具有自旋轨道耦合的低维系统中自旋输运的研究
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摘要
本论文主要讨论了具有自旋轨道耦合的低维系统中的自旋输运问题。在前两章中,我们回顾了自旋电子学的发展进程并简述了基于自旋轨道耦合的自旋输运现象。论文的后几章详细地介绍了我们在这方面取得的如下研究成果：
我们第一次从SU(2)规范场的角度研究了自旋轨道耦合系统中的自旋流。我们发现,普遍定义的自旋流满足协变形式的连续性方程。利用Noether定理,我们得到了守恒的总自旋流。我们认为,自旋密度和自旋流密度会激发出SU(2)规范场,而此规范场又会对自旋流施加自旋力,从而导致了它的不守恒。因此,守恒的总自旋流应该包含由Su(2)规范场给出的贡献。通过引入Su(2)场强张量,我们可以得到作用在自旋密度和自旋流密度上的自旋力。当只有u(1)电磁场时,该自旋力会简单地退化为Stern-Gerlach力。此外,我们还研究了u(1)×su(2)规范场下的轨道流密度。我们指出,由于在一般情况下u(1)和Su(2)规范场的强度可以随空间变化,系统的总角动量并不守恒。因此,普遍定义下的自旋流密度和轨道流密度的不守恒部分并不能相互抵消。
从流体力学出发,我们建立了SU(2)×u(1)场中自旋输运的经典图像。基于此图像,我们给出了自旋流满足的协变形式连续性方程的经典对应。考虑到电子在Su(2)×u(1)场中受到Lorentz力和自旋力的作用,我们写下了单电子运动的经典方程。从该方程中我们可以很容易地得到系统具有无穷长自旋弛豫时间的条件。另一方面,该经典方程表明,即使su(2)规范场不随时间变化,由于电子自旋与Su(2)电磁场的耦合,电子将感受到含时的自旋力。从半经典的Boltzmann方程出发,我们得到了耦合的电荷-自旋扩散方程。我们发现,电子的“振颤”运动是导致其耦合的原因。此外,我们研究了在三种不同形式的自旋轨道耦合下一维弹道系统中的自旋进动。结果表明,自旋进动强烈依赖于电子注入时的自旋极化方向。
我们研究了描述自旋轨道耦合系统对非阿贝尔外场线性响应的SU(2)Kubo公式。我们发现,自旋流满足的协变形式连续性方程保证了SU(2)Kubo公式在两种不同规范固定下的自洽性。我们计算了具有Rashba或Dresselhaus自旋轨道耦合的系统中自旋密度及自旋流密度对SU(2)外场的线性响应。结果表明,当不计入自旋轨道耦合时,如果系统具有平方色散关系,那么即使没有杂质存在,该系统的Su(2)自旋电导率也依然为零。这是由Su(2)李代数生成元之间的反对易关系导致的。此外,我们还将SU(2)Kubo公式推广到了自旋3／2表示。这方便了我们讨论Luttinger模型和耦合的双层二维电子气系统中的自旋输运问题。
我们研究了双层二维电子气中的自旋霍尔电导率以及隧穿自旋流。结果表明,自旋霍尔电导率在能量简并点附近出现峰值,并且无穷小浓度的非磁性杂质并不能将其压制为零。针对这一现象,我们提出了相关的测量方法。另一方面,我们发现,当两层中的杂质强度相同时,隧穿自旋电导率随门压的变化曲线呈现双峰结构。在考虑了两层间存在杂质强度差后,隧穿自旋电导率的一个峰值被压制并改变符号。此时,隧穿自旋流随门压的变化是非对称的。这表明双层系统具有自旋二极管的特性。
我们讨论了量子点中核自旋的低能激发问题。运用相干态路径积分,我们得到了该系统的作用量。将电子自由度积掉后,我们得到了描述核自旋的有效作用量以及核自旋的自旋波传播子。这一初步的结果将有助于我们进一步讨论量子点中电子的自旋退相干过程。
This dissertation focuses on the spin transport in low-dimensional systems with spin-orbit coupling. In the first two chapters, we briefly review the de-velopment of spintronics and introduce some spin-orbit-coupling dependent spin transport phenomena. Then we show the details of our following investigations.
We study the spin current in systems with spin-orbit coupling from the point of view of SU(2) gauge fields for the first time. We find that the naturally defined spin current obeys the covariant continuity equation. By means of the Noether theorem, we obtain the conserved total spin current. We argue that the spin density and spin current density are the sources of SU(2) gauge fields which in turn exert spin force on them. This leads to the nonconservation of the spin current. Thus the conserved spin current should include the contributions from SU(2) gauge fields. By introducing the SU(2) field strength tensor, we can easily obtain the spin force which reduces to the Stern-Gerlach force. We investigate the orbit current in the presence of U(1)×SU(2) gauge fields. We point out that due to the spatially dependent spin-orbit coupling, the total angular momentum does not conserve. Hence the nonconservation parts of the spin and orbit currents can not counteract each other precisely.
Starting from the fluid mechanics, we construct a classical picture for the spin transport in SU(2)×U(1) fields. Based on this picture, we derive the clas-sical analogy of the covariant continuity equation which the spin current obeys. Considering that the electron experiences both Lorentz force and spin force, we obtain the classical equations of motion for an electron moving in SU(2)×U(1) fields. From these equations, we can directly obtain the condition for the occur-rence of the infinite spin relaxation time. On the other hand, these equations show that even though the SU(2) gauge fields do not vary with respect to time, the electron can feel an effective time-dependent spin force due to the coupling between the spin and SU(2) gauge fields. We also formulate the diffusion equa- tions for the charge and spin densities. We find that the Zitterbewegung makes these equations couple to each other. Besides, we study the spin precession in one-dimensional ballistic system with three different forms of spin-orbit coupling. The results manifest that the spin precession strongly depends on the initial con-ditions.
We investigate the SU(2) Kubo formula which describes the linear response to the nonabelian external fields. We find that the covariant continuity equation for the spin current plays a key role in keeping the consistency of the SU(2) Kubo formula with different gauge fixings. We calculate the linear responses of the spin density and spin current density to the SU(2) external fields. It is shown that if the system possesses the parabolic dispersion relation in the absence of the spin-orbit coupling, the SU(2) spin conductivity still vanishes even without the impurities. This is due to the anticommunication relation between the SU(2) generators. Moreover, we generate the SU(2) Kubo formula to the spin 3/2 representation. It facilitates the discussions of the spin transport in the Luttinger model and coupled bilayer two-dimensional gas.
We study the spin Hall conductivity and tunneling spin current in the bi-layer two-dimensional electron gas. The results demonstrate that the spin Hall conductivity shows a sharp peak around the energy degenerate point. This peak can not be suppressed to zero by the infinitesimal concentration of nonmagnetic impurities. We also propose a experimental scheme to detect this magnification effect of the spin Hall conductivity. On the other hand, we find that the tunnel-ing spin conductivity exhibits a double-peak structure in the twin-layer situation. Taking the difference of strengthes of impurity potentials between layers, we find out that the tunneling spin current is asymmetric with respect to the gate voltage. It makes the bilayer system a candidate for the spin diode.
We discuss the low-energy excitation of nuclear spins in quantum dots. Using the path integral approach, we obtain the action of this system. After integrating out the electron degrees of freedom, we derive the effective action describing the nuclear spins and the propagator for the spin wave. This helps us to further investigate the spin decoherence caused by the hyperfine interaction in quantum dots.
我们第一次从SU(2)规范场的角度研究了自旋轨道耦合系统中的自旋流。我们发现,普遍定义的自旋流满足协变形式的连续性方程。利用Noether定理,我们得到了守恒的总自旋流。我们认为,自旋密度和自旋流密度会激发出SU(2)规范场,而此规范场又会对自旋流施加自旋力,从而导致了它的不守恒。因此,守恒的总自旋流应该包含由Su(2)规范场给出的贡献。通过引入Su(2)场强张量,我们可以得到作用在自旋密度和自旋流密度上的自旋力。当只有u(1)电磁场时,该自旋力会简单地退化为Stern-Gerlach力。此外,我们还研究了u(1)×su(2)规范场下的轨道流密度。我们指出,由于在一般情况下u(1)和Su(2)规范场的强度可以随空间变化,系统的总角动量并不守恒。因此,普遍定义下的自旋流密度和轨道流密度的不守恒部分并不能相互抵消。
从流体力学出发,我们建立了SU(2)×u(1)场中自旋输运的经典图像。基于此图像,我们给出了自旋流满足的协变形式连续性方程的经典对应。考虑到电子在Su(2)×u(1)场中受到Lorentz力和自旋力的作用,我们写下了单电子运动的经典方程。从该方程中我们可以很容易地得到系统具有无穷长自旋弛豫时间的条件。另一方面,该经典方程表明,即使su(2)规范场不随时间变化,由于电子自旋与Su(2)电磁场的耦合,电子将感受到含时的自旋力。从半经典的Boltzmann方程出发,我们得到了耦合的电荷-自旋扩散方程。我们发现,电子的“振颤”运动是导致其耦合的原因。此外,我们研究了在三种不同形式的自旋轨道耦合下一维弹道系统中的自旋进动。结果表明,自旋进动强烈依赖于电子注入时的自旋极化方向。
我们研究了描述自旋轨道耦合系统对非阿贝尔外场线性响应的SU(2)Kubo公式。我们发现,自旋流满足的协变形式连续性方程保证了SU(2)Kubo公式在两种不同规范固定下的自洽性。我们计算了具有Rashba或Dresselhaus自旋轨道耦合的系统中自旋密度及自旋流密度对SU(2)外场的线性响应。结果表明,当不计入自旋轨道耦合时,如果系统具有平方色散关系,那么即使没有杂质存在,该系统的Su(2)自旋电导率也依然为零。这是由Su(2)李代数生成元之间的反对易关系导致的。此外,我们还将SU(2)Kubo公式推广到了自旋3／2表示。这方便了我们讨论Luttinger模型和耦合的双层二维电子气系统中的自旋输运问题。
我们研究了双层二维电子气中的自旋霍尔电导率以及隧穿自旋流。结果表明,自旋霍尔电导率在能量简并点附近出现峰值,并且无穷小浓度的非磁性杂质并不能将其压制为零。针对这一现象,我们提出了相关的测量方法。另一方面,我们发现,当两层中的杂质强度相同时,隧穿自旋电导率随门压的变化曲线呈现双峰结构。在考虑了两层间存在杂质强度差后,隧穿自旋电导率的一个峰值被压制并改变符号。此时,隧穿自旋流随门压的变化是非对称的。这表明双层系统具有自旋二极管的特性。
我们讨论了量子点中核自旋的低能激发问题。运用相干态路径积分,我们得到了该系统的作用量。将电子自由度积掉后,我们得到了描述核自旋的有效作用量以及核自旋的自旋波传播子。这一初步的结果将有助于我们进一步讨论量子点中电子的自旋退相干过程。
This dissertation focuses on the spin transport in low-dimensional systems with spin-orbit coupling. In the first two chapters, we briefly review the de-velopment of spintronics and introduce some spin-orbit-coupling dependent spin transport phenomena. Then we show the details of our following investigations.
We study the spin current in systems with spin-orbit coupling from the point of view of SU(2) gauge fields for the first time. We find that the naturally defined spin current obeys the covariant continuity equation. By means of the Noether theorem, we obtain the conserved total spin current. We argue that the spin density and spin current density are the sources of SU(2) gauge fields which in turn exert spin force on them. This leads to the nonconservation of the spin current. Thus the conserved spin current should include the contributions from SU(2) gauge fields. By introducing the SU(2) field strength tensor, we can easily obtain the spin force which reduces to the Stern-Gerlach force. We investigate the orbit current in the presence of U(1)×SU(2) gauge fields. We point out that due to the spatially dependent spin-orbit coupling, the total angular momentum does not conserve. Hence the nonconservation parts of the spin and orbit currents can not counteract each other precisely.
Starting from the fluid mechanics, we construct a classical picture for the spin transport in SU(2)×U(1) fields. Based on this picture, we derive the clas-sical analogy of the covariant continuity equation which the spin current obeys. Considering that the electron experiences both Lorentz force and spin force, we obtain the classical equations of motion for an electron moving in SU(2)×U(1) fields. From these equations, we can directly obtain the condition for the occur-rence of the infinite spin relaxation time. On the other hand, these equations show that even though the SU(2) gauge fields do not vary with respect to time, the electron can feel an effective time-dependent spin force due to the coupling between the spin and SU(2) gauge fields. We also formulate the diffusion equa- tions for the charge and spin densities. We find that the Zitterbewegung makes these equations couple to each other. Besides, we study the spin precession in one-dimensional ballistic system with three different forms of spin-orbit coupling. The results manifest that the spin precession strongly depends on the initial con-ditions.
We investigate the SU(2) Kubo formula which describes the linear response to the nonabelian external fields. We find that the covariant continuity equation for the spin current plays a key role in keeping the consistency of the SU(2) Kubo formula with different gauge fixings. We calculate the linear responses of the spin density and spin current density to the SU(2) external fields. It is shown that if the system possesses the parabolic dispersion relation in the absence of the spin-orbit coupling, the SU(2) spin conductivity still vanishes even without the impurities. This is due to the anticommunication relation between the SU(2) generators. Moreover, we generate the SU(2) Kubo formula to the spin 3/2 representation. It facilitates the discussions of the spin transport in the Luttinger model and coupled bilayer two-dimensional gas.
We study the spin Hall conductivity and tunneling spin current in the bi-layer two-dimensional electron gas. The results demonstrate that the spin Hall conductivity shows a sharp peak around the energy degenerate point. This peak can not be suppressed to zero by the infinitesimal concentration of nonmagnetic impurities. We also propose a experimental scheme to detect this magnification effect of the spin Hall conductivity. On the other hand, we find that the tunnel-ing spin conductivity exhibits a double-peak structure in the twin-layer situation. Taking the difference of strengthes of impurity potentials between layers, we find out that the tunneling spin current is asymmetric with respect to the gate voltage. It makes the bilayer system a candidate for the spin diode.
We discuss the low-energy excitation of nuclear spins in quantum dots. Using the path integral approach, we obtain the action of this system. After integrating out the electron degrees of freedom, we derive the effective action describing the nuclear spins and the propagator for the spin wave. This helps us to further investigate the spin decoherence caused by the hyperfine interaction in quantum dots.
引文
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