f), then it could be expected that a considerable change is observed experimentally for the absorption ratio Λ because of a large change of |b|2. For example, let us consider the spin current at a positive voltage such as the normalized voltage 2. In the spin current shown in Figure 5(d), the electron with the majority spin is dominant; that is, the minority spin hole is dominant, so it is expected that a relation |b|2>1/2 is satisfied for the spin current shown in Figure 5(d). Similar consideration gives a result such that a relation |b|2<1/2 is satisfied for the spin current shown in Figure 5(f). Namely, it may be expected that the absorption ratio Λ decreases due to the change from Figures 5(d) to 5(f), that is, with decreasing the η↑. It may be possible to control the value of η↑(=1-η↓) by experimentally handling the amplitude of the external magnetic field applied parallel to the junction surface. It is not so easy to detect the spin current; however, we believe that present prediction based on the XMCD could be useful to see qualitatively the spin current found in the F/I/S tunneling junction using the copper-oxide high-Tc superconductors.
4. Summary
We have studied the c-axis charge and spin currents in ferromagnet-insulator-superconductor (F/I/S) tunneling junction on the basis of the tunneling Hamiltonian model. As a ferromagnet, Fe metal with bcc structure has been selected for simplicity, and the electronic structure has been calculated on the basis of the spin-polarized band theory using the spin-polarized self-consistent-field atomic data for the atomic energies and the universal tight-binding parameters for the interactions. As a superconductor, HgBa2Ca2Cu3O8+δ (Hg-1223) with δ=0.4 copper-oxide high-Tc superconductor has been selected from the reason of the high Tc of 135 K. The electronic structure of Hg-1223 has been calculated by using the same method done for the ferromagnetic Fe. The band structure calculation for Hg-1223 has clearly showed that the electronic structures near the Fermi level EF are made from Cu 3d and O 2p atomic orbitals and that the most important orbit is the Cu 3dx2-y2 one in CuO2 layers in Hg-1223.
The c-axis charge and spin currents in an F/I/S tunneling junction have been calculated as a function of the magnetization MF of ferromagnet F and the applied voltage V. Using a calculated charge current, the differential conductance has been also calculated. If the insulating layer I shows a magnetic nature, it is surely expected that the tunneling probabilities of the majority and the minority spin electrons differ from each other. We have defined the η↑ and η↓(=1-η↑) giving the tunneling probabilities of majority and the minority spin electrons. The calculated results have showed that the charge current and the resultant differential conductance nearly remain the same as the change of η↑, but the spin current is largely changed due to the change of η↑.
Present model includes the band theory, so that it is possible to see directly which atomic orbital contributes to the F/I/S tunneling phenomena. This is just an advantage of the model we have proposed here. We have clearly found that the most important orbit in the present F/I/S tunneling junction is the Cu 3dx2-y2 atomic orbital in the Hg-1223 high-Tc superconductor. This is a common feature observed in not only the Hg-based superconductors but also the other copper-oxide superconductors such as YBCO and BSCCO. As a method to detect the change of the spin current, we have pointed out to the validity of the experiment of an X-ray magnetic circular dichroism (XMCD).
AppendixDeduction of (<xref ref-type="disp-formula" rid="EEq1">1</xref>)
First, let us consider a tunneling junction consisting of material M1/insulator I/material M2. For an electron tunneling from M1 to M2, the tunneling Hamiltonian H^T1→2 is given by
(A.1)H^T1→2=∑k1∑σ1∑k2∑σ2〈k2σ2|T^|k1σ1〉c^k2σ2†c^k1σ1.
Here, c^k1σ1 is an annihilation operator for an electron with the wavenumber k1 and spin σ1 in material M1 and c^k2σ2† is a creation operator for a k2σ2-electron in M2. 〈k2σ2|T^|k1σ1〉 is the matrix element for an electron tunneling from |k1σ1〉 state to |k2σ2〉 one. We consider here the case in which no spin-flip is found for an electron tunneling. Therefore, the total Hamiltonian H^T defined by H^T1→2+H^T2→1 is given as follows:
(A.2)H^T=∑k1∑k2∑σ(〈k2σ|T^|k1σ〉c^k2σ†c^k1σ+〈k1σ|T^|k2σ〉c^k1σ†c^k2σ).
Next, we consider the case such that the M1 is a ferromagnetic material F and the M2 is a superconductor S, that is, an F/I/S junction. Only a superconductor in which all Cooper-pairs (CPs) are in a spin-singlet state is considered here, so it is instructive to rewrite (A.2) as follows:
(A.3)H^T=H^TF→S+H^TS→F,H^TF→S=∑k1∑k2(Tk1k2↑c^k2↑†c^k1↑+Tk1k2↓*c^-k2↓†c^-k1↓)≡∑k1∑k2h^T;k1k2F→S,H^TS→F=∑k1∑k2(Tk2k1↑c^k1↑†c^k2↑+Tk2k1↓*c^-k1↓†c^-k2↓)≡∑k1∑k2h^T;k1k2S→F,
where Tk1k2σ≡〈k2σ|T^|k1σ〉 and a relation 〈-k2σ|T^|-k1σ〉=〈k1σ|T^|k2σ〉=〈k2σ|T^|k1σ〉* is held. Using Bogoliubov transformation, h^T;k1k2F→S and h^T;k1k2S→F in (A.3) can be written as follows:
(A.4)h^T;k1k2F→S=Tk1k2↑(uk2γ^k2↑†c^k1↑+vk2*γ^-k2↓c^k1↑)+Tk1k2↓*(uk2γ^-k2↓†c^-k1↓-vk2*γ^k2↑c^-k1↓),(A.5)h^T;k1k2S→F=Tk2k1↑(uk2*c^k1↑†γ^k2↑+vk2c^k1↑†γ^-k2↓†)+Tk2k1↓*(uk2*c^-k1↓†γ^-k2↓-vk2c^-k1↓†γ^k2↑†),
where we note that Tk2k1↑=Tk1k2↑* and Tk2k1↓*=Tk1k2↓. It is clear that γ^kσ† and γ^kσ are creation and annihilation operators for a quasiparticle characterized by kσ, respectively, and that the uk and vk are expansion coefficients in the BCS wave function |ΨBCS〉=∏k(uk+vkc^k↑†c^-k↓†)|0〉.
Equation (A.4) tells us that an electron tunneling from k1-electron in F to k2 one in S can be regarded as a combination of four fundamental tunneling processes. Statistical averaging for operators such as γ^k2↑†c^k1↑, γ^-k2↓c^k1↑, γ^-k2↓†c^-k1↓, and γ^k2↑c^-k1↓ gives the following corresponding relations:
(A.6)〈γ^k2↑†c^k1↑〉av=(1-f(±Ek2↑))f(ξk1↑(F)),〈γ^-k2↓c^k1↑〉av=f(±E-k2↓)f(ξk1↑(F)),〈γ^-k2↓†c^-k1↓〉av=(1-f(±E-k2↓))f(ξ-k1↓(F)),〈γ^k2↑c^-k1↓〉av=f(±Ek2↑)f(ξ-k1↓(F)),
where 〈⋯〉av means the statistical average and f(E) is Fermi-Dirac distribution function. Here it is very instructive to point out that the symbol “±" in (A.6) appears when the semiconductor model is adopted for the superconductors [1]. ξk1σ(F) is the energy relative to the Fermi level EF(F) of a k1σ-electron in ferromagnetic layer F, which is obtained from the band structure calculation. Here note that a relation ξ-k1σ(F)=ξk1σ(F) is satisfied for almost all the materials. Ek2σ is a quasiparticle excitation energy defined by ξk2σ(S)2+|Δk2|2. Here it should be noted that the superconducting energy gap Δk2 as a function of k2 does not include explicitly the spin variable σ since the spin-singlet CPs are considered here. ξk2σ(S) is the energy relative to the Fermi level EF(S) of a k2σ-electron in superconducting layer S, which is also obtained from the band structure calculation. We consider here CP with a spin-singlet state, so that the S should be nonmagnetic layer. Namely, the spin variable σ of the one electron energy ξk2σ(S) should be dropped out; that is, a relation ξ-k2σ(S)=ξk2σ(S)=ξk2(S) should be held. A relation Δ-k2=Δk2 is of course satisfied, so that a relation E-k2σ=Ek2σ=Ek2 is obtained for the quasiparticle excitation energy. A similar treatment is also done for operators in (A.5).
From Fermi’s golden rule, we can get the tunneling probabilities Pk1k2F→S and Pk1k2S→F for F→S and S→F tunneling as follows:
(A.7)Pk1k2F→S=2πℏ∑σ|Tk1k2σ|2×{(1-f(Ek2))f(ξk1σ(F))δ(ξk1σ(F)+eV-Ek2)+f(Ek2)f(ξk1σ(F))δ(ξk1σ(F)+eV+Ek2)}≡∑σpk1k2,σF→S(ξk1σ(F),ξk2(S),Δk2,eV),Pk1k2S→F=2πℏ∑σ|Tk1k2σ|2×{(1-f(ξk1σ(F)))f(Ek2)δ(ξk1σ(F)+eV-Ek2)+(1-f(ξk1σ(F)))(1-f(Ek2))×δ(ξk1σ(F)+eV+Ek2)}≡∑σpk1k2,σS→F(ξk1σ(F),ξk2(S),Δk2,eV).
Finally, we can get the charge and spin currents, IT(+)(V) and IT(-)(V), as a function of an applied voltage V. The result is given as follows:
(A.8)IT(±)(V)≡iT,↑(V)±iT,↓(V),
and the tunneling current iT,σ(V) for the σ spin electron is given by
(A.9)iT,σ(V)=e∑k1∑k2(pk1k2,σF→S(ξk1σ(F),ξk2(S),Δk2,eV)-pk1k2,σS→F(ξk1σ(F),ξk2(S),Δk2,eV))=2πeℏ∑k1∑k2|Tk1k2σ|2×{(f(ξk1σ(F))-f(Ek2))δ(ξk1σ(F)+eV-Ek2)+(f(ξk1σ(F))+f(Ek2)-1)×δ(ξk1σ(F)+eV+Ek2)}≡2πeℏ∑k1∑k2|Tk1k2σ|2g(ξk1σ(F),ξk2(S),Δk2,eV).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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