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Tunnel magnetoresistance in ferromagnetic double-barrier planar junctions: coherent tunneling regime
Journal of Magnetism and Magnetic Materials 221 (2000) 373}381
Tunnel magnetoresistance in ferromagnetic double-barrier planar junctions: coherent tunneling regime M. Wilczynski , J. Barnas * Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, Warsaw, Poland Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland Received 18 January 2000; received in revised form 25 July 2000
Abstract Coherent tunneling in a double-barrier system consisting of two external ferromagnetic electrodes and a nonmagnetic central one is studied theoretically within the free-electron approximation. It is shown that the junction resistance depends on the relative orientation of magnetic moments of the ferromagnetic electrodes (so-called tunnel magnetoresistance). The magnetoresistance vs. thickness of the central electrode shows pronounced peaks related to the resonant tunneling through the whole system. Variation of the magnetoresistance with bias voltage is also studied. This variation is generally nonmonotonous. 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetoresistance; Tunnel junctions; Coherent tunneling; Electron transmission
1. Introduction It is well known that electrical resistance of a simple planar ferromagnetic junction depends on its magnetic con"guration [1}3]. The resistance usually drops when magnetic moments of two electrodes rotate from antiparallel to parallel alignment and the resistance drop is described quantitatively by the factor *R/R "(R !R )/R , where R and R denote the junction resistance in the anti parallel and parallel con"gurations, respectively. This resistance drop is known as the normal tunnel magnetoresistance (TMR) e!ect. An inverse e!ect (negative *R/R ) is also possible [4]. From the
* Corresponding author. Tel.: #48-61-8273068; fax: #4861-8273070. E-mail address: [email protected] (J. Barnas).
physical point of view, the TMR e!ect results from spin-dependent electronic band structure of the ferromagnetic electrodes and spin-dependent tunneling rates [5}8]. Tunneling in more complex ferromagnetic junctions is a subject of current interest [9,10]. In a recent paper, Moodera et al. [11] investigated both experimentally and theoretically tunneling in a junction with a thin nonmagnetic layer placed between the barrier and one of the ferromagnetic electrodes (referred to as an interface layer). They observed well-pronounced oscillations in the tunneling current and TMR with increasing thickness of the interface layer, which indirectly proves formation of quantum-well states in this layer. Tunneling in double-barrier junctions was also studied theoretically [12,13]. In a recent paper, we examined TMR in a double-barrier junction in the sequential tunneling regime [14]. The considerations
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were restricted to the case, where the energy relaxation time in the central electrode was su$ciently short to neglect quantum-well-states formation and to assume partial thermalization of electrons. Since the spin relaxation time could be much longer than the energy relaxation time, the two sets of electrons with opposite spin orientations could correspond to di!erent chemical potentials due to spin accumulation e!ect [14]. Full thermalization takes place in the limit of short spin relaxation time (when it is comparable to the energy relaxation time). In the sequential tunneling regime the electron tunneling from the source to sink electrode is a two-stage process. In the "rst stage, one electron tunnels through one of the barriers, while in the second stage, another electron tunnels through the second barrier. When all the three electrodes are ferromagnetic, then TMR exists for arbitrary spin relaxation time. If, however, the central electrode is nonmagnetic, the TMR e!ect exists only when the spin relaxation time in the central electrode is long enough, so that the #owing current leads to spin accumulation in this electrode. The spin accumulation is a result of spin-dependent tunneling rates across the barriers and depends on the magnetic con"guration of the junction. This, in turn, gives rise to a shift of the Fermi energy in the central electrode, which depends on the electron spin and magnetic con"guration of the junction, and consequently, leads to the con"guration-dependent current #owing through the junction, i.e., to the TMR e!ect. In the coherent tunneling regime, on the other hand, an electron is in a coherent state of the whole double-barrier system and spends only a very short time in the central electrode. In this tunneling limit one usually neglects the e!ect of spin accumulation in the central electrode. This is justi"ed when only coherent tunneling processes take place. In real situations, however, both coherent and sequential tunneling processes occur [15] and, therefore, the spin accumulation should be taken into account. In the coherent tunneling regime transmission coe$cients through the double-barrier junction depend on its magnetic con"guration and this dependence gives rise to the TMR e!ect. A special kind of double-barrier junctions are junctions with a mesoscopic central electrode (e.g.
a small grain) [16}19] or junctions in which the central electrode consists of a granular system [20,21]. When the grains are small enough, the single-electron charging e!ects become important and can lead to Coulomb blockade of tunneling current [16}19]. The charging e!ects can also enhance TMR [20,21]. Those e!ects will not be considered in this paper. We restrict ourselves to large-area planar junctions, in which the Coulomb blockade e!ects do not occur. In this paper we consider coherent tunneling in a double-barrier system with a nonmagnetic central electrode. We assume that electron spin is conserved in the tunneling process through the whole system and also neglect any spin accumulation in the central electrode (as well as in the external electrodes). Apart from this we analyze only colinear con"gurations of the electrode magnetizations, which allows us to consider each spin separately. Electronic structure of the electrodes is approximated by a free-electron-like parabolic band, which for ferromagnetic electrodes is spinsplit due to an e!ective exchange "eld. For simplicity, we assume that the band edge in the nonmagnetic electrode coincides with the band edge of the external electrodes in the paramagnetic limit. We also assume that the barriers are of a rectangular shape which transforms into a trapezoidal one when a bias voltage < is applied to the junction. In Section 2 we describe the model and present a general formula for the tunneling current in double-barrier junctions with a nonmagnetic central electrode. Numerical results are presented in Section 3, where the bias dependence of TMR and its variation with the thickness of central electrode are also discussed. Summary and "nal conclusions are in Section 4.
2. Electron transmission across double-barrier junction and tunnel magnetoresistance First, consider transmission of an electron with spin p in a magnetic con"guration k of the junction (k,ap for the antiparallel con"guration and k,p for the parallel one). The transmission coe$cient across the whole double-barrier structure can be derived from the electron wave function with
M. Wilczynski, J. Barnas / Journal of Magnetism and Magnetic Materials 221 (2000) 373}381
standard boundary conditions imposed on the wave function and its "rst derivative at each electrode/barrier interface. When the interfaces are #at, the in-plane component k# of electron wave vector is conserved in the tunneling processes. Let the axis x be normal to the junction and let the interface between the left electrode and the barrier be located at x"0. If we denote the electron wave function in the region j as u (x), then the boundary conditions H at the interface between the regions j and j#1 (the interface located at x ) are u (x )"u (x ) and H H H H> H u (x )"u (x ), where the prime at the wave H H H> H function denotes its "rst derivative and also the same electron mass was assumed for all regions. The index j runs from j"1 for the left electrode to j"5 for the right one, with j"2, 4 corresponding to the barriers and j"3 corresponding to the central electrode. Let the left (right) electrode be the source (drain) one. The transmission coe$cient for an electron incident normally on the barrier from the left electrode and with the total energy e , V measured from the bottom of the electron band of the left electrode in the paramagnetic limit, can be written in the form [22]:
¹ (e )"[¹\¹\#(¹\!1)(¹\!1) N I V #2(¹\¹\(¹\!1)(¹\!1) ;cos( )]\,
(1)
where we neglected transmission of electrons, whose energy e is below the band edge of the V central electrode, as for such electrons the central electrode is e!ectively an additional potential barrier. In the above expression, T and T denote the transmission coe$cients through the "rst (left) and second (right) barriers, respectively, which depend on the electron spin p and magnetic con"guration k of the junction (for clarity of notation this dependence is not indicated explicitly in Eq. (1)). Similarly, in Eq. (1) is a phase factor, to be speci"ed later, which also depends on the electron spin p and magnetic con"guration k (not indicated explicitly in Eq. (1)). For oblique incidence the energy e is the V total energy minus the kinetic energy of in-plane motion.
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In the case under consideration (nonmagnetic central electrode and ferromagnetic external ones) the transmission coe$cients T and T and the phase factor take the forms
1 (k b !k c )#(i #k k a ) \ N N ¹ " , p 4k k D N (2) 1 (k b !k c )#(i #k k a ) \ N I N I ¹" , p 4k k D N I (3)
"2k d # # , (4) where !p) , )p and k (c #k a )!(i #k b ) N N , (5)
"arctan 2k (k a b #c i ) N k (c !k a )#(i !k b ) . (6)
"arctan N I 2k (k a c #b i ) N I The normal components k , k and k of the N N I electron wave vector in the source, drain and central electrodes, respectively, are given by the formulas
1 k " (2m(e #p D), V N N
(7)
1 k " (2m(e #p p D#e<), V N I N I
(8)
d e< , 2m e # (9) V d #d where m is the electron mass and we assumed that the parallel and antiparallel con"gurations di!er in the orientation of magnetic moment of the right electrode (consequently k is independent of N k while k depends on k). Apart from this, d and N I d denote the thickness of the left ("rst) and right (second) barriers, d is the thickness of the central electrode, 2D denotes the spin splitting of the electron band (D"0 in the paramagnetic limit) in ferromagnetic electrodes (assumed the same for both electrodes) and the parameter p describes N electron spin, p "1(!1) for p"!( ), whereas the N 1 k "
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parameter p is related with the magnetic con"gI uration of the junction, p "1(!1) for k,p(ap). I Finally, the following abbreviations are used in Eqs. (2)}(6):
z (x )"z (d #d ) (2m(d #d ) " e<
a "Ai[z (x )]Bi[z (x )] H H H\ H H
d e< , ; E #; !e ! $ V d #d z (x )"z (d #d #d ) (2m(d #d ) " e<
!Bi[z (x )]Ai[z (x )], H H\ H H
(10)
b "D+Ai[z (x )]Bi [z (x )] H H H\ H H !Bi[z (x )]Ai [z (x )],, H H\ H H
(11)
c "D+Ai [z (x )]Bi[z (x )] H H H\ H H !Bi [z (x )]Ai[z (x )],, H H\ H H
(12)
i "D+Ai [z (x )]Bi [z (x )] H H H\ H H !Bi [z (x )]Ai [z (x )], H H\ H H
(13)
for j"2 and j"4, where Ai(z (x)) and Bi(z (x)) are H H the Airy functions while Ai (z (x)) and Bi (z (x)) H H their "rst derivatives. The argument z (x) of the H Airy functions is de"ned as
(2m(d #d ) E #; !e z (x)" $ H V H e<
d x e
(14)
where ; denotes the height of the "rst ( j"2) and H second ( j"4) barriers, respectively, whereas E is $ Fermi energy. The Airy functions and their derivatives in Eqs. (10)}(13) are taken at the points z (x ) H H and z (x ), i.e., at the interfaces. More speci"cally, H H\ those arguments are z (x )"z (0)
"
(2m(d #d ) (E #; !e ), (15) $ V e<
z (x )"z (d )
"
(2m(d #d ) e<
d e< , ; E #; !e ! $ V d #d
(16)
(17)
;(E #; !e !e<). (18) $ V Finally, the parameter D in Eqs. (2), (3) and (11)}(13) is given by D"![2me
#$ \C4 2pme e< ¹ (e ) de j " N I V V I h #N I N #$ # (E !e )¹ (e ) de , (20) $ V N I V V #$ \C4 where the lower integration limit, E , is given by N I E "max+!p D,!d e
3. Numerical results In our numerical calculations, we assumed that the source and drain electrodes were made of the same ferromagnetic material. Additionally, we assumed the same position at <"0 of the electron band in the central electrode and in the external electrodes in the paramagnetic limit (vanishing spin splitting, DP0). Consider "rst the dependence of TMR on the thickness of the central electrode.
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377
Fig. 1. TMR as a function of the central electrode thickness d , calculated for E "2.62 eV, D"1.96 eV, U "U "0.75 eV, $ d "d "0.5 nm and for applied voltage <"20 mV.
Fig. 2. Electric current density j vs. thickness d of the central electrode calculated for the same parameters as in Fig. 1. Solid and dashed curves correspond to the parallel and antiparallel con"gurations, respectively.
The relevant numerical results were obtained for the Fermi energy E "2.62 eV, spin splitting of the $ electron band in the ferromagnetic electrodes D"1.96 eV, barrier height U "U "0.75 eV and the barrier thickness d "d "0.5 nm. In Fig. 1, TMR is shown as a function of the thickness of the central electrode d in the case when the bias voltage <"20 mV is applied to the junction. Several features of the TMR curve are interesting and worth further analysis. First, TMR as a function of d shows well-de"ned peaks, the height of which decreases with increasing d . Sec ond, those peaks are relatively high at small values of thickness d . In the case shown in Fig. 1, there are four such peaks. The "rst peak corresponds to *R/R +35, while the fourth one to *R/R +15. The other peaks are much smaller. One can expect that the periodic variation of TMR with increasing thickness of the central electrode is related to the quantum-well states formed in the central electrode and to resonant tunneling through the whole structure. To understand the origin of this periodicity in TMR and the di!erence in heights between the "rst four peaks and the others, we have plotted the electric current in the parallel and antiparallel con"gurations as a function of the thickness of the central electrode in Fig. 2. The periodic variation with increasing d is now more pronounced and more regular. From the comparison of Figs. 1 and 2, it follows
that the maxima in TMR correspond roughly to the minima in the #owing current. In the antiparallel con"guration the "rst four minima are relatively deep with a small value of the corresponding current (not resolved in Fig. 2), which gives rise to large TMR (the "rst four peaks in Fig. 1). To understand further the origin of periodic variation of electric current and peaks in TMR, we have analyzed the energy dependence of the transmission coe$cient through the double-barrier junction for several different values of the thickness d . We have found that the periodicity in TMR is strongly correlated with the resonance states in the central electrode. Generally, one can expect large TMR when the highest resonance level (below E ) is at the energy $ smaller than D!e< and there is no resonance states in the energy range (D!e<, E ). When < is $ not too high, so this resonance level is above the energy !D, then the existing resonance state participates in the conduction processes only for the spin-majority channel in the parallel con"guration, whereas it does not give rise to current in the antiparallel con"guration for both spin orientations and in the parallel con"guration for the spinminority channel. In the model situation analyzed here this happens when the thickness d of the central electrode corresponds to the "rst four peaks in TMR. Consider for example the case when d is equal to 0.69 nm. The relevant transmission coe$cients for both spin orientations and both magnetic
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Fig. 3. Transmission coe$cients calculated for d "0.69 nm and for both spin orientations in the antiparallel and parallel con"gura tions. The other parameters are the same as in Fig. 1.
con"gurations of the junction are shown in Fig. 3. It is clear from the data that there are two resonance states in the quantum well (central electrode); "rst at e 0.43 eV and the second one at V e 1.74 eV. Both resonance states are below the V band edge of the minority electron band in the ferromagnetic electrodes and, therefore, both are not active for minority electrons in the parallel con"guration and for both spin orientations in the antiparallel con"guration. There are no other resonance states in the relevant energy range. Therefore, there is a large di!erence in the #owing current in both con"gurations and consequently large magnetoresistance. The resonance state is however far from the Fermi level so only electrons with large
in-plane momentum can tunnel resonantly. Therefore, there is a minimum in the electric current in both the con"gurations, while TMR reaches maximum at this point. When the thickness of the central electrode increases, the resonance states go down in energy and new resonance levels appear below E in the energy $ range (D, E ). In Fig. 4, we show the transmission $ coe$cients calculated for d "0.89 nm, i.e., for d corresponding to the "rst minimum in TMR on the right of the "rst maximum (Fig. 1). This value of d corresponds to a local maximum in the #owing current. A new active resonance level appears at this thickness of the central electrode, whose energy is approximately equal to 2.6 eV. This state, lying
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379
Fig. 4. The same as in Fig. 3, but for d "0.89 nm.
slightly below the Fermi energy, contributes signi"cantly to the #owing current for both spin orientations and magnetic con"gurations. Therefore, electric current in both the con"gurations is relatively large (local maximum), whereas the di!erence between the currents in the antiparallel and parallel con"gurations is rather small (local minimum in TMR). The situation for d corresponding to the next large peaks in TMR is qualitatively similar to that described above for d "0.69 nm. The other peaks in TMR are, however, signi"cantly smaller. This is because starting from the "fth peak at least one of the resonance states is always active in both the con"gurations and spin orientations, which signi"cantly reduces TMR. This is because it is impossible to remove simultaneously all the resonance
states from the energy range (D!e<, E ) by slight$ ly increasing or decreasing the thickness of the central electrode. By changing the energy of these states we can only slightly increase or decrease the tunneling current, which gives rise to weak variations of TMR with the central electrode thickness. Let us now consider how TMR varies with increasing voltage applied to the junction. In Fig. 5, we show the bias dependence of TMR for d " 0.69 nm (the other parameters are the same as in Figs. 1}4). The thickness d corresponds to the "rst maximum in Fig. 1. We recall that Fig. 1 was calculated for a relatively small bias voltage, <"20 mV. As one can see in Fig. 5, TMR slowly decreases with increasing bias and then rapidly falls down at about 420 mV to a small value. Such a behavior of TMR is caused by the fact that the
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Fig. 5. Bias dependence of TMR for d "0.69 nm. The other parameters are the same as in Fig. 1.
resonance state lying below D at <"0 begins to participate in the conduction processes also in the antiparallel con"guration when the applied voltage is su$ciently high. From the discussion above we remember that large TMR appears when there is a resonance state below D!e
Fig. 6. The same as in Fig. 5, but for d "0.89 nm.
Fig. 7. Bias dependence of electric current density for d "0.69 nm. The other parameters are the same as in Fig. 1.
4. Summary and concluding remarks We have calculated tunnel magnetoresistance in the coherent tunneling regime in a double-barrier junction with a nonmagnetic central electrode. We showed that in some cases the resonant tunneling can give rise to huge TMR. At this point it is worth to mention that in the sequential tunneling regime nonzero TMR can appear in this junction only when spin-relaxation time in the central electrode is su$ciently long to produce spin accumulation [14]. This TMR is, however, signi"cantly smaller than TMR in the coherent tunneling regime analyzed in this paper. Strong dependence of the transmission coe$cient on the position of quantum-well
M. Wilczynski, J. Barnas / Journal of Magnetism and Magnetic Materials 221 (2000) 373}381
states in the central electrode can lead to negative di!erential resistance. The calculations were performed under several approximations. First, we assumed that electron spin is conserved in tunneling events. It is, however, well known that TMR is reduced when spin-#ip tunneling processes are allowed, despite the fact that such processes usually increase electric current by opening new channels for electron tunneling [23,24]. Other scattering processes leading to incoherent tunneling can reduce the TMR value as well. Second, we used a simple free-electron-like model to describe electronic structure of the junction. In real metallic systems, however, electronic structure is much more complex, which can lead to quantitative modi"cations of the results presented here. We believe that some qualitative features of TMR will survive. The huge TMR value we obtained in our numerical calculations followed from a speci"c position of the electron bands in the ferromagnetic and nonmagnetic electrodes. The condition for such huge TMR may be realized in experimental situations not only in junctions with metallic central electrodes, but also when the metallic central electrode is replaced by a semiconducting one [25]. For example, one may expect such behavior in metallic junctions, in which tunneling current is dominated by d electrons, which may be achieved by taking appropriate barriers [4]. Another approximation used in our description is the zero-temperature limit. We considered this limit since the physical picture is then clear and one can easily explain all the features of the tunneling current and TMR. The description presented in this paper can be easily extended to "nite temperature. Generally, one may expect an increase in the tunneling current with increasing ¹ and a decrease in TMR.
Acknowledgements The work was supported by the Polish State Committee for Scienti"c Research through the Research Project 2P03B 075 14.
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References [1] M. Julliere, Phys. Lett. A 54 (1975) 225. [2] T. Miyazaki, T. Yaoi, S. Ishio, J. Magn. Magn. Mater. 98 (1991) L7. [3] J.S. Moodera, L.R. Kinder, T.M. Wong, R. Meservey, Phys. Rev. Lett. 74 (1995) 3273. [4] J.M. De Teresa, A. Barthelemy, A. Fert, J.P. Contour, R. Lyonnet, F. Montaigne, P. Seneor, A. Vaures, Phys. Rev. Lett. 82 (1999) 4288. [5] J.C. Slonczewski, Phys. Rev. B 39 (1989) 6995. [6] J.M. MacLaren, W.H. Butler, X.-G. Zhang, J. Appl. Phys. 83 (1998) 6521. [7] E.Yu. Tsymbal, D.G. Pettifor, J. Phys.: Condens. Matter 9 (1997) L411. [8] K. Wang, S. Zhang, P.M. Levy, L. Szunyogh, P. Weinberger, J. Magn. Magn. Mater. 189 (1998) L131. [9] A. Vedyayev, N. Ryzhanova, C. Lacroix, L. Giacomoni, B. Dieny, Europhys. Lett. 39 (1997) 219. [10] S. Zhang, P.M. Levy, Phys. Rev. Lett. 81 (1998) 5660. [11] J.S. Moodera, J. Nowak, L.R. Kinder, P.M. Tedrow, R.J.M. van de Veerdonk, B.A. Smits, M. Van Kampen, H.J.M. Swagten, W.J.M. de Jonge, Phys. Rev. Lett. 83 (1999) 3029. [12] X. Zhang, B.-Z. Li, G. Sun, F.-C. Pu, Phys. Lett. A 245 (1998) 133. [13] Z. Zheng, Y. Qi, D.Y. Xing, J. Dong, Phys. Rev. B 59 (1999) 14505. [14] M. Wilczynski, J. Barnas, Acta Phys. Polon. A 97 (2000) 443. [15] G. Iannaccone, B. Pellegrini, Phys. Rev. B 52 (1995) 17406. [16] K. Ohno, H. Shimada, S. Kobayashi, Y. Ootuka, J. Phys. Soc. Japan 65 (1996) 3449. [17] J. Barnas, A. Fert, Phys. Rev. Lett. 80 (1998) 1058. [18] J. Barnas, A. Fert, Europhys. Lett. 44 (1998) 85. [19] J. Barnas, A. Fert, J. Magn. Magn. Mater. 192 (1999) L391. [20] L.F. Schelp, A. Fert, F. Fettar, P. Holody, S.F. Lee, J.L. Maurice, F. Petro!, A. Vaures, Phys. Rev. B 56 (1997) R5747. [21] S. Mitani, S. Takahashi, K. Takanashi, K. Yakushiji, S. Maekawa, H. Fujimori, Phys. Rev. Lett. 81 (1998) 2799. [22] K. Miyamoto, H. Yamamoto, J. Appl. Phys. 84 (1998) 311. [23] S. Zhang, P.M. Levy, A.C. Marley, S.S.P. Parkin, Phys. Rev. Lett. 79 (1997) 3744. [24] J.S. Moodera, J. Nowak, R.J.M. van de Veerdonk, Phys. Rev. Lett. 80 (1998) 2941. [25] P. Bruno, J. Wunderlich, J. Appl. Phys. 84 (1998) 978.
Tunnel magnetoresistance in ferromagnetic double-barrier planar junctions: coherent tunneling regime M. Wilczynski , J. Barnas * Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, Warsaw, Poland Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland Received 18 January 2000; received in revised form 25 July 2000
Abstract Coherent tunneling in a double-barrier system consisting of two external ferromagnetic electrodes and a nonmagnetic central one is studied theoretically within the free-electron approximation. It is shown that the junction resistance depends on the relative orientation of magnetic moments of the ferromagnetic electrodes (so-called tunnel magnetoresistance). The magnetoresistance vs. thickness of the central electrode shows pronounced peaks related to the resonant tunneling through the whole system. Variation of the magnetoresistance with bias voltage is also studied. This variation is generally nonmonotonous. 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetoresistance; Tunnel junctions; Coherent tunneling; Electron transmission
1. Introduction It is well known that electrical resistance of a simple planar ferromagnetic junction depends on its magnetic con"guration [1}3]. The resistance usually drops when magnetic moments of two electrodes rotate from antiparallel to parallel alignment and the resistance drop is described quantitatively by the factor *R/R "(R !R )/R , where R and R denote the junction resistance in the anti parallel and parallel con"gurations, respectively. This resistance drop is known as the normal tunnel magnetoresistance (TMR) e!ect. An inverse e!ect (negative *R/R ) is also possible [4]. From the
* Corresponding author. Tel.: #48-61-8273068; fax: #4861-8273070. E-mail address: [email protected] (J. Barnas).
physical point of view, the TMR e!ect results from spin-dependent electronic band structure of the ferromagnetic electrodes and spin-dependent tunneling rates [5}8]. Tunneling in more complex ferromagnetic junctions is a subject of current interest [9,10]. In a recent paper, Moodera et al. [11] investigated both experimentally and theoretically tunneling in a junction with a thin nonmagnetic layer placed between the barrier and one of the ferromagnetic electrodes (referred to as an interface layer). They observed well-pronounced oscillations in the tunneling current and TMR with increasing thickness of the interface layer, which indirectly proves formation of quantum-well states in this layer. Tunneling in double-barrier junctions was also studied theoretically [12,13]. In a recent paper, we examined TMR in a double-barrier junction in the sequential tunneling regime [14]. The considerations
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 5 1 4 - X
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were restricted to the case, where the energy relaxation time in the central electrode was su$ciently short to neglect quantum-well-states formation and to assume partial thermalization of electrons. Since the spin relaxation time could be much longer than the energy relaxation time, the two sets of electrons with opposite spin orientations could correspond to di!erent chemical potentials due to spin accumulation e!ect [14]. Full thermalization takes place in the limit of short spin relaxation time (when it is comparable to the energy relaxation time). In the sequential tunneling regime the electron tunneling from the source to sink electrode is a two-stage process. In the "rst stage, one electron tunnels through one of the barriers, while in the second stage, another electron tunnels through the second barrier. When all the three electrodes are ferromagnetic, then TMR exists for arbitrary spin relaxation time. If, however, the central electrode is nonmagnetic, the TMR e!ect exists only when the spin relaxation time in the central electrode is long enough, so that the #owing current leads to spin accumulation in this electrode. The spin accumulation is a result of spin-dependent tunneling rates across the barriers and depends on the magnetic con"guration of the junction. This, in turn, gives rise to a shift of the Fermi energy in the central electrode, which depends on the electron spin and magnetic con"guration of the junction, and consequently, leads to the con"guration-dependent current #owing through the junction, i.e., to the TMR e!ect. In the coherent tunneling regime, on the other hand, an electron is in a coherent state of the whole double-barrier system and spends only a very short time in the central electrode. In this tunneling limit one usually neglects the e!ect of spin accumulation in the central electrode. This is justi"ed when only coherent tunneling processes take place. In real situations, however, both coherent and sequential tunneling processes occur [15] and, therefore, the spin accumulation should be taken into account. In the coherent tunneling regime transmission coe$cients through the double-barrier junction depend on its magnetic con"guration and this dependence gives rise to the TMR e!ect. A special kind of double-barrier junctions are junctions with a mesoscopic central electrode (e.g.
a small grain) [16}19] or junctions in which the central electrode consists of a granular system [20,21]. When the grains are small enough, the single-electron charging e!ects become important and can lead to Coulomb blockade of tunneling current [16}19]. The charging e!ects can also enhance TMR [20,21]. Those e!ects will not be considered in this paper. We restrict ourselves to large-area planar junctions, in which the Coulomb blockade e!ects do not occur. In this paper we consider coherent tunneling in a double-barrier system with a nonmagnetic central electrode. We assume that electron spin is conserved in the tunneling process through the whole system and also neglect any spin accumulation in the central electrode (as well as in the external electrodes). Apart from this we analyze only colinear con"gurations of the electrode magnetizations, which allows us to consider each spin separately. Electronic structure of the electrodes is approximated by a free-electron-like parabolic band, which for ferromagnetic electrodes is spinsplit due to an e!ective exchange "eld. For simplicity, we assume that the band edge in the nonmagnetic electrode coincides with the band edge of the external electrodes in the paramagnetic limit. We also assume that the barriers are of a rectangular shape which transforms into a trapezoidal one when a bias voltage < is applied to the junction. In Section 2 we describe the model and present a general formula for the tunneling current in double-barrier junctions with a nonmagnetic central electrode. Numerical results are presented in Section 3, where the bias dependence of TMR and its variation with the thickness of central electrode are also discussed. Summary and "nal conclusions are in Section 4.
2. Electron transmission across double-barrier junction and tunnel magnetoresistance First, consider transmission of an electron with spin p in a magnetic con"guration k of the junction (k,ap for the antiparallel con"guration and k,p for the parallel one). The transmission coe$cient across the whole double-barrier structure can be derived from the electron wave function with
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standard boundary conditions imposed on the wave function and its "rst derivative at each electrode/barrier interface. When the interfaces are #at, the in-plane component k# of electron wave vector is conserved in the tunneling processes. Let the axis x be normal to the junction and let the interface between the left electrode and the barrier be located at x"0. If we denote the electron wave function in the region j as u (x), then the boundary conditions H at the interface between the regions j and j#1 (the interface located at x ) are u (x )"u (x ) and H H H H> H u (x )"u (x ), where the prime at the wave H H H> H function denotes its "rst derivative and also the same electron mass was assumed for all regions. The index j runs from j"1 for the left electrode to j"5 for the right one, with j"2, 4 corresponding to the barriers and j"3 corresponding to the central electrode. Let the left (right) electrode be the source (drain) one. The transmission coe$cient for an electron incident normally on the barrier from the left electrode and with the total energy e , V measured from the bottom of the electron band of the left electrode in the paramagnetic limit, can be written in the form [22]:
¹ (e )"[¹\¹\#(¹\!1)(¹\!1) N I V #2(¹\¹\(¹\!1)(¹\!1) ;cos( )]\,
(1)
where we neglected transmission of electrons, whose energy e is below the band edge of the V central electrode, as for such electrons the central electrode is e!ectively an additional potential barrier. In the above expression, T and T denote the transmission coe$cients through the "rst (left) and second (right) barriers, respectively, which depend on the electron spin p and magnetic con"guration k of the junction (for clarity of notation this dependence is not indicated explicitly in Eq. (1)). Similarly, in Eq. (1) is a phase factor, to be speci"ed later, which also depends on the electron spin p and magnetic con"guration k (not indicated explicitly in Eq. (1)). For oblique incidence the energy e is the V total energy minus the kinetic energy of in-plane motion.
375
In the case under consideration (nonmagnetic central electrode and ferromagnetic external ones) the transmission coe$cients T and T and the phase factor take the forms
1 (k b !k c )#(i #k k a ) \ N N ¹ " , p 4k k D N (2) 1 (k b !k c )#(i #k k a ) \ N I N I ¹" , p 4k k D N I (3)
"2k d # # , (4) where !p) , )p and k (c #k a )!(i #k b ) N N , (5)
"arctan 2k (k a b #c i ) N k (c !k a )#(i !k b ) . (6)
"arctan N I 2k (k a c #b i ) N I The normal components k , k and k of the N N I electron wave vector in the source, drain and central electrodes, respectively, are given by the formulas
1 k " (2m(e #p D), V N N
(7)
1 k " (2m(e #p p D#e<), V N I N I
(8)
d e< , 2m e # (9) V d #d where m is the electron mass and we assumed that the parallel and antiparallel con"gurations di!er in the orientation of magnetic moment of the right electrode (consequently k is independent of N k while k depends on k). Apart from this, d and N I d denote the thickness of the left ("rst) and right (second) barriers, d is the thickness of the central electrode, 2D denotes the spin splitting of the electron band (D"0 in the paramagnetic limit) in ferromagnetic electrodes (assumed the same for both electrodes) and the parameter p describes N electron spin, p "1(!1) for p"!( ), whereas the N 1 k "
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parameter p is related with the magnetic con"gI uration of the junction, p "1(!1) for k,p(ap). I Finally, the following abbreviations are used in Eqs. (2)}(6):
z (x )"z (d #d ) (2m(d #d ) " e<
a "Ai[z (x )]Bi[z (x )] H H H\ H H
d e< , ; E #; !e ! $ V d #d z (x )"z (d #d #d ) (2m(d #d ) " e<
!Bi[z (x )]Ai[z (x )], H H\ H H
(10)
b "D+Ai[z (x )]Bi [z (x )] H H H\ H H !Bi[z (x )]Ai [z (x )],, H H\ H H
(11)
c "D+Ai [z (x )]Bi[z (x )] H H H\ H H !Bi [z (x )]Ai[z (x )],, H H\ H H
(12)
i "D+Ai [z (x )]Bi [z (x )] H H H\ H H !Bi [z (x )]Ai [z (x )], H H\ H H
(13)
for j"2 and j"4, where Ai(z (x)) and Bi(z (x)) are H H the Airy functions while Ai (z (x)) and Bi (z (x)) H H their "rst derivatives. The argument z (x) of the H Airy functions is de"ned as
(2m(d #d ) E #; !e z (x)" $ H V H e<
d x e
(14)
where ; denotes the height of the "rst ( j"2) and H second ( j"4) barriers, respectively, whereas E is $ Fermi energy. The Airy functions and their derivatives in Eqs. (10)}(13) are taken at the points z (x ) H H and z (x ), i.e., at the interfaces. More speci"cally, H H\ those arguments are z (x )"z (0)
"
(2m(d #d ) (E #; !e ), (15) $ V e<
z (x )"z (d )
"
(2m(d #d ) e<
d e< , ; E #; !e ! $ V d #d
(16)
(17)
;(E #; !e !e<). (18) $ V Finally, the parameter D in Eqs. (2), (3) and (11)}(13) is given by D"![2me
#$ \C4 2pme e< ¹ (e ) de j " N I V V I h #N I N #$ # (E !e )¹ (e ) de , (20) $ V N I V V #$ \C4 where the lower integration limit, E , is given by N I E "max+!p D,!d e
3. Numerical results In our numerical calculations, we assumed that the source and drain electrodes were made of the same ferromagnetic material. Additionally, we assumed the same position at <"0 of the electron band in the central electrode and in the external electrodes in the paramagnetic limit (vanishing spin splitting, DP0). Consider "rst the dependence of TMR on the thickness of the central electrode.
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377
Fig. 1. TMR as a function of the central electrode thickness d , calculated for E "2.62 eV, D"1.96 eV, U "U "0.75 eV, $ d "d "0.5 nm and for applied voltage <"20 mV.
Fig. 2. Electric current density j vs. thickness d of the central electrode calculated for the same parameters as in Fig. 1. Solid and dashed curves correspond to the parallel and antiparallel con"gurations, respectively.
The relevant numerical results were obtained for the Fermi energy E "2.62 eV, spin splitting of the $ electron band in the ferromagnetic electrodes D"1.96 eV, barrier height U "U "0.75 eV and the barrier thickness d "d "0.5 nm. In Fig. 1, TMR is shown as a function of the thickness of the central electrode d in the case when the bias voltage <"20 mV is applied to the junction. Several features of the TMR curve are interesting and worth further analysis. First, TMR as a function of d shows well-de"ned peaks, the height of which decreases with increasing d . Sec ond, those peaks are relatively high at small values of thickness d . In the case shown in Fig. 1, there are four such peaks. The "rst peak corresponds to *R/R +35, while the fourth one to *R/R +15. The other peaks are much smaller. One can expect that the periodic variation of TMR with increasing thickness of the central electrode is related to the quantum-well states formed in the central electrode and to resonant tunneling through the whole structure. To understand the origin of this periodicity in TMR and the di!erence in heights between the "rst four peaks and the others, we have plotted the electric current in the parallel and antiparallel con"gurations as a function of the thickness of the central electrode in Fig. 2. The periodic variation with increasing d is now more pronounced and more regular. From the comparison of Figs. 1 and 2, it follows
that the maxima in TMR correspond roughly to the minima in the #owing current. In the antiparallel con"guration the "rst four minima are relatively deep with a small value of the corresponding current (not resolved in Fig. 2), which gives rise to large TMR (the "rst four peaks in Fig. 1). To understand further the origin of periodic variation of electric current and peaks in TMR, we have analyzed the energy dependence of the transmission coe$cient through the double-barrier junction for several different values of the thickness d . We have found that the periodicity in TMR is strongly correlated with the resonance states in the central electrode. Generally, one can expect large TMR when the highest resonance level (below E ) is at the energy $ smaller than D!e< and there is no resonance states in the energy range (D!e<, E ). When < is $ not too high, so this resonance level is above the energy !D, then the existing resonance state participates in the conduction processes only for the spin-majority channel in the parallel con"guration, whereas it does not give rise to current in the antiparallel con"guration for both spin orientations and in the parallel con"guration for the spinminority channel. In the model situation analyzed here this happens when the thickness d of the central electrode corresponds to the "rst four peaks in TMR. Consider for example the case when d is equal to 0.69 nm. The relevant transmission coe$cients for both spin orientations and both magnetic
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Fig. 3. Transmission coe$cients calculated for d "0.69 nm and for both spin orientations in the antiparallel and parallel con"gura tions. The other parameters are the same as in Fig. 1.
con"gurations of the junction are shown in Fig. 3. It is clear from the data that there are two resonance states in the quantum well (central electrode); "rst at e 0.43 eV and the second one at V e 1.74 eV. Both resonance states are below the V band edge of the minority electron band in the ferromagnetic electrodes and, therefore, both are not active for minority electrons in the parallel con"guration and for both spin orientations in the antiparallel con"guration. There are no other resonance states in the relevant energy range. Therefore, there is a large di!erence in the #owing current in both con"gurations and consequently large magnetoresistance. The resonance state is however far from the Fermi level so only electrons with large
in-plane momentum can tunnel resonantly. Therefore, there is a minimum in the electric current in both the con"gurations, while TMR reaches maximum at this point. When the thickness of the central electrode increases, the resonance states go down in energy and new resonance levels appear below E in the energy $ range (D, E ). In Fig. 4, we show the transmission $ coe$cients calculated for d "0.89 nm, i.e., for d corresponding to the "rst minimum in TMR on the right of the "rst maximum (Fig. 1). This value of d corresponds to a local maximum in the #owing current. A new active resonance level appears at this thickness of the central electrode, whose energy is approximately equal to 2.6 eV. This state, lying
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379
Fig. 4. The same as in Fig. 3, but for d "0.89 nm.
slightly below the Fermi energy, contributes signi"cantly to the #owing current for both spin orientations and magnetic con"gurations. Therefore, electric current in both the con"gurations is relatively large (local maximum), whereas the di!erence between the currents in the antiparallel and parallel con"gurations is rather small (local minimum in TMR). The situation for d corresponding to the next large peaks in TMR is qualitatively similar to that described above for d "0.69 nm. The other peaks in TMR are, however, signi"cantly smaller. This is because starting from the "fth peak at least one of the resonance states is always active in both the con"gurations and spin orientations, which signi"cantly reduces TMR. This is because it is impossible to remove simultaneously all the resonance
states from the energy range (D!e<, E ) by slight$ ly increasing or decreasing the thickness of the central electrode. By changing the energy of these states we can only slightly increase or decrease the tunneling current, which gives rise to weak variations of TMR with the central electrode thickness. Let us now consider how TMR varies with increasing voltage applied to the junction. In Fig. 5, we show the bias dependence of TMR for d " 0.69 nm (the other parameters are the same as in Figs. 1}4). The thickness d corresponds to the "rst maximum in Fig. 1. We recall that Fig. 1 was calculated for a relatively small bias voltage, <"20 mV. As one can see in Fig. 5, TMR slowly decreases with increasing bias and then rapidly falls down at about 420 mV to a small value. Such a behavior of TMR is caused by the fact that the
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Fig. 5. Bias dependence of TMR for d "0.69 nm. The other parameters are the same as in Fig. 1.
resonance state lying below D at <"0 begins to participate in the conduction processes also in the antiparallel con"guration when the applied voltage is su$ciently high. From the discussion above we remember that large TMR appears when there is a resonance state below D!e
Fig. 6. The same as in Fig. 5, but for d "0.89 nm.
Fig. 7. Bias dependence of electric current density for d "0.69 nm. The other parameters are the same as in Fig. 1.
4. Summary and concluding remarks We have calculated tunnel magnetoresistance in the coherent tunneling regime in a double-barrier junction with a nonmagnetic central electrode. We showed that in some cases the resonant tunneling can give rise to huge TMR. At this point it is worth to mention that in the sequential tunneling regime nonzero TMR can appear in this junction only when spin-relaxation time in the central electrode is su$ciently long to produce spin accumulation [14]. This TMR is, however, signi"cantly smaller than TMR in the coherent tunneling regime analyzed in this paper. Strong dependence of the transmission coe$cient on the position of quantum-well
M. Wilczynski, J. Barnas / Journal of Magnetism and Magnetic Materials 221 (2000) 373}381
states in the central electrode can lead to negative di!erential resistance. The calculations were performed under several approximations. First, we assumed that electron spin is conserved in tunneling events. It is, however, well known that TMR is reduced when spin-#ip tunneling processes are allowed, despite the fact that such processes usually increase electric current by opening new channels for electron tunneling [23,24]. Other scattering processes leading to incoherent tunneling can reduce the TMR value as well. Second, we used a simple free-electron-like model to describe electronic structure of the junction. In real metallic systems, however, electronic structure is much more complex, which can lead to quantitative modi"cations of the results presented here. We believe that some qualitative features of TMR will survive. The huge TMR value we obtained in our numerical calculations followed from a speci"c position of the electron bands in the ferromagnetic and nonmagnetic electrodes. The condition for such huge TMR may be realized in experimental situations not only in junctions with metallic central electrodes, but also when the metallic central electrode is replaced by a semiconducting one [25]. For example, one may expect such behavior in metallic junctions, in which tunneling current is dominated by d electrons, which may be achieved by taking appropriate barriers [4]. Another approximation used in our description is the zero-temperature limit. We considered this limit since the physical picture is then clear and one can easily explain all the features of the tunneling current and TMR. The description presented in this paper can be easily extended to "nite temperature. Generally, one may expect an increase in the tunneling current with increasing ¹ and a decrease in TMR.
Acknowledgements The work was supported by the Polish State Committee for Scienti"c Research through the Research Project 2P03B 075 14.
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