铁及铁合金的反常霍尔效应和磁性研究
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摘要
人们对磁的研究在过去二十年时间里有了全新的发展。磁性研究不再限于宏观的、静态的物质磁性,而是拓展到研究单电子的、输运状态下的自旋变化。这一领域的主要推动力来自于人们希望能利用电子的自旋自由度来承载信息和进行计算。实验上,人们发现了隧道磁阻(TMR)和庞磁阻效应(CMR),自旋霍尔效应,自旋扭矩,反常霍尔效应,自旋量子霍尔效应等一系新的现象,并由此拓展为一个包含多种器件结构的设计、多种机理的研究和多种材料的制备的庞大领域-自旋电子学。
在自旋电子学的阵容里,反常霍尔效应(anomalous Hall effect, AHE)无疑是一朵奇葩。因为这一现象1889年就已经被霍尔本人发现。但是关于其机理是来自于非本征机制还是本征机制的争论一直延续到了今天。现存的几种机理包括1954年Kaplus和Luttinger提出的能带结构和自旋轨道耦合决定的本征机制、1955年Smit提出的由散射和自旋轨道耦合导致的斜散射机制(skew scattering)以及1971年Berger提出的side jump非本征机制。其复杂的理论模型和实验现象使得简单的实验不能结束这一争论。今天,人们发现AHE的机理同样适用于自旋霍尔效应。这使得对其机理来自于本征还是非本征、其随温度和尺度的变化如何的讨论尤为重要。在本文中,作者对AHE的经典体系-Fe单晶进行了研究。在GaAs(001)和MgO(001)两种衬底上外延生长了Fe单晶,利用改变厚度的办法来调节非本征散射的大小。值得说明的是,这一方法在改变非本征散射的同时在一定范围内能保持本征机制不变。这就克服了传统的通过掺杂改变非本征散射的做法在改变非本征散射强度的同时也改变了本征机制的缺陷。正是这一特点使得我们得到了许多新的结果,包括：
1.证明了长期以来大家认为的反常霍尔电阻率与非本征散射强度(纵向电阻率)之间函数关系式不能描述铁在有限温度下的实验数据。进而根据实验数据确立了新的函数关系。新的函数关系式能够统一描述以往在不同体系、不同温度出现的复杂实验现象,函数中的各项分别与skew scattering、side jump和本征机制联系起来。更为重要的是,它对以前理论上理解不够清楚的有限温度情况的实验结果做了总结,这必将促进理论的进一步完善。
2.通过对样品的不同设计,显示了对界面的调节、衬底的变化和掺杂的不同对本征和非本征机制的影响。证明了之前确立的函数关系式适用于Fe基合金。对低温下反常霍尔电导率略微偏离函数关系式的现象进行了研究讨论。这种偏离可能表示了Fe-磁性金属合金中含有未被考虑的反常霍尔效应。
3.基于前面的进展,研究了反常霍尔效应在低维情况的变化,观察到了本征效应随厚度降低而降低,非本征效应几乎保持不变的现象。对可能的原因进行了讨论。这部分实验结果为第一性原理计算提供了参照对象。除此之外,还观察到了铁磁材料在超薄时的局域化现象。结合之前的研究工作,确定在GaAs衬底上的Fe薄膜非本征散射机制中包含很强的side jump机制,这支持在厚膜中得出的函数关系中有一项是side jump机制的推论。MgO衬底上的Fe薄膜side jump却很少。
本论文的最后一个部分是作者对薄膜磁光科尔效应的数值计算。与实验结果对比,表明纳米尺度下Fe的磁光科尔效应常数可能大于体材料下的情况。而Fe和Au的其他光学常数则变化不大。计算结果还表明对于光学常数略微偏离各向同性的情况,比如某些MBE外延生长的材料,磁光科尔效应仍可用来确定各项异性常数的大小。
In the past two decades, the study on magnetism is prominent in the area of condensed matter physics. It is not limited to static and macro magnetism but expanded to spin dynamics in transport, even in the single electron limit. Due to the promising future of using the freedom of spin in data storage and calculation, this area draws intensive attention and investment. Experimentally, the tunneling magneto-resistance (TMR) and Colossal magnetoresistance (CMR), spin Hall effect, anomalous Hall effect, spin torque and quantum spin Hall effect are typical new phenomena found in this area. It has been developed into a comprehensive discipline which contains device fabrication, mechanism study and materials development, and has a specific name as spintronics. Among various topics, anomalous Hall effect is one of the most fascinating problem because it is discovered by Edwin Hall himself in 1889. But the debate about its origin has been lasting to date. The existing mechanisms includes an intrinsic mechanism proposed by Kaplus and Luttinger, an extrinsic mechanism called skew scattering which is found by Smit and another extrinsic one-side jump found by Berger. The complicated theoretical model and experimental results make it difficult to settle this debate. Today people found that the mechanism of anomalous Hall effect also works in spin Hall effect, which highlights the importance of understanding its mechanism and behavior with temperature and dimension.
In this thesis, the author investigates the classical system of AHE-single crystal of Fe. Fe films are grown epitaxially on GaAs(001) and MgO(001). Its resistivity is tuned by film thickness. This way of tuning resistivity has the benefit of keeping the intrinsic mechanism unchanged within certain extend, while the extrinsic mechanisms vary with resistivity. In contrast, the old way of tuning resistivity-adding impurities-has the disadvantage that while the extrinsic mechanisms are changing, the intrinsic one also varies because of the alloying. It is this difference that brings some undiscovered results:
1. We prove that the widely used function describing the anomalous Hall resistivity and the strength of scattering (the longitudinal resistivity) is not suitable describing our experimental data. Then we established a new function based on experiments. It is capable describing AHE in various materials at different temperature. The terms in the function are interpreted to have connection to the skew scattering, side jump and intrinsic mechanisms. Moreover, it summaries the experimental results at finite temperature, which is not fully understand in theory. This progress should stimulate further discussions in theory.
2. Through tuning of the interface and impurity level of our samples, the AHE at different interface, substrates and impurities is measured and discussed. We show that the previously established new function also works in Fe-based alloys. The intrinsic and extrinsic mechanisms are separated and discussed consequently. In some special cases the experimental data deviates from the new function, which indicates in the case of magnetic impurities, some mechanisms of AHE remains unclear.
3. Based on our progress, the AHE in ultra-thin Fe films are investigated. The intrinsic contribution is observed to decrease in thinner films. The situation for extrinsic contributions depends on whether localization effect happens. This part of results provides experimental facts for theory to explore. Furthermore, the localization correction to anomalous Hall conductivity is studied within the established framework of extrinsic mechanisms. Side jump mechanism is found to dominant in ultra-thin Fe films on GaAs, which support the interpretation that one term in our new function is connected to side jump mechanism. Meanwhile the side jump contribution in Fe on MgO is found to be minor.
The last chapter of this thesis is some calculation about magneto-optical Kerr effect in multi-layer system. Combining with experiment, we show that the magneto-optical constant of nano-scale Fe may be larger than that of the bulk Fe. Other optical constants like reflective index of Fe and Au remains the same. The calculation results demonstrate that in optically anisotropic materials, for instance some materials prepared by molecular beam epitaxy, the magneto-optical Kerr effect can still be used to measure the magneto-crystalline anisotropy.
在自旋电子学的阵容里,反常霍尔效应(anomalous Hall effect, AHE)无疑是一朵奇葩。因为这一现象1889年就已经被霍尔本人发现。但是关于其机理是来自于非本征机制还是本征机制的争论一直延续到了今天。现存的几种机理包括1954年Kaplus和Luttinger提出的能带结构和自旋轨道耦合决定的本征机制、1955年Smit提出的由散射和自旋轨道耦合导致的斜散射机制(skew scattering)以及1971年Berger提出的side jump非本征机制。其复杂的理论模型和实验现象使得简单的实验不能结束这一争论。今天,人们发现AHE的机理同样适用于自旋霍尔效应。这使得对其机理来自于本征还是非本征、其随温度和尺度的变化如何的讨论尤为重要。在本文中,作者对AHE的经典体系-Fe单晶进行了研究。在GaAs(001)和MgO(001)两种衬底上外延生长了Fe单晶,利用改变厚度的办法来调节非本征散射的大小。值得说明的是,这一方法在改变非本征散射的同时在一定范围内能保持本征机制不变。这就克服了传统的通过掺杂改变非本征散射的做法在改变非本征散射强度的同时也改变了本征机制的缺陷。正是这一特点使得我们得到了许多新的结果,包括：
1.证明了长期以来大家认为的反常霍尔电阻率与非本征散射强度(纵向电阻率)之间函数关系式不能描述铁在有限温度下的实验数据。进而根据实验数据确立了新的函数关系。新的函数关系式能够统一描述以往在不同体系、不同温度出现的复杂实验现象,函数中的各项分别与skew scattering、side jump和本征机制联系起来。更为重要的是,它对以前理论上理解不够清楚的有限温度情况的实验结果做了总结,这必将促进理论的进一步完善。
2.通过对样品的不同设计,显示了对界面的调节、衬底的变化和掺杂的不同对本征和非本征机制的影响。证明了之前确立的函数关系式适用于Fe基合金。对低温下反常霍尔电导率略微偏离函数关系式的现象进行了研究讨论。这种偏离可能表示了Fe-磁性金属合金中含有未被考虑的反常霍尔效应。
3.基于前面的进展,研究了反常霍尔效应在低维情况的变化,观察到了本征效应随厚度降低而降低,非本征效应几乎保持不变的现象。对可能的原因进行了讨论。这部分实验结果为第一性原理计算提供了参照对象。除此之外,还观察到了铁磁材料在超薄时的局域化现象。结合之前的研究工作,确定在GaAs衬底上的Fe薄膜非本征散射机制中包含很强的side jump机制,这支持在厚膜中得出的函数关系中有一项是side jump机制的推论。MgO衬底上的Fe薄膜side jump却很少。
本论文的最后一个部分是作者对薄膜磁光科尔效应的数值计算。与实验结果对比,表明纳米尺度下Fe的磁光科尔效应常数可能大于体材料下的情况。而Fe和Au的其他光学常数则变化不大。计算结果还表明对于光学常数略微偏离各向同性的情况,比如某些MBE外延生长的材料,磁光科尔效应仍可用来确定各项异性常数的大小。
In the past two decades, the study on magnetism is prominent in the area of condensed matter physics. It is not limited to static and macro magnetism but expanded to spin dynamics in transport, even in the single electron limit. Due to the promising future of using the freedom of spin in data storage and calculation, this area draws intensive attention and investment. Experimentally, the tunneling magneto-resistance (TMR) and Colossal magnetoresistance (CMR), spin Hall effect, anomalous Hall effect, spin torque and quantum spin Hall effect are typical new phenomena found in this area. It has been developed into a comprehensive discipline which contains device fabrication, mechanism study and materials development, and has a specific name as spintronics. Among various topics, anomalous Hall effect is one of the most fascinating problem because it is discovered by Edwin Hall himself in 1889. But the debate about its origin has been lasting to date. The existing mechanisms includes an intrinsic mechanism proposed by Kaplus and Luttinger, an extrinsic mechanism called skew scattering which is found by Smit and another extrinsic one-side jump found by Berger. The complicated theoretical model and experimental results make it difficult to settle this debate. Today people found that the mechanism of anomalous Hall effect also works in spin Hall effect, which highlights the importance of understanding its mechanism and behavior with temperature and dimension.
In this thesis, the author investigates the classical system of AHE-single crystal of Fe. Fe films are grown epitaxially on GaAs(001) and MgO(001). Its resistivity is tuned by film thickness. This way of tuning resistivity has the benefit of keeping the intrinsic mechanism unchanged within certain extend, while the extrinsic mechanisms vary with resistivity. In contrast, the old way of tuning resistivity-adding impurities-has the disadvantage that while the extrinsic mechanisms are changing, the intrinsic one also varies because of the alloying. It is this difference that brings some undiscovered results:
1. We prove that the widely used function describing the anomalous Hall resistivity and the strength of scattering (the longitudinal resistivity) is not suitable describing our experimental data. Then we established a new function based on experiments. It is capable describing AHE in various materials at different temperature. The terms in the function are interpreted to have connection to the skew scattering, side jump and intrinsic mechanisms. Moreover, it summaries the experimental results at finite temperature, which is not fully understand in theory. This progress should stimulate further discussions in theory.
2. Through tuning of the interface and impurity level of our samples, the AHE at different interface, substrates and impurities is measured and discussed. We show that the previously established new function also works in Fe-based alloys. The intrinsic and extrinsic mechanisms are separated and discussed consequently. In some special cases the experimental data deviates from the new function, which indicates in the case of magnetic impurities, some mechanisms of AHE remains unclear.
3. Based on our progress, the AHE in ultra-thin Fe films are investigated. The intrinsic contribution is observed to decrease in thinner films. The situation for extrinsic contributions depends on whether localization effect happens. This part of results provides experimental facts for theory to explore. Furthermore, the localization correction to anomalous Hall conductivity is studied within the established framework of extrinsic mechanisms. Side jump mechanism is found to dominant in ultra-thin Fe films on GaAs, which support the interpretation that one term in our new function is connected to side jump mechanism. Meanwhile the side jump contribution in Fe on MgO is found to be minor.
The last chapter of this thesis is some calculation about magneto-optical Kerr effect in multi-layer system. Combining with experiment, we show that the magneto-optical constant of nano-scale Fe may be larger than that of the bulk Fe. Other optical constants like reflective index of Fe and Au remains the same. The calculation results demonstrate that in optically anisotropic materials, for instance some materials prepared by molecular beam epitaxy, the magneto-optical Kerr effect can still be used to measure the magneto-crystalline anisotropy.
引文
[1]A. Kundt, Wied. Ann.49,257 (1893).
[2]E. M. Pugh, Phys. Rev.36,1503. (1930).
[3]R. Karplus and J. M. Luttinger, Phys. Rev.95,1154 (1954).
[4]J. Smit, Physica 21,877 (1955).
[5]J. Smit, Physica 24,39 (1958).
[6]Nagaosa et al, arXiv:0904.4154 (2009).
[7]J. Kondo, Prog. Theor. Phys.27,772 (1962).
[8]A. Fert and A. Friederich, Phys. Rev. B 13,397 (1974).
[9]F. E. Maranzana, Phys. Rev.160,421-429 (1967).
[10]L. Berger, Phys. Rev. B 2,4559 (1970).
[11]A. Crepieux and P. Bruno, Phys. Rev. B 64,014416 (2001).
[12]J. Ye et al., Phys. Rev. Lett.83,3737 (1999).
[13]G. Sundaram et al., Phys. Rev. B 59,14 915 (1999).
[14]T. Jungwirth et al., Phys. Rev. Lett.88,207208 (2002).
[15]M. Onoda et al., J. Phys. Soc. Jpn.71,19 (2002).
[16]Yugui Yao, Phys. Rev. B 75,020401 (2007).
[17]Shigeki Onda, Naoyuki Sugimoto and Naoto Nagaosa, Phys. Rev. Lett. 97,126602 (2006).
[18]N. A. Sinitsyn, J. Phys. Condens. Matter 20,023201 (2008).
[19]W. Jellinghaus and M.P. DeAndres, Ann. Physik,462,189 (1961).
[20]P. N. Dheer, Phys. Rev.156,637 (1967).
[21]A. Fert et al., Phys. Rev. Lett.28,303 (1972).
[22]J. Kotzler et al., Phys. Rev. B 72,060412 (2005).
[23]C. G. Zeng et al., Phys. Rev. Lett.96,037204 (2006).
[24]J. M. Lavine Phys. Rev.123,1273 (1961).
[25]Yugui Yao, et al, Phys. Rev. B 75,020401(R) (2007).
[26]T. Miyasato et al, Phys. Rev. Lett.99,086602 (2007).
[1]K. Fuchs, Proc. Camb. Phil, Soc.,34,100 (1983).
[2]C. R. Tellier, Size Effects, Amsterdan-Oxford-New York, (1982).
[3]S. F. Zhang, Phys. Rev. B 51,3632 (1995).
[4]A. Gerber et, al. Phys. Rev. B 65,054426 (2002).
[1]W. Jellinghaus, M. P. DeAndres, Ann. Physik 7,189 (1961).
[2]P. N. Dheer, Phy. Rev.156,637 (1967).
[3]C. G. Zeng, Y. G. Yao, Q. Niu, H. H. Weitering, Phys. Rev. Lett.96,037204 (2006).
[4]J. Kotzler, W. Gil, Phys. Rev. B 72,060412(R) (2005).
[5]J. Lavine, Phys. Rev.123,1273 (1961).
[6]A. Fert,O. Jaoul, Phys. Rev. Lett.28,303 (1972).
[1]Y. Shiomi, Y. Onose and Y. Tokura, Phys. Rev. B 79,100404(R) (2009).
[2]N. A. Sinitsyn, J. Phys. Condens. Matter 20,023201 (2008).
[3]A. K. Majumudar and L. Berger, Phys. Rev. B.7,4203 (1973).
[4]A. Fert and O. Jaoul, Phys. Rev. Lett.28,303 (1972).
[1]Y. Yao, Q. Niu et al., Phys. Rev. Lett.92 037204 (2004).
[2]C. G. Zeng et al., Phys. Rev. Lett.96,037204 (2006).
[3]Y. Yao, Y. Liang et al., Phys. Rev. B 75 020401(R) (2007).
[4]L. F. Yin et al, Phys. Rev. Lett.97,067203 (2006).
[5]Kh. Zakery et al, Phys. Rev. B 73,052405 (2006).
[6]Shigeki Onoda, Naoyuki Sugimoto, and Naoto Nagaosa,97,126602 (2006).
[7]T. Miyasato et al, Phys. Rev. Lett.99,086602 (2007).
[8]S. Sangiao et al, Phys. Rev. B 79,014431 (2009).
[9]A. Langenfeld and P. WAol°r, Phys. Rev. Lett.67,739 (1991).
[10]G. Bergmann and F. Ye, Phys. Rev. Lett.67,735 (1991).
[11]V. K. Dugaev, A. Crepieux, and P. Bruno, Phys. Rev. B 64,104411 (2001).
[12]P. Mitra, R. Misra, A. F. Hebard, K. A. Muttalib and P. Wolfle. Phys. Rev. Lett.99,046804 (2007).
[1]M. Faraday, Trans. Roy. Soc. (London) 5 (1846) 592.
[2]J. Kerr, Philos. Mag.3,339 (1877). J. Kerr, Philos. Mag.5,161(1878).
[3]E. R. Moog, S. D. Bader, Superlattices Microstruct.1,543 (1985); S.D. Bader, E.R. Moog, P. GruKnberg, J. Magn. Magn. Mater.53, L295 (1986).
[4]J. Zak, E. R. Moog, C. Liu, and S. D. Bader, Universal approach to magnetooptics, J. Magn. Magn. Mater.89,107(1990)
[5]J. Zak, E. R. Moog, C. Liu, et al. Magnetooptics of multilayers with arbitrary magnetization directions, Phys. Rew. B 43 (8):6423-6429 (1991)
[6]Z. Q. Qiu and S. D. Bader, Surface magneto-optic Kerr effect (SMOKE), J. Magn. Magn. Mater.200,664-678(1999)
[2]E. M. Pugh, Phys. Rev.36,1503. (1930).
[3]R. Karplus and J. M. Luttinger, Phys. Rev.95,1154 (1954).
[4]J. Smit, Physica 21,877 (1955).
[5]J. Smit, Physica 24,39 (1958).
[6]Nagaosa et al, arXiv:0904.4154 (2009).
[7]J. Kondo, Prog. Theor. Phys.27,772 (1962).
[8]A. Fert and A. Friederich, Phys. Rev. B 13,397 (1974).
[9]F. E. Maranzana, Phys. Rev.160,421-429 (1967).
[10]L. Berger, Phys. Rev. B 2,4559 (1970).
[11]A. Crepieux and P. Bruno, Phys. Rev. B 64,014416 (2001).
[12]J. Ye et al., Phys. Rev. Lett.83,3737 (1999).
[13]G. Sundaram et al., Phys. Rev. B 59,14 915 (1999).
[14]T. Jungwirth et al., Phys. Rev. Lett.88,207208 (2002).
[15]M. Onoda et al., J. Phys. Soc. Jpn.71,19 (2002).
[16]Yugui Yao, Phys. Rev. B 75,020401 (2007).
[17]Shigeki Onda, Naoyuki Sugimoto and Naoto Nagaosa, Phys. Rev. Lett. 97,126602 (2006).
[18]N. A. Sinitsyn, J. Phys. Condens. Matter 20,023201 (2008).
[19]W. Jellinghaus and M.P. DeAndres, Ann. Physik,462,189 (1961).
[20]P. N. Dheer, Phys. Rev.156,637 (1967).
[21]A. Fert et al., Phys. Rev. Lett.28,303 (1972).
[22]J. Kotzler et al., Phys. Rev. B 72,060412 (2005).
[23]C. G. Zeng et al., Phys. Rev. Lett.96,037204 (2006).
[24]J. M. Lavine Phys. Rev.123,1273 (1961).
[25]Yugui Yao, et al, Phys. Rev. B 75,020401(R) (2007).
[26]T. Miyasato et al, Phys. Rev. Lett.99,086602 (2007).
[1]K. Fuchs, Proc. Camb. Phil, Soc.,34,100 (1983).
[2]C. R. Tellier, Size Effects, Amsterdan-Oxford-New York, (1982).
[3]S. F. Zhang, Phys. Rev. B 51,3632 (1995).
[4]A. Gerber et, al. Phys. Rev. B 65,054426 (2002).
[1]W. Jellinghaus, M. P. DeAndres, Ann. Physik 7,189 (1961).
[2]P. N. Dheer, Phy. Rev.156,637 (1967).
[3]C. G. Zeng, Y. G. Yao, Q. Niu, H. H. Weitering, Phys. Rev. Lett.96,037204 (2006).
[4]J. Kotzler, W. Gil, Phys. Rev. B 72,060412(R) (2005).
[5]J. Lavine, Phys. Rev.123,1273 (1961).
[6]A. Fert,O. Jaoul, Phys. Rev. Lett.28,303 (1972).
[1]Y. Shiomi, Y. Onose and Y. Tokura, Phys. Rev. B 79,100404(R) (2009).
[2]N. A. Sinitsyn, J. Phys. Condens. Matter 20,023201 (2008).
[3]A. K. Majumudar and L. Berger, Phys. Rev. B.7,4203 (1973).
[4]A. Fert and O. Jaoul, Phys. Rev. Lett.28,303 (1972).
[1]Y. Yao, Q. Niu et al., Phys. Rev. Lett.92 037204 (2004).
[2]C. G. Zeng et al., Phys. Rev. Lett.96,037204 (2006).
[3]Y. Yao, Y. Liang et al., Phys. Rev. B 75 020401(R) (2007).
[4]L. F. Yin et al, Phys. Rev. Lett.97,067203 (2006).
[5]Kh. Zakery et al, Phys. Rev. B 73,052405 (2006).
[6]Shigeki Onoda, Naoyuki Sugimoto, and Naoto Nagaosa,97,126602 (2006).
[7]T. Miyasato et al, Phys. Rev. Lett.99,086602 (2007).
[8]S. Sangiao et al, Phys. Rev. B 79,014431 (2009).
[9]A. Langenfeld and P. WAol°r, Phys. Rev. Lett.67,739 (1991).
[10]G. Bergmann and F. Ye, Phys. Rev. Lett.67,735 (1991).
[11]V. K. Dugaev, A. Crepieux, and P. Bruno, Phys. Rev. B 64,104411 (2001).
[12]P. Mitra, R. Misra, A. F. Hebard, K. A. Muttalib and P. Wolfle. Phys. Rev. Lett.99,046804 (2007).
[1]M. Faraday, Trans. Roy. Soc. (London) 5 (1846) 592.
[2]J. Kerr, Philos. Mag.3,339 (1877). J. Kerr, Philos. Mag.5,161(1878).
[3]E. R. Moog, S. D. Bader, Superlattices Microstruct.1,543 (1985); S.D. Bader, E.R. Moog, P. GruKnberg, J. Magn. Magn. Mater.53, L295 (1986).
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