关注：

1） GW方法、hse方法计算设置时能否同时设置加U、加自旋耦合计算、加磁矩计算

2） 自旋轨道耦合的物理含义

3） 计算参数设置

 Spin-orbit coupling in a Fe monolayer

http://cms.mpi.univie.ac.at/wiki/index.php/Spin-orbit_coupling_in_a_Fe_monolayer

参数解释：

1. Setting LNONCOLLINEAR=.TRUE. in the INCAR file allows to perform fully  non-collinear magnetic structure calculations.

2. LSORBIT=.TRUE. switches on spin-orbit coupling and automatically sets  LNONCOLLINEAR= .TRUE.. This option works only for PAW potentials and is not supported for ultrasoft pseudopotentials.

（1） If spin-orbit coupling is not included,

the energy does not depend on the direction of the magnetic moment, i.e. rotating all magnetic moments by the same  angle results exactly in the same energy. Hence there is no need to define the spin quantization axis, as long as spin-orbit coupling is not included.

（2） Spin-orbit coupling,

however, couples the spin to the crystal structure. Spin orbit coupling is switched on by selecting

LSORBIT = .TRUE.
 SAXIS =   s_x s_y s_z (quantisation axis for spin)
 GGA_COMPAT = .FALSE. ! apply spherical cutoff on gradient field

where the default for  SAXIS= (the notation implies an infinitesimal small positive number  in direction).

The flag GGA_COMPAT (see Sec. 6.42)  is optional and should be set when small energy differences in the sub meV regime need to be calculated (often the case for magnetic anisotropy calculations).

All magnetic moments are now given with respect to the axis

, where we have adopted the convention that all magnetic moments and spinor-like quantities written or read by VASP are given with respect to this axis.

This includes the MAGMOM line in the INCAR file, the total and local magnetizations in the OUTCAR and PROCAR file, the spinor-like orbitals in the WAVECAR file, and the magnetization density in the CHGCAR file.

With respect to the cartesian lattice vectors the components of the magnetization are (internally) given by

(公式略)

Hence now the magnetic moment is parallel to the vector SAXIS.

Thus there are two ways to rotate the spins in an arbitrary direction【为什么要任意方向旋转自旋？】, either by changing the initial magnetic moments MAGMOM or by changing SAXIS.

To initialise calculations with the magnetic moment  parallel to a chosen vector , it is therefore possible to either specify (assuming a single atom in the cell)  MAGMOM = x y z   ! local magnetic moment in x,y,z SAXIS =  0 0 1   ! quantisation axis parallel to z

or  

MAGMOM = 0 0 total_magnetic_moment   ! local magnetic moment parallel to SAXIS SAXIS =  x y z   ! quantisation axis parallel to vector (x,y,z)

Both setups should in principle yield exactly the same energy, but for implementation reasons the second method is usually more precise.

（2）The second method

also allows to read a preexisting WAVECAR file (from a collinear or non collinear run), and to continue the calculation with a different spin orientation.

When a non collinear WAVECAR file is read, the spin is assumed to be parallel to  SAXIS (hence VASP will initially report a magnetic moment in the z-direction only).

The recommended procedure for the calculation of magnetic anisotropies【各向异性，铁磁性，反铁磁性】 is therefore (please check the section on LMAXMIX6.63):

（a) Start with a collinear calculation【什么是线性计算】 and calculate a WAVECAR and CHGCAR file.

(b) Add the tags

 LSORBIT = .TRUE.
ICHARG = 11      ! non selfconsistent run, read CHGCAR
 LMAXMIX = 4      ! for d elements increase LMAXMIX to 4, f: LMAXMIX = 6
 ! you need to set LMAXMIX already in the collinear calculation
SAXIS =  x y z   ! direction of the magnetic field
NBANDS = 2 * number of bands of collinear run
GGA_COMPAT = .FALSE. ! apply spherical cutoff on gradient field

VASP reads in the WAVECAR and CHGCAR files, aligns the spin quantization axis parallel to SAXIS, which implies that the magnetic field is now parallel to SAXIS, and performs a non selfconsistent calculation.    

• By comparing the energies for different orientations the magnetic anisotropy can be determined.

• Please mind, that a completely selfconsistent calculation (ICHARG=1) is in principle also possible with VASP, but this would allow the the spinor wavefunctions to rotate from their initial orientation  parallel to SAXIS until the correct groundstate is obtained, i.e. until the magnetic moment is parallel to the easy axis.

• In practice this rotation will be slow, since reorientation of the spin gains little energy. Therefore if the convergence criterion is not too tight, sensible results might be obtained even for fully selfconsistent calculations (in the few cases we have tried selfconsistentcy worked without problems).

   

   

   

• Be very careful with symmetry. We recommend to switch off symmetry (ISYM=0) altogether, when spin orbit coupling is selected. Often the k-point set changes from one to the other spin orientation, worsening the transferability of the results  (also the WAVECAR file can not be reread properly if the number of k-points changes). The flag GGA_COMPAT is usually required and should be set, since magnetic anisotropy energies are often in the sub meV regime (see Sec. 6.42).

• Generally be extremely careful, when using spin orbit coupling and, specifically, magnetic anisotropies: energy differences are tiny, k-point convergence is tedious and slow, and the computer time might be huge. Additionally, this feature-- although long implemented in VASP-- is still in a late beta stage, as you might deduce from the frequent updates. No promise,  that your results will be useful! Here is a small summary from the README file:

  • 20.11.2003:

    The present GGA routine breaks the symmetry slightly for non orthorhombic cells.              

    A spherical cutoff is now imposed on the gradients and all intermediate results               in reciprocal space.              

     

  • This changes the GGA results slightly (usually by 0.1 meV per atom), but is important               for magnetic anisotropies.

  •  

  • 05.12.2003:  continue... Now VASP.4.6 defaults to the old behavior GGA_COMPAT=.TRUE.,               the new behavior can be obtained by setting GGA_COMPAT=.FALSE.               in the INCAR file.

  • 12.08.2003: MAJOR BUG FIX in symmetry.F and paw.F:               for non-collinear calculations the symmetry routines did not work properly

   

   

   

   

• If you have read the previous lines, you will realize that it is recommended to set GGA_COMPAT=.FALSE. for non collinear calculations in VASP.4.6 and VASP.5.2, since this improves the numerical precision of GGA calculations.

   

  3.GGA_COMPAT-tag GGA_COMPAT = .TRUE. | .FALSE.

  Default
  GGA_COMPAT	=	.TRUE.

  
  
  

  For gradient corrected functionals the  exchange correlation functional might break the symmetry of the Bravais lattice slightly for non cubic cells (this includes primitive fcc and bcc lattices). The origin of this problem is subtle and relates to the fact that the gradient field breaks the lattice  symmetry for non-cubic lattices. To fix this, a spherical cutoff is applied to the gradient field for GGA_COMPAT = .FALSE., e.g. for all reciprocal lattice vectors that exceed a certain cutoff length the gradient field as well as the charge density is set to zero before calculating the exchange correlation energy and potential.  The cutoff is determined automatically so that the cutoff sphere is fully inscribed in the parallelepiped defined by the FFT grid in the reciprocal space.

  This flag restores the full lattice symmetry for gradient corrected functionals, and we therefore recommend to set   GGA_COMPAT = .FALSE.for all gradient corrected calculations. For compatibility reasons, the default is GGA_COMPAT = .TRUE. until VASP.5.2. However, setting the flag usually changes the energy only in the sub meV energy range (0.1 meV), and for most results it does matter little how GGA_COMPAT is set. The most important exception are magnetic anisotropies, for which we strongly recommend to set  GGA_COMPAT = .FALSE..

   

   

   

   

   

   

   

   

   

   

4.LNONCOLLINEAR-tag

Supported as of VASP.4.5.

Setting LNONCOLLINEAR=.TRUE. in the INCAR file allows to perform fully  non-collinear magnetic structure calculations.

VASP is capable of reading WAVECAR and CHGCAR  files from previous  non-magnetic or collinear  calculations, 【先前自洽的非磁性或共线/线性计算，是必须的】it is however not possible to rotate the magnetic field locally on selected atoms.

Hence, in practice, we recommend to perform non collinear calculations in two steps: 

（1） First, calculate the non magnetic groundstate and generate a WAVECAR and CHGCAR file.

（2）Second, read the WAVECAR and CHGCAR file, and

supply initial  magnetic moments by means of the MAGMOM tag (compare Sec. 6.13).

For a non-collinear setup, three values must be supplied for each ion in the MAGMOM line. The  three entries correspond to the initial local magnetic moment for each ion in x, y and z direction respectively.

The line

MAGMOM = 1 0 0   0 1 0

initialises the magnetic moment【矩】 on the first atom in the x-direction, and on the second atom in the y direction.

Mind, that the MAGMOM line supplies initial magnetic moments only if ICHARG=2, or if the CHGCAR file contains only charge but no magnetisation density.

5. MAGMOM

MAGMOM-tag

MAGMOM= [real array]

Specifies the initial magnetic moment for each atom, if and only if  ICHARG=2,  or if the CHGCAR file contains no magnetisation density (ICHARG=1).

If one is searching for a spin polarised (magnetic or antiferromagnetic) solution,  it is usually safest to start from larger local magnetic moments, because in  some cases, the default values might not be sufficiently big.

 A save default is usually the experimental magnetic moment multiplied by 1.2 or 1.5. It is important to emphasize that the MAGMOM tag is  used only, if the CHGCAR file holds no information on the magnetisation density, and if the initial charge density is not calculated from the orbitals supplied in  the WAVECAR file.

This means that the MAGMOM tag is useful for two kind of calculations 

（1） Calculations starting from scratch with no WAVECAR and CHGCAR file. 

（2）Calculations starting from  a non magnetic WAVECAR and CHGCAR file (ICHARG=1).[非磁性计算]

 Often such calculations converge more reliably to the desired magnetic configuration than calculations of the first kind.

Hence, if you have problems to converge to a desired magnetic solution,

try to calculate first

a . the non magnetic groundstate,

b. and continue from the generated WAVECAR and CHGCAR file.

For the continuation job, you need to set  

ISPIN=2
ICHARG=1

in the INCAR file.

Starting from VASP.4.4.4, VASP also determines whether the magnetic moments  supplied in the MAGMOM line break the symmetry.

If they do, the corresponding symmetry operations are removed  and not applied during the symmetrization of charges and forces.

This means that antiferromagnetic calculations can be performed by  specifying anti-parallel magnetic moments for the atoms in the cell MAGMOM = 1 -1 The above is as an example consider AF bcc Cr with the POSCAR file: Cr: AF 2.80000  1.00000   .00000   .00000   .00000  1.00000   .00000   .00000   .00000  1.00000  2Kartesisch   .00000   .00000   .00000   .50000   .50000   .50000

With the MAGMOM line specified above, VASP should converge to the proper groundstate.

In this example, the total net magnetisation  is matter of factly zero,

but it is possible to determine the local magnetic moments by using the RWIGS or LORBIT tags (see sections 6.346.33).

网络摘录：

计算磁各向异性的推荐步骤是：

1）首先计算线性磁矩以产生WAVECAR 和 CHGCAR文件（注意加入LMAXMIX）。

2）然后INCAR中加入：

LSORBIT = .TRUE.

ICHARG = 11           ! non selfconsistent run, read CHGCAR

                                 !或 ICHARG ==1 优化到易磁化轴，但此时应提高EDIFF的精度

LMAXMIX = 4         ! for d elements increase LMAXMIX to 4, f: LMAXMIX = 6

                                   ! you need to set LMAXMIX already in the collinear calculation

SAXIS = x y z              ! direction of the magnetic field  如 0 0  1

NBANDS = 2 * number of bands of collinear run                 ！ grep NBANDS OUTCAR

ISYM=0                         ！switch off symmetry (ISYM=0) when spin orbit coupling is selected

GGA_COMPAT=.FALSE.              ！ it improves the numerical precision of GGA for non collinear calculations

  LORBMOM=.TRUE.                       !计算轨道磁矩

 继续计算，VASP会读取WAVECAR 和 CHGCAR将自旋量子化方向（磁场方向）平行于SAXIS方向。

最后可以比较各个方向磁矩时能量的不同。

注意： 第二步使用自洽计算(ICHARG=1)原则上也是可以的，但是初始平行于SAXIS的磁场发生旋转，直到达到基态，如平行于易磁化轴，但这个过程会很慢且能量变化很小，而且如果收敛标准不是很严格的话，自洽计算会在未达到基态就停止。

注意： VASP的输入输出的磁矩和类自旋量都会按照这个SAXIS方向，包括INCAR中的

 MAGMOM行，OUTCAR和PROCAR文件中的总磁矩和局域磁矩，WAVECAR中的类自旋轨道和CHGCAR中的磁性密度。

2. 概念辨析：自旋极化、自旋轨道耦合

某些原子结构的电子不能完全配对，就会出现自旋极化。
主要是计算态密度，看在费米面附近的自旋向上和向下的态密度是否相同，如果不一样就有自旋极化。

http://zh.wikipedia.org/wiki/%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E8%80%A6%E5%90%88

在量子力学里，一个粒子因为自旋与轨道运动而产生的作用，称为自旋-轨道作用（英语：Spin–orbit interaction），自旋-轨道效应或自旋-轨道耦合。最著名的例子是电子能级的位移。电子移动经过原子核的电场时，会产生电磁作用．电子的自旋与这电磁作用的耦合，形成了自旋-轨道作用。谱线分裂实验明显地侦测到电子能级的位移，证实了自旋-轨道作用理论的正确性。另外一个类似的例子是原子核壳层模型（shell model）能级的位移。

半导体或其它新颖材料常常会涉及电子的自旋-轨道效应。自旋电子学专门研究与应用这方面的问题。

摘录：

伴自旋共舞轨道    辨凝聚区分超流

http://blog.sciencenet.cn/blog-2344-660237.html

“刻苦专研，厚积薄发，运用路径积分和格林函数等量子场论方法，深入研究冷原子气体系统的基态、激发谱和量子相变。发现自旋-轨道耦合对费米系统凝聚和超流的相反影响，引起国际同行的广泛关注。”

这是中国科学院金属研究所年终总结大会上2012年度优秀青年学者奖（基础研究类）的颁奖词。该奖颁给我们磁性材料与磁学研究部青年学者周可召副研究员。颁奖词是对他潜心科学研究，取得有影响力的研究成果的肯定。“伴自旋共舞轨道  辨凝聚区分超流”是对这项研究成果的写照。在此，借科学网这个平台再次向他表示祝贺！

中国科学院金属研究所2012年度优秀青年学者奖（基础研究类）推荐理由：

近年来他一直运用场论方法（路径积分和格林函数）研究冷原子气体系统的基态、激发谱、量子相变以及凝聚和超流现象。一个重要的进展是，研究了自旋轨道耦合（SOC）对一个费米系统凝聚和超流的影响。运用路径积分方法和Hubbard-Stratonovich变换，推出了关于配对场的有效作用量，在鞍点（平均场）近似下得到了系统的基态能量并由此推出能隙和粒子数方程。根据非对角长程序的概念，发现系统的总凝聚密度应该等于单重态和三重态的总贡献，更正了 Phys. Rev. A 84， 033633（2011）中认为凝聚密度仅含单重态的贡献这个错误。运用phase-twist方法计算了超流密度张量，这种方法与通过线性响应理论研究系统的流关联函数来计算超流密度张量的方法是一致的。由于自旋轨道耦合的存在，系统的各向异性明显地体现在超流密度变为一个张量。分析了在这样一个不满足伽利略变换不变性的系统中朗道关于超流密度计算公式的局限性。通过求解能隙和粒子数方程，得到能隙和化学势以及凝聚密度和超流密度随系统参数的变化行为。结果表明，在零温下自旋轨道耦合增强了凝聚，但抑制了超流。由于能隙以及凝聚密度均与费米面附近粒子的配对有关，因此费米面上的态密度决定了能隙（配对场）和凝聚的大小。计算发现只当自旋轨道耦合大于某个特征值时，凝聚密度才有明显地增加，与态密度的行为一致。自旋轨道耦合对超流的抑制和三重态配对场的存在紧密相关。由于自旋三重态的行为与正常态类似（零温时，自旋磁化率不为零），因此，自旋轨道耦合抑制超流是通过自发地产生三重态配对来实现的，与杂质通过耗散的方式抑制超流截然不同。以上所述现象在三维和二维中均存在。鉴于自旋轨道耦合对凝聚和超流的这种相反作用，被推荐人预言在杂质和自旋轨道耦合的共同调制下，系统将更易于出现玻色玻璃态。被推荐人发现的这种由自旋轨道耦合引起的凝聚和超流的相反行为已经引起了国际同行的很多关注。论文发表在国际著名学术刊物Phys. Rev. Lett. 108, 025301 (2012)。2011年10月在arXiv上贴出后至今，已有20次引用。

论文链接：

http://prl.aps.org/abstract/PRL/v108/i2/e025301

http://arxiv.org/abs/1110.3565

名词解释：

自旋轨道耦合：根据量子力学，一个粒子（如电子）因为自旋与轨道移动而产生的相互作用，称为自旋-轨道耦合。自旋-轨道耦合可以使电子的能级产生位移。电子移动经过原子核的电场时，会产生电磁作用，电子的自旋与这电磁作用的耦合，形成了自旋-轨道作用。类似的例子是原子核的壳层模型能级的位移。近年来，与自旋轨道耦合相关的量子自旋霍尔效应、自旋流、几何相位等是热点问题。

超流：当液氦（4He）的温度降到2.17K 时，液氦从原来的正常流体突然转变为具有一系列极不寻常的性质的“超流体”。液氦能沿极细的毛细管管径约0.1微米）流动而几乎不呈现任何粘滞性。当液氦由容器中通过多孔塞（或极细的毛细管）流出时，容器内的液氦的温度升高，为机械致热效应。其逆过程称为热机械效应，即：当升高容器内的温度时，其中液氦的液面将上升，若容器本身是一毛细管，则将观察到液氦从上口喷出，故也称喷泉效应。超流是一种宏观范围内的量子效应。由于玻色—爱因斯坦凝聚，许多分子都转到动量为零的状态，这就使得它们在坐标空间中还是在容器中的液体，而此时液体的流动性发生了突变。