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Theoretical study on temperature-bias phase diagram of MgO-based magnetic tunnel junctions

Lü Jie Fang He-Nan Lü Tao-Tao Sun Xing-Yu

Lü Jie, Fang He-Nan, Lü Tao-Tao, Sun Xing-Yu. Theoretical study on temperature-bias phase diagram of MgO-based magnetic tunnel junctions. Acta Phys. Sin., 2021, 70(10): 107302. doi: 10.7498/aps.70.20201905
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Theoretical study on temperature-bias phase diagram of MgO-based magnetic tunnel junctions

Lü Jie, Fang He-Nan, Lü Tao-Tao, Sun Xing-Yu
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  • MgO-based magnetic tunnel junction is a hot issue in the field of spin electronic devices, and its temperature and bias voltage play quite an important role in practical applications. Therefore, it is desiderated to obtain the temperature-bias phase diagram of MgO-based magnetic tunnel junction. This paper develops a theory which is suitable for magnetic tunnel junctions with single crystal barrier. In this theory, the single crystal barrier is regarded as a periodic grating, and the tunneling process is treated by optical diffraction theory, so the coherence of the tunneling electron can be well taken into account. Most importantly, the theory can handle both the temperature effect and bias effect of MgO-based magnetic tunnel junctions. According to the present theory, the temperature-bias phase diagram of MgO-based magnetic tunnel junctions is calculated under different half the exchange splittings, chemical potentials and periodic potentials. The theoretical results show that the extreme phase point of tunneling magnetoresistance (TMR) can move to high temperature region through regulating half the exchange splitting Δ of ferromagnetic electrode of MgO-based magnetic tunnel junction. This will be beneficial to the applications of magnetic tunnel junctions at room temperature. Moreover, the chemical potential μ can change the bias corresponding to the maximum phase point of TMR. As is well known, the chemical potential will vary with the material of ferromagnetic electrode. Therefore, if the material of ferromagnetic electrode is chosen with a proper chemical potential, we can obtain a large TMR under high bias voltage. In other words, the output voltage can be considerably increased. This will be favorable for the preparation of high power devices. In addition, it is found that the phase diagram of TMR is significantly dependent on periodic potential v( K h). As a result, the effects of temperature and bias voltage in the MgO-based magnetic tunnel junctions can be optimized by regulating half the exchange splitting Δ, chemical potential μ, and periodic potential v( K h). The present work provides a solid theoretical foundation for the applications of MgO-based magnetic tunnel junctions.
      Corresponding author: Fang He-Nan, fanghn@njupt.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11704197)

    在上世纪90年代, 磁性隧道结的势垒层大多选取非晶Al2O3来制备. 然而, 由于非晶Al2O3势垒层中存在严重的无序散射, 所以很大程度上抑制了磁性隧道结隧穿磁阻(tunneling magnetoresistance, TMR) 的大小. 尽管很多课题组对Al—O结的物理结构等因素进行了优化, 使得Al—O结低温下的TMR可以达到107%[1], 但是仍然难以满足下一代磁存储器件的要求. 2001年, Butler和Mathon等利用第一性原理对基于单晶MgO势垒层和Fe电极的磁性隧道结进行了理论计算, 预测其可能实现超过1000%的TMR. 随后, 人们对单晶MgO基磁性隧道结展开了广泛的实验研究. 2004年, Yuasa等[2]在室温的Fe/MgO/Fe磁性隧道结中实现了180%的TMR. 2007年Lee等[3]在CoFeB/MgO/CoFeB磁性隧道结中将TMR在室温下提升至604%, 低温下提升至1144%. 上述结果表明单晶MgO基磁性隧道结有潜力成为下一代磁存储器件的基本元件.

    单晶MgO基磁性隧道结除了具有较高的TMR外, 还展现了许多新奇的物理效应, 其中最重要的是偏压特性与温度特性. MgO基磁性隧道结温度特性的实验结果大多表现为: 反平行电导(antiparallel conductance, GAP)和TMR均随温度升高而显著降低, 而平行电导(parallel conductance, GP)随温度单调变化且幅度较小[4-10]. 另外, 也有实验发现平行电导GP会随温度振荡[11]. MgO基磁性隧道结偏压特性的实验结果大多表现为: TMR随偏压增大单调减小[12,13]. 然而, 有部分实验发现TMR会随偏压非单调变化[14]. 此外, 研究结果表明温度特性与偏压特性是耦合在一起的, 即温度对偏压特性有影响, 偏压对温度特性也有影响[14,15]. 上述结果表明温度和偏压对MgO基磁性隧道结TMR的影响比较复杂, 因此在实际应用中需要寻找最优的温度和偏压来实现最大的TMR. 在物理上, 这相当于在以温度和偏压作为双变量的相图上找到TMR最大的(或极大的)相点. 为此, 首先需要通过理论计算得到上述相图.

    本课题组之前已经发展出一个适用于单晶势垒层磁性隧道结的理论. 该理论基于传统的光学衍射理论, 将单晶势垒层视作周期性光栅, 很好地计入了单晶势垒层对隧穿电子的散射所带来的相干性. 此理论可以较好地解释MgO基磁性隧道结的基本特性, 特别地, 上述偏压特性和温度特性的物理机理均可以在该理论框架下得到阐明. 因此, 本文拟对上述理论进行拓展来同时计入温度和偏压对MgO基磁性隧道结的影响, 进而计算得到以温度和偏压作为双变量的TMR相图.

    单晶势垒层的周期势可以写作:

    $U({{r}}) = \sum\limits_{{l_{\rm{3}}} = 0}^{n - 1} {\sum\limits_{{{{R}}_{\rm{h}}}} {v({{r}} - {{{R}}_{\rm{h}}} - {l_{\rm{3}}}{{{a}}_{\rm{3}}})} } , $

    (1)

    其中v(r)表示势垒层每个格点处的原子势; n表示势垒层总层数; Rh = l1a1+l2a2, 其中a1a2是势垒层面内初基矢量, l1l2是相应的整数; a3是层间的初基矢量, l3是其相应的整数. 定义$e_z \!=\! \dfrac{ a_1 \times a_2}{| a_1 \times a_2|}$, 即z轴的正方向为从上电极指向下电极, 如图1所示.

    Figure 1.  Diagram of MgO-based magnetic tunnel junction.

    根据文献[16,17], 温度对隧穿磁阻效应的影响来源于晶格畸变对势垒层周期势的修正, 偏压对隧穿磁阻效应的影响来源于外加势场与势垒周期势的叠加. 因此, 在物理上, 两者均是通过改变势垒层势场进而影响隧穿磁阻效应的. 考虑到上述影响后, 根据Bethe理论、双束近似方法以及文献[16-18], 自旋向上子能带到自旋向上子能带通道的透射系数如下:

    $ \begin{split} {T_{ \uparrow \uparrow }}({{k}}) =\;& \frac{1}{{8{{{k}}_z}}}\big\{ {{p}}_ + ^z{{\rm{e}}^{{\rm{i}}\left[ {{{p}}_ + ^z - {{({{p}}_ + ^z)}^*}} \right]d}} + {{p}}_{\rm{ - }}^z{{\rm{e}}^{{\rm{i}}\left[ {{{p}}_{\rm{ - }}^z - {{({{p}}_{\rm{ - }}^z)}^*}} \right]d}} \\ &+ {{q}}_ + ^z{{\rm{e}}^{{\rm{i}}\left[ {{{q}}_ + ^z - {{({{q}}_ + ^z)}^*}} \right]d}} + {{q}}_ - ^z{{\rm{e}}^{{\rm{i}}\left[ {{{q}}_ - ^z - {{({{p}}_ - ^z)}^*}} \right]d}}\quad \quad \\ &+ {{p}}_ + ^z{{\rm{e}}^{{\rm{i}}\left[ {{{p}}_ + ^z - {{({{p}}_ - ^z)}^*}} \right]d}} + {{p}}_ - ^z{{\rm{e}}^{{\rm{i}}\left[ {{{p}}_ - ^z - {{({{p}}_ + ^z)}^*}} \right]d}} \\ &- {{q}}_ + ^z{{\rm{e}}^{{\rm{i}}\left[ {{{q}}_ + ^z - {{({{q}}_ - ^z)}^*}} \right]d}} - {{q}}_ - ^z{{\rm{e}}^{{\rm{i}}\left[ {{{q}}_ - ^z - {{({{q}}_ + ^z)}^*}} \right]d}} \quad \quad \\ & + {{\rm{c}}.{\rm{c}}.} \big\},\\[-12pt] \end{split} $

    (2)

    式中, k表示入射电子的波矢量, kz是其z方向分量; d表示势垒层厚度; ${{p}}_ \pm $${{q}}_ \pm $为透射电子波函数分波的波矢, 可表示为:

    $ {{p}}_ \pm ^z = {[{{{k}}^2} - {{k}}_{\rm{h}}^{\rm{2}} \pm 2m{\hbar ^{ - 2}}v({{{K}}_{\rm{h}}}) + 2me{\hbar ^{ - 2}}{V_{\rm{0}}}]^{1/2}},\tag{3a} $

    $ \begin{split} {{q}}_ \pm ^z =\;& [{{{k}}^2} - {({{{k}}_{\rm{h}}} + {{{K}}_{\rm{h}}})^{\rm{2}}} \pm 2m{\hbar ^{ - 2}}v({{{K}}_{\rm{h}}}) \\ &+ 2me{\hbar ^{ - 2}}{V_{\rm{0}}}]^{1/2},\end{split}\tag{3b} $

    其中, khk的面内分量; KhRh相应的倒格矢; m是电子的质量, $\hbar $是普朗克常量; e是电子电荷; V0是外加偏置电压.

    $\begin{split} v({{{K}}_{\rm{h}}}) =\;& \left\{ {1 + 2\frac{\sigma }{{1 - \sigma }}\cos \left[ {{{{K}}_{\rm{h}}} \cdot {\alpha _{\rm{0}}}\left({\rm{1 - }}\frac{T}{{{T_{\rm{c}}}}}\right)} \right]} \right\}\\ &\times(1 - \sigma ){v_{\rm{0}}}({{{K}}_{\rm{h}}}),\end{split}$

    (4)

    式中$\sigma $是缺陷浓度; ${\alpha _{\rm{0}}}$为绝对零温下势垒层的应变; ${T_{\rm{c}}}$为回复温度, 它的物理含义是当温度$T = {T_{\rm{c}}}$时, 应变刚好消失; ${v_{\rm{0}}}({{{K}}_{\rm{h}}})$是理想单晶势垒层原子势的傅里叶变换,

    ${v_{\rm{0}}}({{{K}}_{\rm{h}}}) = {\varOmega ^{ - 1}}\int {{\rm{d}}{{r}}v({{r}}){{\rm{e}}^{ - {\rm{i}}{{{K}}_h} \cdot {{r}}}}} , $

    (5)

    其中$\varOmega $是单晶势垒层原胞的体积$\varOmega = ({{{a}}_{\rm{1}}} \times {{{a}}_{\rm{2}}}) \cdot {{{a}}_{\rm{3}}}$.

    ${T_{ \uparrow \uparrow }}$, 可以得到自旋向上子能带到自旋向上子能带的电导${G_{ \uparrow \uparrow }}$:

    $\begin{split} {G_{ \uparrow \uparrow }} = \;&\frac{{e\hbar }}{{16{{\rm{\pi }}^3}m{V_0}}}\int_{\sqrt {{{k}}_{{\rm{F}} \uparrow }^2 - 2me{\hbar ^{ - 2}}} {V_0}}^{{{{k}}_{{\rm{F}} \uparrow }}} {\rm{d}}{{k}}\int_0^{{\rm{\pi }}/2} {{\rm{d}}\theta } \\ & \times\int_0^{2{\rm{\pi }}} {{\rm{d}}\varphi {{{k}}^3}\sin (2\theta ){T_{ \uparrow \uparrow }}({{k}},\theta,\varphi )} ,\end{split}$

    (6)

    ${{{k}}_{{\rm{F}} \uparrow }} = \sqrt {2m{\hbar ^{ - 2}}(\mu + \varDelta )}, $

    (7)

    其中$\theta $表示kez之间的角度, $\varphi $表示kha1之间的角度, ${{{k}}_{{\rm{F}} \uparrow }}$表示自旋向上电子的费米波矢量, μΔ分别表示铁磁电极的化学势和半交换劈裂能. 类似地, 可以得到${G_{ \downarrow \uparrow }}$, ${G_{ \uparrow \downarrow }}$以及${G_{ \downarrow \downarrow }}$, 进而可以得到平行电导${G_{\rm{P}}} = {G_{ \uparrow \uparrow }} + {G_{ \downarrow \downarrow }}$, 反平行电导${G_{{\rm{AP}}}} = $$ {G_{ \uparrow \downarrow }} + {G_{ \downarrow \uparrow }}$, 以及${\rm{TMR}} = ({G_{\rm{P}}}/{G_{{\rm{AP}}}}){\rm{ - 1}}$. 下文将上述公式应用于单晶MgO基磁性隧道结: 根据文献[19], 此时${{{K}}_{\rm{h}}} = 2.116 \times {10^{10}}\;{{\rm{m}}^{ - 1}}$; 因为在已有文献中MgO势垒层厚度范围通常为1—3 nm, 所以势垒层厚度d在本文中设定为2 nm; 晶格畸变的相关参数选取为$\sigma = {\rm{0}}{\rm{.08}}$, ${{{K}}_{\rm{h}}} \cdot {\alpha _0} = {\text{π}}/3$以及${T_{\rm{c}}} = 800\;{\rm{ K }}$.

    首先, 计算了在不同的铁磁电极半交换劈裂能Δ下的温度-偏压相图, 结果如图2所示. 其中, 化学势μ = 11 eV, 原子势的傅里叶变换v(Kh) = 15.3 eV. 图2(a)图2(c)分别对应于Δ = 8 eV, Δ = 9 eV和Δ = 10 eV. 从图2可以看出, TMR最大的相点均处于低偏温区域, 这与已有的MgO基磁性隧道结的实验结果相符[20]. 此外, 由图2(b)图2(c)可知, 当Δ = 9 eV和Δ = 10 eV时, TMR分别在温度为240 K和400 K时出现了与最大值接近的极大值点. 在物理上, 由公式(2)、公式(3)、公式(4)和公式(7)可知, 这来源于隧穿电导随温度的振荡效应[17]. 该结果表明可以通过调节铁磁电极的半交换劈裂能使TMR的极大值处于室温区域, 这将有利于磁性隧道结在室温下的实际应用.

    Figure 2.  Phase diagram of temperature and bias with variation of half the exchange splitting of the ferromagnetic electrodes Δ: (a) Δ = 8 eV; (b) Δ = 9 eV; (c) Δ = 10 eV.

    其次, 计算了在不同的化学势μ下的温度-偏压相图, 结果如图3所示, 其中半交换劈裂能Δ=9 eV, 原子势的傅里叶变换v(Kh) = 15.3 eV. 图3(a)图3(c)分别对应μ = 10 eV, μ = 11 eV和μ = 12 eV. 从图3可以看出TMR最大的相点对应的偏压随化学势会显著变化, 在物理上, 由(2)式、(3)式和(7)式可知, 这来源于化学势μ对透射系数振荡项相位的影响. 由上可知, 如果通过改变铁磁电极材料来改变化学势, 则可以令MgO基磁性隧道结既能够在低偏压下应用于低功率器件, 又能够在高偏压下实现大的TMR, 进而提高输出电压(${V_{\rm out}} \equiv V \times ({G_{\rm{P}}} - {G_{\rm AP}})/{G_{\rm{P}}}$)来应用于高功率磁性隧道结器件.

    Figure 3.  Phase diagram of temperature and bias with variation of chemical potential μ: (a) μ = 10 eV; (b) μ = 11 eV; (c) μ = 12 eV.

    最后, 计算了在不同v(Kh)下的温度-偏压相图, 结果如图4所示. 其中, 化学势μ = 11 eV, 铁磁电极半交换劈裂能Δ = 9 eV. 图4(a)图4(c)分别对应于v (Kh) = 12.3 eV, v (Kh) = 15.3 eV和v (Kh) = 18.3 eV. 从图4可以看出, TMR的相图显著依赖于v (Kh)的变化. 这是因为, 根据(2)式、(3)式和(4)式, 相比于半交换劈裂能Δ和化学势μ, 透射系数对v (Kh)的变化更敏感. 该结果说明, 如果将单晶MgO势垒层换作其他的单晶势垒层材料, 将会显著地改变TMR的温度-偏压相图. 因此, 势垒层材料的选择是优化磁性隧道结温度和偏压特性的重要因素.

    Figure 4.  Phase diagram of temperature and bias with variation of v (Kh): (a) v(Kh) = 12.3 eV; (b) v(Kh) = 15.3 eV; (c) v(Kh) = 18.3 eV.

    由前文可知, 图2(c)中TMR极大的相点处于室温区域与低偏压区域, 即在本文选取的参数范围之内, Δ = 10 eV, μ = 11 eV, v (Kh) = 15.3 eV最有利于MgO基磁性隧道结的实际应用.

    本文基于传统光学衍射方法构建了适用于单晶势垒层磁性隧道结的理论. 该理论由于充分考虑了周期性势垒层对隧穿电子的衍射效应, 所以很好地计入了隧穿电子波的相干性. 另外, 此理论可以引入温度和偏压对隧穿磁阻效应的影响, 有利于同时处理温度效应和偏压效应. 利用上述理论, 分别计算了不同铁磁电极半交换劈裂能Δ、化学势μ和势垒层周期势v (Kh)下的MgO基磁性隧道结的温度-偏压TMR相图. 结果表明, 可以通过调节半交换劈裂能Δ使得TMR极大的相点处于室温区域中. 此外, 可以通过调节化学势μ改变TMR最大的相点对应的偏压. 同时, 研究发现, 温度-偏压TMR相图显著依赖于势垒层周期势v (Kh). 上述结果说明, 通过调节铁磁电极半交换劈裂能Δ、化学势μ和势垒层周期势v (Kh)可以优化MgO基磁性隧道结的温度特性和偏压特性, 进而有利于MgO基磁性隧道结的实际应用.

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  • 图 1  MgO基磁性隧道结示意图

    Figure 1.  Diagram of MgO-based magnetic tunnel junction.

    图 2  不同铁磁电极半交换劈裂能Δ下的温度-偏压相图 (a) Δ = 8 eV; (b) Δ = 9 eV; (c) Δ = 10 eV

    Figure 2.  Phase diagram of temperature and bias with variation of half the exchange splitting of the ferromagnetic electrodes Δ: (a) Δ = 8 eV; (b) Δ = 9 eV; (c) Δ = 10 eV.

    图 3  不同的化学势μ下的温度-偏压相图 (a) μ = 10 eV; (b) μ = 11 eV; (c) μ = 12 eV

    Figure 3.  Phase diagram of temperature and bias with variation of chemical potential μ: (a) μ = 10 eV; (b) μ = 11 eV; (c) μ = 12 eV.

    图 4  不同v(Kh)下的温度-偏压相图 (a) v(Kh) = 12.3 eV; (b) v(Kh) = 15.3 eV; (c) v(Kh) = 18.3 eV

    Figure 4.  Phase diagram of temperature and bias with variation of v (Kh): (a) v(Kh) = 12.3 eV; (b) v(Kh) = 15.3 eV; (c) v(Kh) = 18.3 eV.

  • [1]

    韩秀峰 2008 物理 37 392Google Scholar

    Han X F 2008 Physics 37 392Google Scholar

    [2]

    Yuasa S, Nagahama T, Fukushima A, Suzuki Y, Ando K 2004 Nat. Mater. 3 868Google Scholar

    [3]

    Ikeda S, Hayakawa J, Ashizawa Y, Lee Y M, Miura K, Hasegawa H, Tsunoda M, Matsukura F, Ohno H 2008 Appl. Phys. Lett. 93 082508

    [4]

    Parkin S S P, Kaiser C, Panchula A, Rice P M, Hughes B S, Mahesh Y S H 2004 Nat. Mater. 3 862Google Scholar

    [5]

    Ma Q L, Wang S G, Zhang J, Wang Y, Ward R C C, Wang C, Kohn A, Zhang X G, Han X F 2009 Appl. Phys. Lett. 95 052506Google Scholar

    [6]

    Faure-Vincent J, Tiusan C, Jouguelet E, Canet F, Sajieddine M, Bellouard C, Hehn M, Montaigne F, Schuhl A 2003 Appl. Phys. Lett. 82 4507Google Scholar

    [7]

    Miao G X, Chetry K B, Gupta A, Bulter W H, Tsunekawa K, Djayaprawira D Xiao G 2006 J. Appl. Phys. 99 08T305Google Scholar

    [8]

    Ishikawa T, Marukame T, Kijima H, Matsuda K I, Uemura T, Arita M, Ymamoto M 2006 Appl. Phys. Lett. 89 192505Google Scholar

    [9]

    Yuasa S, Fukushima A, Kubota H, Suzuki Y, Ando K 2006 Appl. Phys. Lett. 89 042505Google Scholar

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Metrics
  • Abstract views:  4934
  • PDF Downloads:  71
  • Cited By: 0
Publishing process
  • Received Date:  12 November 2020
  • Accepted Date:  15 December 2020
  • Available Online:  11 May 2021
  • Published Online:  20 May 2021

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