Output voltage calculations in double barrier magnetic tunnel junctions with asymmetric voltage behavior

Journal of Magnetism and Magnetic Materials 324 (2012) 2844–2848

Contents lists available at SciVerse ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Output voltage calculations in double barrier magnetic tunnel junctions with asymmetric voltage behavior ¨ Arthur Useinov a,n, Oleg Mryasov b, Jurgen Kosel c a

Spintronics Theory Group, Physical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal (Jeddah) 23955, Saudi Arabia b MINT Center, University of Alabama, Tuscaloosa, AL, USA c Sensing, Magnetism and Microsystems Group, Physical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal (Jeddah) 23955, Saudi Arabia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 December 2011 Available online 7 May 2012

In this paper we study the asymmetric voltage behavior (AVB) of the tunnel magnetoresistance (TMR) for single and double barrier magnetic tunnel junctions (MTJs) in range of a quasi-classical free electron model. Numerical calculations of the TMR–V curves, output voltages and I–V characteristics for negative and positive values of applied voltages were carried out using MTJs with CoFeB/MgO interfaces as an example. Asymmetry of the experimental TMR–V curves is explained by different values of the minority and majority Fermi wave vectors for the left and right sides of the tunnel barrier, which arises due to different annealing regimes. Electron tunneling in DMTJs was simulated in two ways: (i) Coherent tunneling, where the DMTJ is modeled as one tunnel system and (ii) consecutive tunneling, where the DMTJ is modeled by two single barrier junctions connected in series. & 2012 Elsevier B.V. All rights reserved.

Keywords: Magnetic tunnel junction (MTJ) Tunneling magnetoresistance (TMR) Spinelectronic

1. Introduction Electronic circuits utilizing spintronic principles of operation are becoming increasingly important in modern devices. In this paper we present theoretical investigations of the magnetic tunnel junctions, which are used for many applications such as read-heads of hard disks, magnetic random access memory, bio-sensors, resonant tunneling diodes, etc. [1–3]. In the case of an ideal homojunction (both sides of the junction are exactly the same), the branches of the TMR–Va curve must be symmetric with respect to negative and positive values of the applied voltage Va. Therefore, a symmetric or asymmetric voltage behavior (AVB) can be used one criteria for the quality of the deposited layers [4]. In most cases, AVB arises due to different ratios of minority to majority Fermi wave vectors for the left and right sides of the tunnel contact, respectively [5]. This difference is caused by minute variations of the material compositions, lattice mismatch of the deposited layers, layer thicknesses, sputtering conditions, impurities, annealing regimes etc. Different studies have previously shown AVB of the tunnel magnetoresistance of SMTJ and DMTJ [4–9]. Ikeda et al. [10] were among those who demonstrated symmetric TMR curves in SMTJs

n

Corresponding author. E-mail addresses: [email protected], [email protected] (A. Useinov). 0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.04.025

based on CoFeB/MgO/CoFeB structures with very high TMR ratios of 355% at a temperature of 300 K and 578% at 5 K. Important experimental results were published recently [6,7], where the annealing effects on the voltage dependence of TMR (including AVB) and output voltages were investigated. Feng et al. [6] showed that TMR asymmetry in DMTJs can be explained using a model with two SMTJs in series having a common magnetic layer. Theoretical classification and numerical calculations in range of a ballistic electron transport approximation of these MTJs were done in Ref. [11], where the importance of the middle layer thickness and the difference between coherent (CHT) and consecutive (CST) tunneling models for resonant and non-resonant tunneling cases in DMTJs have been shown. In this work we present further details for both tunneling models, which include numerically calculated TMR ratios, AVBs of the TMR– Va curves for DMTJs and SMTJs and their output voltages as well as their I–Va curves. The numerical calculations were based on previously developed models [5,11,12], where the electron transport through the tunnel structures is considered as three-dimensional problem, taking into account momentum conservation law and assuming electron spin conservation during the tunneling. In the model it is assumed that the Fermi energy of the conduction electrons in metals EF, exchange energy Eex, barrier heights UB1(2) and applied voltages are much larger than the thermal energy kBTE0.025 eV at 300 K making the model applicable at room temperature. The results of our simulations are in good agreement with experimental data [6–8].

A. Useinov et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2844–2848

2. Theoretical formalism Theoretical models were developed for SMTJs consisting of a left ferromagnetic layer FML, an insulating layer I(L) with thickness L and a right ferromagnetic layer FMR as well as for DMTJs consisting of FML, I1(L1), FMM(LM), I2(L2), FMR layers (see Fig. 1), where LM is the thickness of the middle layer. The transport problem can be considered in range of the twocurrent model, where the normalized TMR is determined as TMR ¼ ðJ P J AP Þ=J AP  TMR1 max

ð1Þ

P(AP)

are current densities in case of parallel (P) and where J antiparallel (AP) magnetization alignments of the middle magnetic layer FMM in case of the DMTJ, or P and AP magnetization alignments in FMC in case of series connection SMTJ-1 and SMTJ-2 (assuming that the magnetizations in FML and in FMR are strongly pinned in one direction). The current densities for the compositions are proportional to the integral of the product of the tunnel transmission coefficient (TTC) and the cosine of the incidence angle of the electron   trajectory cos yL,s  xs [11]: L

¼ J PðAPÞ s

e2 ðkF,s Þ2 V 4p2 _

Z

1:0 X CR

xs DPðAPÞ dxs , s

ð2Þ

L

where kF s is the Fermi wave vector of FML and s ¼m(k) is the electron spin index, V is the voltage drop at the single or double barrier system. In case of a SMTJ we assume V¼Va. In case of a DMTJ modeled by CST and equal barrier heights for SMTJ-1 and SMTJ-2: V¼V1(2) ¼ Va/2. In case of a DMTJ modeled by CHT: V¼Va. DsPðAPÞ is the tunnel transmission coefficient (TTC), which is a function of the barrier geometry, V and the values of the transverse components (parallel along the normal of interface L

L

j

I/FM) of the electron wave vectors k?,s ¼ kF,s cosðyL,s Þ and k?,s ¼ j

ks ðVÞcosðyj,s Þ with j¼R (in case of SMTJ) or M, R (in case of DMTJ). The lower limit XCR for the SMTJ is the critical restriction. XCR ¼0 in case electrons which tunnel from the minority (min) to the majority (maj) conduction sub-band, while, in the opposite case, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L R X CR ¼ 1ðkF,m =kk ðVÞÞ2 . These conditions arise according to the conservation law of the Fermi wave vector projection on the junction plane, which is in case of SMTJs: L

L

R

k99,s ¼ kF,s sinðyL,s Þ ¼ ks ðVÞsinðyR,s Þ

ð3Þ

L

where k99,s are the longitudinal components of the wave vector L kF,s .

The right-side wave vector voltage dependence is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R ks ðVÞ ¼ ðkF,s Þ2 þ cM R V

ð4Þ

2845

where MR ¼m/m0; m is the effective electron mass for the magnetic layers, which is close to the free electron mass m0 and c is a dimensional factor: c ¼ 2m20 e 1020  0:2625 [A˚  2 eV  1]. _

Eq. (2) is applied for positive voltages and the solution for negative one can be derived using symmetric relations of the system, i.e. the parameters of the electronic states of the magnetic LðRÞ RðLÞ LðRÞ RðLÞ layers have to be reversed kFmmaj -kFmmaj , kFkmin -kFkmin . P(AP) The definition of the current densities J , given below, represents the CST model with the total resistance Rtot ¼R1 þR2, where R1 and R2 are the resistances from SMTJ-1 and SMTJ-2, respectively (Fig. 1a). JPðAPÞ ¼

J PðAPÞ JPðAPÞ Va 1 2 V 1 J 2PðAPÞ þ V 2 J 1PðAPÞ

,

ð5Þ

PðAPÞ PðAPÞ ¼ J PðAPÞ and JPðAPÞ are the JPðAPÞ n n,m þ J n,k , n¼ 1,2, where the J 1 2

current densities for the first and second barriers. In case of equal PðAPÞ J PðAPÞ =ðJ PðAPÞ þ JPðAPÞ Þ, and J PðAPÞ must be barriers JPðAPÞ ¼ 2JPðAPÞ 1 2 1 2 1,s , J 2,s found separately, according Eq. (2), using the transmission coeffi-

cient DsPðAPÞ for the SMTJ. The analytical equation for the TTC was taken from [11] and adjusted by adding the parameter e while considering the values of the effective masses for the insulators mI (MI  mI/m0) and for the magnetic layers separately. e is a parameter, which considers tunnel leakages, suggesting that tunnel leakages exist independently of the applied voltage. The TTC was taken as follows: Ds ¼

L R 4k?,maj k?,min ðT= Þ2 L R L R 2 ðk?,s bk?,s Þ þ ðk?,maj k?,min

p

g

a þ wÞ2

ð6Þ

Eq. (6) is a unified formula for the TTC and considers all cases: min -maj and maj - min for P and AP alignments; L

L

R

k?,maj ¼ maxfkF,s ,kF,s ðVÞg  cosðyl,s Þ, R k?,min

¼

L R minfkF,s ,kF,s ðVÞg

 cosðyR,s Þ;

a ¼ Aiðq1 ÞBiðq2 ÞBiðq1 ÞAiðq2 Þ, b ¼ TfAiðq1 ÞBi0 ðq2 ÞBiðq1 ÞAi0 ðq2 Þg,

g ¼ TfAi0 ðq1 ÞBiðq2 ÞBi0 ðq1 ÞAiðq2 Þg, w ¼ T 2 fAi0 ðq1 ÞBi0 ðq2 ÞBi0 ðq1 ÞAi0 ðq2 Þg, are the linear combinations of the Airy functions and their R

R

derivatives. Note when k?,s ¼ ks ðVÞcosðyR,s Þ, cosðyR,s Þ can be repreqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L R sented as cosðyR,s Þ ¼ 1ðkF,s =ks ðVÞÞ2 ð1x2s Þ; T ¼ ½c  V=L1=3 ; L

2

L

2

L

2

q1,s ¼ LTf½kF,s ðkF,m þ kF,k Þ=2cB uB =ðc  V þ eÞ, q2,s ¼ q1,s þLT, where uB ¼ k2B =cM I is the potential barrier height with kB being the complex k-wave vector inside the barrier. When V-0 and e ¼0 then T-0 and q1, q2-N. This case has a non-stable solution for the current in the bias region close to zero. Therefore we chose

e C1:0  104 . SMTJ-2

SMTJ-1

k FL ↓ ,↑

J 2,P(AP) ↑

J 1,P(AP) ↑ R1

P(AP)

FM

L

I(L)

k FL ↓,↑

FM

C

R2

I(L) FM

k FM↓ ,↑

k FR↓,↑

J ↑P(AP)

J 2,P(AP) ↓

J 1,↓

3. Results and Discussions

DMTJ

k FR↓ ,↑

k FR↓( L,↑)

P(AP)

J↓ R

FM

L

I1(L1) FM M I 2 (L2 ) FM R

Fig. 1. Schematic view for two tunneling models of DMTJs. (a) Consecutive tunneling model where FMC is a common magnetic layer FMC  FMR(L). (b) Coherent tunneling model.

3.1. Consecutive tunneling in DMTJs. Fig. 2 shows the non-normalized TMR–Va curves for Ta ¼ 300 1C and uB ¼5.2 eV (curve 1) as well as for Ta ¼350 1C and uB ¼5.1 eV (curve 2). The numerical results with the highest TMR ratio of around 120% reproduce the values of the experimental data measured by Feng et al. The inset shows the comparison between the numerically obtained normalized data and the experimental one. Out of the available experimental curves we considered two cases: One with strong asymmetric behavior (see TMR curve with annealing temperature Ta ¼300 1C and red squares in inset) and

2846

A. Useinov et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2844–2848

120

1

TMR (%)

100 80

2

Normalized TMR

1.00

140

0.75

Exp. curve 2, Ta = 350 oC

0.50

0.50 0.25 -2

60

-1

40

0

Va (V)

1

2

0.00

DMTJs

0 1

2

3

4

Vout (V)

0

Va (V) Fig. 2. Numerical result of TMR as a function of the applied voltage in a DMTJ. The dashed extension of the TMR curves show the behavior at higher voltages. In real systems, a voltage 4 2 V may cause damage. For both cases: m¼ mI ¼ m0, ˚ The inset shows normalized experimental data of Feng et al. [6] L1(2) ¼25 A. compared to our numerical data (black solid and dash-dot lines). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 1 Values used for the numerical results in Fig. 2. (A˚  1)

L kFk

(A˚  1)

0.75

1

0.50

2

0.25 0.00 CoFe/CoFeB DMTJ

(A˚  1)

R kFk

CoFeB DMTJ

(A˚  1)

0.0 1.04 1.009

0.725 0.73

1.009 1.0

0.73 0.68

Curve2 SMTJ-1 SMTJ-2

0.95 0.99

0.7045 0.7077

0.99 0.97

0.7077 0.6463

-1.5 -1.0 -0.5 0.0

0.5

1.0

1.5

Va (V)

one that represents the most symmetrical case (obtained at Ta ¼350 1C, blue rhombs). The data for the DMTJs represents the case of consecutive non-resonant tunneling where numerical values of the wave vectors describing these curves are collected in Table 1, (Va is positive in direction from SMTJ-1 to SMTJ-2). Here, the initial values of the wave vectors for the left side of SMTJ-2 are equal to the initial values of the wave vectors on the right side of SMTJ-1 due to the common FMC layer. LðRÞ The curves are most sensitive to initial values of kF,maj and 1 ˚ even a difference of 70.001A results in considerable differences (a few percents of TMR change). According to our model, the AVB of the TMR has its origin mostly in the disparity between ‘‘spin-down-Fermi-level’’ and ‘‘spin-up-Fermi-level’’ (characterized by kF,min and kF,maj, respecLðRÞ

tively), which, as a consequence, causes also different kF,min = LðRÞ

kF,maj a 1 ratios for each magnetic layer (note that in case LðRÞ

kF,min =kF,maj ¼ 1 the magnetic material becomes nonmagnetic and the exchange splitting [(kF,maj)2  (kF,min)2]/cM of the conduction spin-sub-bands disappeared). In Fig. 3, the values of the output voltages of MTJs obtained experimentally are compared with the numerical results of our model. The output voltage is defined as: V out ¼ VðJP JAP Þ=J AP

CoFeB SMTJ

1.0

0.5 R kFm

Curve1 SMTJ-1 SMTJ-2

LðRÞ

1

SMTJs

0.25

2

20

L kFm

Exp. curve 1, Ta = 300 oC

0.75

ð7Þ

The dependence of Vout(V) on Va of the SMTJs and DMTJ are shown in Fig. 3a (solid red and dash-dot green lines) and Fig. 3b (blue squares, Ta ¼300 1C) respectively. Both cases show asymmetric behavior with respect to the applied voltage. An example of symmetrical Vout(V) behavior was found by Gan et al. [7] (see Fig. 3c). The experimental data was obtained with different kinds of free magnetic layers at Ta ¼350 1C (see black

Fig. 3. Numerical results of output voltage for SMTJs and DMTJs. The curves were derived based on the initial parameters shown in Table 1. Experimental results from Feng et al. [6] are collected in (a), (b), curves 1, 2 and Gan’s data [7] with different free layers at (c). Numerical data are shown as dashed or solid lines and experimental data are shown as scattered symbols. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

circles and red squares, which correspond to Co40Fe40B20 and Co50Fe50/Co40Fe40B20 as a middle layers, respectively). The higher output voltage arises for a Co50Fe50/Co40Fe40B20 layer, which, however, causes small branch asymmetry as well. Considering this case the analytical model of Vout(V) for DMTJs shows symmetric behavior because of mutual asymmetry compensation of the SMTJs’ output voltage branches. This can be demonstrated using our numerical results, which correspond to Feng’s data with Ta ¼350 1C (see Fig. 3a, curves for SMTJ-1, 2: dashed orange and violet lines; and Fig. 3b, black dashdot line for DMTJ). Numerical and experimental data (curve 2, Fig. 3a) for the SMTJ shows the maximum peak of Vout for positive voltage. However, Vout(V) in case of the DMTJ (black circles, Fig. 3b) max shows almost symmetric behavior V max out ðV a Þ ffi V out ð þV a Þ. As a result DMTJ’s output voltage may have symmetrical behavior while the corresponding SMTJs may not. The effect achieved due to a non-equal (in relation to FML and FMR) conducL

L

tion band spin asymmetry for the FMM layer: kF,min = kF,maj ¼ R R C C kF,min =kF,maj a kF,min =kF,maj ,

L kF,maj

R ¼ kF,maj

C kF

RðLÞ

assuming (here ¼ kF means k-vectors for right side of the SMTJ-1, see Fig. 1a) the system is symmetric with respect to Fermi-level splitting in the common layer but not symmetric in general. This means that a specific output voltage characteristic of a DMTJ could be achieved by tailoring the fabrication of the layers, i.e. after fabricating and measuring the first SMTJ, the parameters for the following layers are determined and implemented. Fig. 4 shows the I–V characteristic for the parameters of curve 1 in Fig. 2. Other I–V curves have almost the same pattern for this

A. Useinov et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2844–2848 10

1.00

1

15

Normalized TMR

Current Density (μA/m2)

0.1

10

0.01 1E-3

5

1E-4

-4

-2

0

2

4

0 P

J AP J

-5 -10 -15

Curve 1 Curve 2

0.75 0.50 0.25 0.00 -1.0

-4

2847

-2

0

Va (V)

2

-0.5

0.0

0.5

1.0

Va (V)

4

Fig. 5. Numerical results of possible asymmetric voltage behavior of TMR for a DMTJ in coherent mode.

Fig. 4. Numerical results of the I–V curve for the DMTJ, which is a series connection of two SMTJs. The inset shows the logarithmic scale of the same plot.

3.2. Coherent tunneling in DMTJ The results of the previous chapter include experimental data, which were found by Feng et al. [6] and Gan et al. [7] for DMTJs. The thickness of the middle magnetic layer in those works is about 30 A˚ or higher, which is too thick for conduction electrons to tunnel through all two barriers at once, and the model of coherent tunneling cannot be applied. Numerical [11] and experimental estimations [9,13] predict that the coherent tunneling model in DMTJs with ferromagnetic metal layers can be applied ˚ For LM ¼ 13–15 A˚ DMTJs have to be when LM is less than 13 A. described in terms of both CHT and CST models coexisting together. The shape of the TMR–Va curve in this transitional case is very sensitive to most model’s parameters and different experimental and numerical results have been reported [9,11,13]. Moreover CHT may have resonant as well as nonresonant behaviors. Consider the case of non-resonant coherent tunneling through ˚ The current densities J sPðAPÞ will be the DMTJ assuming LM  10 A. derived directly from Eq. (2) with the TTC DPðAPÞ corresponding s to the entire double barrier system [11]. Moreover, Eq. (4) has to M

be extended with the following additional conditions: k?,s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M M ks ðVÞcosðyM,s Þ, ks ðVÞ ¼ ðkF,s Þ2 þ cMM V , (MM  mM/m0). The lower limit XCR in Eq. (2) also need to be changed: XCR ¼0 when the electrons tunnel from the min into the maj conduction sub-band and, in the opposite case, X CR ¼ cosðymin Þ, where ymin ¼ minfy1 , y2 g, with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   L M y1 ¼ arc cos 1ðkF,maj =kmin Þ2  , rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   L R 1ðkF,maj =kmin Þ2  ,

y2 ¼ arc cos

Finally, Eq. (3) has to be extended too: L k99,s

L

M

R

¼ kF,s sinðyL,s Þ ¼ ks ðVÞsinðyM,s Þ ¼ ks ðVÞsinðyR,s Þ

Fig. 5 shows the normalized curves of TMR vs. Va in range of the CHT model for the DMTJ with the following initial parameters,

1.00

1.0

0.75 0.50

0.5

0.25 0.00

Output Voltage (V)

Normalized Output Voltage

type of junctions including I–V for SMTJs. It is worth to note there is high sensitivity of the current density to the barrier thickness and applied voltage. Due to the series connection of two STMJs with high barrier thicknesses of 25 A˚ the current density is as low as a few nA/m2 for Va ¼0.5 V, while in case of a single ˚ the current density is SMTJ with the same voltage and L¼12 A, about 1.0 nA/mm2.

0.0 -1.0

-0.5

0.0

0.5

1.0

V a (V) Fig. 6. Numerical results of normalized (and non-normalized) output voltage curves for a DMTJ, where electrons tunnel through the system in a coherent mode.

˚ kLFm ¼ 1:071 A˚ curve 1: L1(2) ¼18 A, 1

1

1

L , kFk ¼ 0:51 A˚

1

1

M , kFm ¼ 1:12 A˚

,

1

M R R kFk ¼ 0:55 A˚ , kFm ¼ 1:081 A˚ , kFk ¼ 0:49 A˚ with uB ¼4.95 eV (uB ¼EF þVB, where EF ¼3.75 eV is Fermi energy, VB ¼1.2 eV is the barrier height above Fermi level), an initial barrier’s slopes was

˚ kLFm ¼ 1:018 A˚ assumed to be 0.015 eV. Curve 2: L1(2) ¼ 28 A, L kFk L

¼ 0:9981 A˚

1

,

M kFm

¼ 1:0 A˚

1

,

M kFk

¼ 0:99 A˚

1

,

R kFm

1 1

¼ 1:02 A˚

, ,

1

kFk ¼ 0:9964 A˚ with uB ¼4.76 eV and an initial barrier’s slope assumed to be 0.1 eV. Common parameters for both curves were: ˚ Some parameters were MR(L) ¼MM ¼0.85, MI ¼ 0.38 and LM ¼10 A. taken initially different in order to observe asymmetric TMR behavior (here we do not consider asymmetric behavior within a case of non-equal barriers). Curve 1 in Fig. 5 represents the case with high splitting of the Fermi surface within each magnetic layer/surface. In this case there is a large difference between JP and JAP which can take on values of eight orders or above yielding giant TMR amplitudes. With respect to practical applications, due to different conductance states for P and AP cases, the DMTJ with coherent tunneling may serve as magnetic diode or switch, which has been suggested by results of another papers [14]. However, the presented model disregards possible leakages like spin-flip electron processes, charging effects, disorder effects at metal/ insulator interfaces, zero-bias anomaly, etc [15–18]. It was found, for example, that factors like spin-flip processes, roughness or disorder [16–18] can diminish the difference between the JP and JAP values. Due to large difference of the current densities we also

2848

A. Useinov et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2844–2848

normalized the output voltage V Norm ¼ V out ð 7 V a Þ=V max out out ð þV a Þ, see red solid line in Fig. 6. Curve 2 in Fig. 5 represents the case of low Fermi surface splitting in all interfaces. The maximal TMR value in this case of 417% was derived at negative voltages. The blue dashed line depicted in Fig. 6, represents the output voltage without normalization. In both cases the location of the V max out ð 7V a Þ is shifted towards low voltages and the curves have a more complicated shape compared with the CST model (Fig. 3). In the CHT model Vout(Va) and the dependency of TMR on Va M are more sensitive to kF,min/kF,maj, kF , LM as well as the heights and thicknesses of the barriers than in the CST model.

4. Conclusion In this paper we study the typical behaviors of TMR–Va, I-V and asymmetric output voltage dependencies of the single and double barrier magnetic tunnel junctions based on two simple models of coherent and consecutive tunneling. The electrical current is proportional to the tunnel transmission coefficient. In this work, mainly asymmetric output voltage behaviors were explored for two types of DMTJs. For the first type of the DMTJ, when the system is represented by two SMTJs connected in series, we found that, in order to provide symmetric output voltage branches, the conditions of equal kF,min/kF,maj ratios have to be fulfilled for left and right magnetic layers while it can be different for the common FM layer. LðRÞ LðRÞ C C Only in the case of kF,min =kF,maj ¼ kF,min =kF,maj the output voltage of each individual SMTJ and the entire system will be symmetric. We found that DMTJs, in range of the coherent tunneling model, will have different output voltage characteristics compared to the model of consecutive tunneling, where the peaks will be shifted into the region of low voltages. Moreover within the coherent tunneling model two cases of high and low Fermi level splitting were considered, where high splitting gives giant difference of conductive states between P and AP configurations, which could be used for applications in magnetic electronics. References [1] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. Molna´r, M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Spintronics: a spin-based electronics vision for the future, Science 294 (5546) (2001) 1488–1495.

[2] Y. Huai, Spin-transfer torque MRAM (STTMRAM): challenges and prospects, AAPPS Bulletin 18 (6) (2008) 33. [3] K. Inomata, Y. Saito, K. Nakajima, M. Sagoi, Double tunnel junctions for magnetic random access memory devices, Journal of Applied Physics 87 (2000) 6064. [4] S. Colis, G. Gieres, L. Bar, J Wecker, Low tunnel magnetoresistance dependence versus bias voltage in double barrier magnetic tunnel junction, Applied Physics Letters 83 (2003) 948. [5] A. Useinov, J. Kosel, Spin asymmetry calculations of the TMR–V curves in single & double-barrier magnetic tunnel junctions, IEEE Transactions on Magnetics 47 (10) (2011) 2724–2727. [6] G. Feng, S. van Dijken, J.F. Feng, J.M.D. Coey, T. Leo, D.J. Smith, Annealing of CoFeB/MgO based single and double barrier magnetic tunnel junctions: tunnel magnetoresistance, bias dependence, and output voltage, Journal of Applied Physics 105 (2009) 033916. [7] H.D. Gan, S. Ikeda, W. Shiga, J. Hayakawa, K. Miura, H. Yamamoto, H. Hasegawa, F. Matsukura, T. Ohkubo, K. Hono, H. Ohno, Tunnel magnetoresistance properties and film structures of double MgO barrier magnetic tunnel junctions, Applied Physics Letters 96 (2010) 192507. [8] J. Shin, B. Min, Jin Pyo Hong, Kyung-Ho Shin, Effects of Ru diffusion in exchange-biased MgO magnetic tunnel junctions prepared by in situ annealing, Applied Physics Letters 95 (2009) 222501. [9] T. Nozaki, A. Hirohata, N. Tezuka, S. Sugimoto, K. Inomata, Bias voltage effect on tunnel magnetoresistance in fully epitaxial MgO double-barrier magnetic tunnel junctions, Applied Physics Letters 86 (2005) 082501. [10] S. Ikeda, J. Hayakawa, Y.M. Lee, R. Sasaki, T. Meguro, F. Matsukura, H. Ohno, Dependence of tunnel magnetoresistance in MgO based magnetic tunnel junctions on Ar pressure during MgO sputtering, Japanese Journal of Applied Physics 44 (2005) L1442. [11] A.N. Useinov, N.Kh. Useinov, L.R. Tagirov, J. Kosel, Resonant tunnel magnetoresistance in double barrier planar magnetic tunnel junction, Physical Review B 84 (2011) 085424. [12] A. Useinov, R. Deminov, N. Useinov, L. Tagirov, Tunneling magneto-resistance in ferromagnetic planar hetero-nanojunctions, Physica Status Solidi B 247 (2010) 1797–1801. [13] L. Jiang, H. Naganuma, M. Oogane, Y. Ando, Large tunnel magnetoresistance of 1056% at room temperature in MgO based double barrier magnetic tunnel junction, Applied Physics Express 2 (2009) 083002. [14] M. Chshiev, D. Stoeffler, A. Vedyayev, K. Ounadjela, Magnetic diode effect in double barrier tunnel junctions, Europhysics Letters 58 (2001) 10. [15] J. Bernos, M. Hehn, F. Montaigne, C. Tiusan, D. Lacour, M. Alnot, B. Negulescu, G. Lengaigne, E. Snoeck, F.G. Aliev, Impact of electron-electron interactions induced by disorder at interfaces on spin-dependent tunneling in Co–Fe–B/ MgO/Co–Fe–B magnetic tunnel junctions, Physical Review B 82 (2010) 060405. (R). [16] J. Mathon, A. Umerski, Theory of tunneling magnetoresistance in a disordered Fe/MgO/Fe(001) junction, Physical Review B 74 (2006) 140404. (R). [17] G. Tkachov, K. Richter, Spin-polarized tunneling through randomly transparent magnetic junctions: reentrant magnetoresistance approaching the Jullie re limit, Physical Review B 77 (2008) 184427. [18] P. Xu, V. Karpan, K. Xia, M. Zwierzycki, I. Marushchenko, P. Kelly, Influence of roughness and disorder on tunneling magnetoresistance, Physical Review B 73 (2006) 180402. (R).