IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 12, DECEMBER 2013 5635

Magnetic Polarization of the Tunneling CurrentImara L. Fernandes and Guillermo G. Cabrera

Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, UNICAMP, 13083-859 Campinas, SP, Brazil

In this work, we theoretically study the spin-dependent transport in a magnetic tunnel junction (MTJ). Using a simple model andballistic transport, the magnetic polarization of the tunneling current on this system is studied by focusing on the tunneling of andelectrons. We investigate the tunneling of these electrons through potential barriers, which represents the insulating layer between theferromagnetic electrodes. We also examine how the conductance depends on voltage applied between the electrodes and on the effectivemass of the electrons. The conductance is controlled by the transmission coefficient of the tunnel effect, and qualitatively it is knownthat tunneling probability of the electrons is lower than the electrons. We also estimate the effect of the tunneling magnetoresistance(TMR) and it is strongly influenced by the effective mass of the electrons. The electrons do not contribute significantly to the TMR.

Index Terms—Magnetoresistance, tunneling.

I. INTRODUCTION

I N the last decades, research on magnetic tunnel junctions(MTJs) has aroused considerable interest [1]–[3]. Most of

the interest is due to their potential applications in spintronic de-vices such as magnetic sensors and magnetoresistive random-access memory (MRAM) [4], [5]. A MTJ consists of two ferro-magnetic metal layers (electrodes) separated by a thin insulatinglayer (tunnel barrier). If a bias voltage is applied, the electronstunnel through the insulating barrier from one metal electrodeto the other. One interesting effect of MTJs is that the tunnelingcurrent depends on the relative orientation of the magnetizationof the two ferromagnetic layers. This effect is known as tunnelmagnetoresistance (TMR) [6], [7] effect and is defined as

(1)

where and are the conductance in the anti-parallel(AP) and parallel (P) state for the magnetizations, respectively.The theoretical model that is widely used to describe the TMR

was proposed by Julliere, who studied the spin polarized tun-neling between two ferromagnetic layer separated by an insu-lator in 1975 [6]. He reported a conductance change of 14%at 4.2 K on application of a magnetic field sweep in Fe/Ge/Cotunnel junctions. Julliere did the interpretation of his data using asimple model based on the assumption that the spin is conservedin the tunneling process, so the conductance occurs in two inde-pendent spin channels. According to this, electrons originatingfrom one spin state of the first electrode are accepted by unfilledstates of the same spin of the second electrode and the tunnelingcurrent in each channel is proportional to the product of the spinpolarization for the metals on either side of the barrier. In the APconfigurations, the roles of the spin bands are interchanged. Forthis model the TMR is given by

(2)

Manuscript received February 17, 2013; revised April 25, 2013; acceptedApril 30, 2013. Date of current version November 20, 2013. Correspondingauthor: I. L. Fernandes (e-mail: imaraf@ifi.unicamp.br).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2272214

where and are the spin polarization in the left and rightmagnetic layer, respectively, and it is given as a function of thedensity of states of the up and down spin subbands:

(3)

The Julliere simple model can be subjected to criticism. Inferromagnetic electrodes, which typically are transition metals,both and electrons contribute to the electronic conduction,but only the band is highly polarized, with very different bandstructure for spin up and spin down electrons. In contrast, theband has a small polarization, but yields the dominant contri-bution for the conductance. Since orbitals are of more local-ized character, one qualitatively expects a small contribution tothe tunneling current. It is then uncertain which polarization oneshould use in Julliere’s equation (2). In addition, Jullieremodel isindependent of the geometry and the structure of the barrier layerand independent of the voltage. Most important, boundary con-ditions at the barrier have a prominent role, with the matchingof the incident and transmitted waves at the barrier interfaces.In this paper, we will address some of the above issues, using asimplemodel that includes the following features: 1) Band struc-ture is simply parameterized with an effective mass. For theband, effective masses are very different for the up and downspin sub-bands. Note that this condition is more general than theStoner model, since we are assuming that the exchange split-ting is not rigid, thus mimicking more realistic band properties.Different effective masses for different spins will cause a mis-match of wave functions when themagnetic electrodes are in theAP configuration, since the roles of both spins are interchangedfrom onemagnetic layer to the other. The band is poorly polar-ized, and effective masses of similar values are assigned to bothspins ( electrons dominate the conductance but not the magne-toresistance). 2) Dependence on the applied voltage is includedin the barrier profile, using a trapezoidal potential barrier. 3) Tosolve the scattering problem, electrons are represented by planewaves (extended states), while for electrons we employed lo-calizedwave packets.We use a approach in the ballisticregime (Landauer theory), to calculate the conductance.Our results are at variance from expectations based on Julliere

theory, and this will be discussed in the next sections. An inver-sion of the magnetoresistance is possible for some values of the

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5636 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 12, DECEMBER 2013

Fig. 1. Schematic diagram of the potential in the MTJ. is the Fermi level,is the applied voltage, is the barrier height, and is the barrier thickness.

parameters. This paper is organized as follows. In Section II,we introduce the model and formalism for the calculation of theconductance. In Section III, the corresponding results are pre-sented. Finally, Section IV summarizes the results of our work.

II. MODEL AND FORMALISM

We propose a model to provide the simplest insight intothe essential physics for the transport in MTJ by studying thetunneling probability of and electrons through a potentialbarrier. The transmission probability depends on the effectivemass, which is different for each spin subband. The electronicstructure for ferromagnets is characterized by a hybridizationbetween extended bands with more localized bands. Thebands have similar DOS for both spins, therefore the effectivemasses have similar values. They can be represented by waveplanes with small wave vector, therefore, they are modulated byfree-electron. The bands have very different band structures forspin up and down, and very different DOS for the both spins, andso, the effective masses are very different. Since the electronshave higher effective mass and localized states, they can berepresented by a wave packet with larger wave vectors. They aremodulated by Gaussian wave function, centered far away fromthe barrier and moving toward it with a constant average energy.Themagnetic momentum in the left electrode is assumed con-

stant while themagnetic momentum in the right electrode can beparallel or anti-parallel in relation to the left electrode. The junc-tion is biased with electrical potential and the electrons flowalong the axis, which is perpendicular to the junction plane.The barrier potential profile is represented by a linear function,as shown in Fig. 1:

(4)

where is the Heaviside step function, is the appliedvoltage, is the barrier height, is the elementary charge, andis the barrier thickness.The wave function of electrons in the system can be found

using the Schödinger equation. By solving the equation, thewave function satisfying each region is given as

(5)

where the wave vector for the regions I and III areand ,

respectively. In region II, and areAiry functions and the function is given by

,where . The coefficients , , and arecalculated by matching (5) with the boundary conditions at

and :

(6)

(7)

where and are the effective masses outside the barrier,region I and region III, respectively, and is the effectivemass inside the barrier (region II). Therefore, the transmissioncoefficient for this barrier is given by

(8)

where the parameter is given byand

(9)

The electrons are modulated by Gaussian wave functionand in the -space is given by

(10)

where , is the width of the wave packet in the realspace and is the Fermi wave number. The tunneling prob-ability of a Gaussian wave packet tunnels through a potentialbarrier is given by

(11)

where is the wave packet incident in the -space. Finallyfor the analysis of the conductance , we use the Landauerformula [8]

(12)

where is made the sum over the channels and is thetransmission coefficient. We have assumed that during the tun-neling process the electrons preserve their spin orientation soan electron can only tunnel to a subband of same spin orien-tation. Therefore, in the parallel magnetization configurationthe two channels are electrons tunneling from a majority sub-band (M) to a majority subband and electrons tunneling from

FERNANDES AND CABRERA: MAGNETIC POLARIZATION OF THE TUNNELING CURRENT 5637

a minority subband (m) to a minority subband. The two chan-nels in the anti-parallel magnetization configuration are elec-trons tunneling from a majority subband to a minority subbandand electrons tunneling from a minority subband to a majoritysubband.

III. RESULTS AND DISCUSSION

We evaluated (1) and (12) numerically for various effectivemass of the carriers for and electrons, choosing the Fermi en-ergy of iron 7.64 eV [9], a barrier height of 9.4 eV, and barrierthickness of 10 . The effective mass inside the barrier (Re-gion II) is set by , where is the mass of the freeelectron [10], and in the regions I and III they are consideredas a parameter in function of the mass of the majorityand minority carriers. For the parallel configuration, theeffective masses are set by , for the con-tribution of majority carriers and , for thecontribution of minority carriers and for the anti-parallel config-uration the parameters are set by , for thecontribution of majority carriers and ,for the contribution of minority carriers. The density of states(DOS) for majority and minority subbands at Fermi level foriron are and [9], respectively. There-fore, using the DOS for the three-dimensional free electron [11]the ratio between effective mass of the majority carriers and mi-nority carriers is kept constant 2.3. The maximumvalue of the TMR by Julliere model (2) depends essentially onDOS of the ferromagnetic electrodes and the TMR calculatedfor a symmetric MTJ composed of Fe ferromagnetic electrodes,was about 45.5%.

A. Dependence on the Effective Mass

In this part, we are interested in understanding how the con-ductance and the TMR depend on the effective mass of theelectrons. The electrons are modulated by Gaussian functions(10), the spacial width is chosen andis the Fermi wave number associated with each subband. Thisparticular value of was chosen to get a localized dis-tribution of the wave packet, in relation to the width of the po-tential barrier. The ratio between effective mass of the minoritycarriers and the mass of free-electrons was variedin the range 1.5 to 6 and keeping the ratio ,as discussed earlier. The conductance is evaluated as a func-tion of the ratio for bias applied . Asshown in Fig. 2(a), one can be seen that the conductance de-creases with increasing for both configurations. Amechanism which could contribute to this dependency is themismatching of the wave function in different regions due thedifference in mass between these regions. This mismatching ofthe wave function would make more difficult the tunneling ofelectrons between the electrodes. As , one mayobserve that , which indicates that the inverse of theTMR may occur.The dependence of the TMR is shown in Fig. 2(b).

One can see that the TMR increases with increasing ,and it is negative for . The inverse TMR ef-fect indicates that the polarization of the tunneling current can

Fig. 2. (a) Conductance of electrons as a function of the effective mass forminority carriers (in units of the free electron mass ) for the AP and P con-figurations. (b) Corresponding magnetoresistance as a function of . Param-eters for the figures are: , , , and

. Both curves are taken at .

TABLE IEFFECTIVE MASS AND FERMI WAVE NUMBER FOR MINORITY

AND MAJORITY CARRIERS

be very different from the polarization of the volume Fermilevel. The maximum value found for TMR was approximately15% and it is lower than expected by the Julliere model (45.5%). A decrease in TMR is expected because the Julliere modeldoes not consider characteristics of the insulating layer and scat-tering processes, and we are representing the by localizedwave packets with different effective mass for majority and mi-nority carriers and we are also including the characteristics ofthe insulating by changing the height of the barrier by applyinga voltage. It is seen that TMR approaches to a constant when

increases to a large value.

B. Dependence on the Voltage

In the second part of this work, we are interested in esti-mating the contribution of and electrons in the TMR andtheir dependency on the voltage, which was varied in the range0 to 1.5 V. The electrons are modulated by free-electrons andthe effective mass of the minority and majority carriers are setby and , respectively. Forelectrons the transmission coefficient is given by (8) with

wave vector and is theeffective mass of majority (minority) carriers. As discussed inSection III-A, the electrons are modulated by Gaussian func-tions and the effective mass of the minority carriers is set by twovalues and and they have beenchosen to illustrate the bias voltage dependence of the TMR.The effective mass and the Fermi wave number associated witheach sub-band are shown in Table I.

5638 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 12, DECEMBER 2013

Fig. 3. Magnetoresistance as a function of voltage for and electrons. Pa-rameters for the examples are , , and ;for electrons we used and and for electrons

and . The value of the minority mass isindicated in the figure.

For and electrons, the conductance has the same behavior;it increases with increasing the bias voltage. Biasing the MTJthe tunneling probability increases due to the contribution ofthe tunneling of electrons from occupied states of one electrodeto the empty states above the Fermi energy of the other elec-trode. The TMR is evaluated as a function of the bias applied;the result in Fig. 3 shows that the electrons do not contributesignificantly for the TMR. As expected, for electrons the mag-nitude of the TMR decreases with increasing the bias voltage.Due to the conductance be dependent on the voltage, it results inthe variation of TMR versus the applied voltage. Furthermore,as discussed earlier with Fig. 2(b), the signal of the TMR de-pends on the effective mass of carrier.

IV. CONCLUSION

In this paper, we have investigated the spin-dependent trans-port in MTJ based on a simple model proposed. We concludethat both and electrons contribute to the transport, however,only the electrons contribute to the TMR. The effects of theeffective mass of the electrons on the conductance and on TMR

were studied. It can be observed that the effective mass of thecarriers under certain conditions could lead to a conductance inparallel configuration lower than in anti-parallel configurationand, therefore, to an inversion of TMR. As the Julliere’s for-mula overestimates the value of the TMR, our model with theinclusion of extended states for electrons and localized wavepackets for electrons with different effective masses for ma-jority and minority carriers could estimate a lower value.

ACKNOWLEDGMENT

This work was supported in part by Coordination for the Im-provement of Higher Education (CAPES) and by São PauloResearch Foundation (FAPESP) through project No. FAPESP2011/19298-4. This paper was presented at the X Latin Amer-ican Workshop On Magnetism, Magnetic Materials And TheirApplications (X-LAW3M 2013) (IEEE Trans. Magn., vol. 49,no 8, pt. 1, Aug. 2013).

REFERENCES

[1] G. A. Prinz, “Magnetoelectronics,” Science, vol. 282, no. 5394, pp.1660–1663, 1998.

[2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S.von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger,“Spintronics: A spin-based electronics vision for the future,” Science,vol. 294, no. 5546, pp. 1488–1495, 2001.

[3] E. Y. Tsymbal, O. N. Mryasov, and P. R. LeClair, “Spin-dependenttunnelling in magnetic tunnel junctions,” J. Phys.: Condens. Matter,vol. 15, no. 4, p. R109, 2003.

[4] Y. Li, F. Qian, J. Xiang, and C. M. Lieber, “Nanowire electronic andoptoelectronic devices,” Materials Today, vol. 9, no. 10, pp. 18–27,2006.

[5] S. Kokado and K. Harigaya, “A theoretical investigation of ferromag-netic tunnel junctions with 4-valued conductances,” J. Phys.: Condens.Matter, vol. 15, no. 50, p. 8797, 2003.

[6] M. Julliere, “Tunneling between ferromagnetic films,” Phys. Lett. A,vol. 54, no. 3, pp. 225–226, 1975.

[7] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, “Largemagnetoresistance at room temperature in ferromagnetic thin filmtunnel junctions,” Phys. Rev. Lett., vol. 74, no. 16, pp. 3273–3276,Apr. 1995.

[8] R. Landauer, “Electrical resistance of disordered one-dimensional lat-tices,” Philos. Mag., vol. 21, no. 172, pp. 1478–6435, Nov. 1970.

[9] J. Callaway and C. S. Wang, “Energy bands in ferromagnetic iron,”Phys. Rev. B, vol. 16, no. 5, pp. 2095–2105, Sep. 1977.

[10] A. M. Bratkovsky, “Tunneling of electrons in conventional and half-metallic systems: Towards very large magnetoresistance,” Phys. Rev.B, vol. 56, no. 5, p. 2344, 1997.

[11] N. W. Ashcroft and D. N. Mermin, Solid State Physics, 1sted. Toronto, ON, Canada: Thomson Learning, 1976.