Quantum size effects on spin-transfer torque in a double barrier magnetic tunnel junction with a nonmagnetic-metal (semiconductor) spacer

Journal of Magnetism and Magnetic Materials 410 (2016) 18–22

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Quantum size effects on spin-transfer torque in a double barrier magnetic tunnel junction with a nonmagnetic-metal (semiconductor) spacer Reza Daqiq, Nader Ghobadin Department of Physics, Malayer University, Malayer, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 9 January 2016 Received in revised form 23 February 2016 Accepted 4 March 2016 Available online 5 March 2016

We study the quantum size effects of an MgO-based double barrier magnetic tunnel junction with a nonmagnetic-metal (DBMTJ-NM) (semiconductor (DBMTJ-SC)) spacer on the charge current and the spin-transfer torque (STT) components using non-equilibrium Green's function (NEGF) formalism. The results show oscillatory behavior due to the resonant tunneling effect depending on the structure parameters. We find that the charge current and the STT components in the DBMTJ-SC demonstrate the magnitude enhancement in comparison with the DBMTJ-NM. The bias dependence of the STT components in a DBMTJ-NM shows different behavior in comparison with spin valves and conventional MTJs. Therefore, by choosing a specific SC spacer with suitable thickness in a DBMTJ the charge current and the STT components significantly increase so that one can design a device with high STT and faster magnetization switching. & 2016 Elsevier B.V. All rights reserved.

Keywords: Quantum size effects DBMTJ Charge current STT components

1. Introduction A spin-polarized current can induce magnetization switching in a magnetic multilayer. This spin transfer torque (STT) effect has been investigated both theoretically [1,2] and subsequent experimentally [3,4]. The magnetization switching has been observed in magnetic tunnel junction (MTJ) based on AlO [5] and MgO as the insulator [6,7]. The critical current densities for switching a typical tri-layer MTJ as CoFeB/MgO/CoFeB are usually of the order 107 A/cm2 [8]. The STT effect has attracted lots of interest in recent years due to its potential applications in spintronic devices for instance STT- MRAM [9], spin-torque nano-oscillators, spin-torque microwave nano-detectors and spin-torque memristor [10]. Double barrier MTJs (DBMTJs) are significant in nano-electronics due to the formation of resonant states inside the middle ferro-magnet (FM) layer in FM/I/FM/I/FM structure [11–14] where I symbolize the insulator. For specific thickness of the middle FM layer the resonant tunneling effect has been shown in a DBMTJ [14]. For enhancement of the STT switching, resonant tunneling DBMTJs based on MgO-semiconductor hetero-structures has also been proposed [15]. The tunnel magneto-resistance (TMR) effect has been studied theoretically in Fe/MgO/Fe/MgO/Fe DBMTJ [16] and subsequent experimentally in CoFeB/MgO/CoFeB/MgO /CoFeB DBMTJ [17]. n

Corresponding author.

http://dx.doi.org/10.1016/j.jmmm.2016.03.016 0304-8853/& 2016 Elsevier B.V. All rights reserved.

Also, the TMR has been investigated in a structure including a nonmagnetic-metal spacer [18]. Confinement of an electron inside the quantum well between two barriers causes to oscillation in transport quantities due to the resonant tunneling effect. Quantum effects only go in through the explanation of the basis states resulting from the confinement. The object of this work is to study the quantum size effects of the DBMTJs on the charge current and the STT components. Our device is based on a nonmagnetic-metal (NM) (semiconductor (SC)) spacer as the quantum well between two MgO barriers (insulators) and enclosed by two semi-infinite ferromagnetic CoFeB contacts (Fig. 1). This paper is organized as follows. Section 2, describes the model and formalism for charge current and STT components through CoFeB/ MgO/NM(SC)/MgO/CoFeB DBMTJ. Results and discussion are presented in Section 3. Finally, we conclude our findings in Section4.

2. Model and formalism We use the non-equilibrium Green's function (NEGF) formalism [11,19–22] with a single band effective mass Hamiltonian and self-energy ΣL, R so that Hamiltonian is given by:

(

)

^ ∆ H =H0 I + σ ⃗. m

(1)

^ )∆ is the The first term is the spin-independent part and (σ ⃗. m spin-dependent part of the Hamiltonian where the direction of the

R. Daqiq, N. Ghobadi / Journal of Magnetism and Magnetic Materials 410 (2016) 18–22

19

In the Eqs. (4–9), G (E ) is the Green's function of the device at energy E . The spectral function is A so that its diagonal elements give us the local density of state (LDOS). ΓL, R, Σin L, R and fL, R are the broadening matrix, the in-scattering function and the Fermi function of the left and the right FM contacts respectively. The inscattering function is Σin L, R , describing the rate at which electrons come into the device from the contacts. The current operator between two lattice points' j and j ± 1 is defined as:

Iop=

i Hj, j ±1G nj ±1, j −G nj, †j ±1H †j ±1, j ℏ

(

)

(10)

In this regard, the charge currents IC and the spin currents IS⃗ are calculated by: Fig. 1. Schematic illustration of FM/I/NM(SC)/I/FM junction consisting of two semiinfinite FM contacts surrounding the middle region of the left and right insulators and the NM (SC). tI and tNM (SC ) denote the thickness in the left (right) I and middle NM (SC) spacer, respectively. The magnetization of the fixed (left) FM, ML is along the z direction and that of the free (right) FM, MR is rotated by angle θ = π /2 around the y axis relating to ML . Difference in the bottom of the conduction band of the NM (SC) and FM is UW .

^ , I is the 2  2 identity matrix, ∆ is the spin-splitting magnet is m and σ⃗ is the Pauli spin vector. The 2  2 self-energy matrices ΣL, R of the left and the right FM contacts are given by:

⎤ ⎡ − t exp (ik↑ a) 0 FM j ⎥ Σj ( i, i, k∥ )=⎢ ⎢ 0 − tFM exp (ik ↓j a)⎥⎦ ⎣

(2)

The chemical potential of the right (μR ) and the left (μL ) FM contacts are defined as:

qV =μL −μR

(3)

In the above equations a is the lattice spacing, k↑j , ↓ are the wave vectors of the spin up and the spin down electrons [19], q is the charge of an electron, V is the applied voltage between two FM contacts and k∥ is the transverse wave vector. In the Eq. (2), ⋆ ⋆ 2 is the effectFM =ℏ2/2mFM a is the coupling parameter where mFM tive mass of the electron inside the FM contacts and ℏ is the reduced Planck constant. In this model, the periodic boundary conditions are assumed in the transverse direction so that transverse modes are decoupled similar to individual 1-D wires. We will talk about DBMTJ properties in the coherent regime. The correlation function G n is calculated from the NEGF equations [21] (Note that the electron density is the diagonal elements of the correlation function):

G (E )=[EI − H−Σ]−1

(4)

A (E )=i ⎡⎣ G−G†⎤⎦

(5)

ΓL, R (E )=i ⎡⎣ ΣL, R −Σ†L, R ⎤⎦

(6)

Σin L, R (E )=ΓL, R (E ) fL, R (E )

(7)

G n (E )=GΣinG†

(8)

where in Σ = ΣL +ΣR , Σin=Σin L +ΣR

(9)

IC =q

∫ dEReal ⎡⎣ Trace ( Iop ) ⎤⎦

(11)

IS⃗ =q

∫ dEReal ⎡⎣ Trace { σI⃗ op } ⎤⎦

(12)

The currents are summed over the first 50 transverse modes [8]. Finally, the in-plane and perpendicular components of the STT are defined as τ∥=Is, z and τ⊥=Is, y , respectively.

3. Results and discussions We assume that the fixed (left) magnet M⃗ L is along z^ and the free (right) magnet M⃗ R is along x^ direction (i.e. θ =π /2 in Fig. 1). The parameters used here for the FM contacts are the Fermi energy EF =2.25 eV , the spin-splitting ∆=2.15 eV , the barrier height of the insulator UI =0.77 eV and the effective mass for electrons inside the * =0.8me [22]. FM contacts mFM The parameters used here for the barriers (insulators), the NM layer and the SC layer are the effective mass for electrons inside the insulator mI*=0.18me [22], the effective mass for electrons in* =me , the well height of the NM, UW =0eV , the side the NM layer mNM effective mass for electrons inside the SC (e.g. ZnO) layer * =0.29me [23,15], ( me is the mass of the free electron) and the mSC well height of the SC, UW =−0.23 eV [24,15], respectively. The thickness of the insulator is tI =0.5 nm in the two types of the DBMTJs. The junction area is 70×160 nm2 [25]. A constant bias voltage of V = 0.5 V is applied in Figs. 2, 3.The potential drops linearly in the barriers (insulators) and the middle SC layer while in the middle NM layer the potential is constant. Fig. 2(a) shows the charge current for a DBMTJ with the NM spacer (DBMTJ-NM) as a function of the thickness tNM . With increasing tNM the charge current oscillates and some peaks appear. At the same time, the magnitude of these peaks decreases as their width gets broader. The STT components as a function of the thickness tNM of the NM layer also illustrate oscillatory behavior as shown in Fig. 2(b), (c). The magnitude of the peaks decreases as their width gets broader with increasing tNM . The origin of the oscillatory behavior is explained as follows. There are several quasi-bound states inside the quantum well between two confining barriers. When the energy of the electrons matches with the energy of a quasi-bound state a resonant condition fulfills. Consequently, the transmission of the electrons through the DBMTJ-NM strongly increases and a peak appears. Conversely, the position of the quantum well states depends on the well thickness tNM . As a result, the position of the resonant peaks varies with variation in tNM and the dips between the peaks emerge. These dips have formed due to the gaps between the discrete resonant levels in the quantum well. Therefore, the charge current and the STT components show oscillatory behavior. This oscillatory behavior decreases gradually for increasing well thicknesses duo to decreasing the gaps between the well levels.

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IC (mA)

3 2 (a) 1 0

||

(mA)

3 2 (b) 1

(mA)

0 0 -1 (c) -2 0.1

0.5

0.9

1.3

1.7

2.1

tNM (nm) Fig. 2. The charge current (a) and the STT components (b), (c) for a DBMTJ-NM as a function of the thickness of the NM layer with tI =0.5 nm, V = 0.5 V and UW =0eV .

IC (mA)

15 10

(a)

5 0

||

(mA)

15 10

(b)

5 0

(mA)

0 -3

(c)

-6 -9 0.1

0.5

0.9

1.3

1.7

2.1

tSC (nm) Fig. 3. The charge current (a) and the STT components (b), (c) for a DBMTJ-SC as a function of the thickness of the SC layer with tI =0.5 nm, V = 0.5 V and UW =0.23 eV .

The charge current and the STT components as a function of the thickness tSC for a DBMTJ with the SC spacer (DBMTJ-SC) are shown in Fig. 3(a)–(c). Both the charge current and the STT components oscillate with increasing tSC similar to Fig. 2. Nevertheless,

the length period of oscillations for a DBMTJ-NM is shorter than a DBMTJ-SC so that the number of the resonant peaks in the former is greater than that in the latter. Also, the magnitude of the charge current and the STT components in the DBMTJ-SC has considerably

IC (mA)

5 (a) 0

(b) 0

||

(mA)

-5 5

(mA)

-5 2

tNM= 0.7 nm

tNM= 1.0 nm

tNM= 1.8 nm

(c)

0

-2 -1

-0.5

0

0.5

1

Voltage (V) Fig. 4. Bias dependence of the charge current (a) and the STT components (b), (c) for three different thicknesses (tNM =0.7 nm, 1.0 nm and 1.8 nm ) of the NM layer with tI =0.5 nm and UW =0 eV .

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21

IC (mA)

40 20

(a)

0 -20

||

(mA)

-40 40 20 0 -20 -40 10

(mA)

(b)

5

tSC= 0.6 nm

tSC= 1.0 nm

tSC= 1.8 nm

(c)

0 -5 -10 -1

-0.5

0

0.5

1

Voltage (V) Fig. 5. Bias dependence of the charge current (a) and the STT components (b), (c) for three different thicknesses (tSC =0.6 nm, 1.0 nm and 1.8 nm ) of the SC layer with tI =0.5 nm and UW =0.23 eV .

enhanced in comparison with the DBMTJ-NM for the different thicknesses of the middle spacer. For instance, the in-plane component of the STT for the DBMTJ-SC with tSC =0.6 nm is about 15 mA at the first resonant peak, which is about 6 times the value for the DBMTJ-NM with tNM =0.3 nm . The quantum well depth in the SC layer is lower than that in the NM layer due to the conduction band offset UW . Consequently, the SC layer has lesser resonant levels and the number of the resonant peaks decreases. The transmission through the DBMTJ-SC increases because of the shorter barrier height. Moreover, the effective mass in the middle SC layer is different from that in the middle NM layer. Fig. 4(a)–(c) shows the bias dependence of the charge current and the STT components for three different thicknesses (tNM =0.7 nm, 1.0 nm and 1.8 nm ) of the NM layer in a DBMTJ-NM. These thicknesses correspond to the positions of large and small STT in the Fig. 2. For the positive and the negative bias, the magnitude of the charge current for the thickness tNM =0.7 nm is larger than the one for the other thicknesses. For positive bias, the STT components for the thickness tNM =0.7 nm are also larger than those for the other thicknesses. For negative bias, the STT components are not as large as those at positive bias for DBMTJ-NM. The behavior of the STT components differs from that of spin valves in which the bias dependence of the interlayer exchange coupling shown oscillatory behavior for increasing spacer thicknesses. For positive bias, the STT components show a gradually decreasing behavior without any oscillation in contrast the oscillatory behavior of the interlayer exchange coupling observed in spin valves [26]. Moreover, the bias dependence of the STT components in a DBMTJ-NM differs from that of a conventional symmetric MTJ as reported in previous publications [25,27] due to the existence of the quantum well states in a DBMTJ-NM structure. For instance, the perpendicular component had shown quadratic behavior in a symmetric MTJ while here it shows gradually decreasing behavior. Furthermore we study the bias dependence of the charge current and the STT components for three different thicknesses (tSC =0.6 nm, 1.0 nm and 1.8 nm ) of the SC layer in a DBMTJ-SC, as shown in Fig. 5. The magnitude of the charge current and the STT components for the thickness tSC =0.6 nm are larger than those for the other thicknesses and they have enhanced in comparison with the DBMTJ-NM. Due to difference in the positions of the resonant peaks for the two types of the DBMTJs we have chosen the thickness tSC =0.6 nm for a resonant position of the SC layer in contrast the thickness tNM =0.7 nm in the NM layer (Fig. 4).

4. Conclusions The quantum size effects on the charge current and the STT components are studied in CoFeB/MgO/NM(SC)/MgO/CoFeB DBMTJ using the NEGF formalism. Both the charge current and the STT components have oscillated with increasing tNM (SC ) due to the resonant tunneling effect throughout the system. The results show that the magnitude of the charge current and the STT components in the DBMTJ-SC is larger than the DBMTJ-NM. Therefore, by choose a specific SC spacer (e.g. ZnO) as a replacement for the NM spacer and right thickness for the spacer in a DBMTJ the charge current and the STT components have increased. The bias dependence of the STT components shows different behavior in comparison with spin valves and conventional MTJs due to the existence of quantum well states in a DBMTJ-NM. Finally, the results may lead to design the high STT devices with faster magnetization switching.

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