10.4 Electronic State Calculation of the Surface where Heavy Metals are Laid on Si(111)(シリコン(111)表面 重元素積層表面の電子状態計算)

Types of Layered Surfaces, Symmetry, Generating Elements of the Space Group

積層表面の種類、対称性、空間群の生成元

surface structure	space group	generators
$1\times 1$Surface	P 3 m 1 C$_{3v}^{1}$ (No. 156)	C3+, IC212(Reflection in the yz-Plane：yz面での鏡映)
$\sqrt{3}\times \sqrt{3}$Surface	P 3 1 m C$_{3v}^{2}$ (No. 157)	C3+, IC211(Reflection in the xz-Plane：xz面での鏡映)

Lattice Constant of Basement Stratum :

基板層の格子定数 :

7.25562268 a.u.=3.8395086 Å= $\frac{1}{\sqrt{2}}$ (5.429885 Å) = $\frac{1}{\sqrt{2}}$ (10.26 a.u.)

Surface with ESM (Effective Screening Medium) Model :

ESM(有効遮蔽媒質)モデルを用いた表面 :

(Otani and Sugino, PRB,2006,73,115407[56])

Atomic Positions in the Unit Cell $1\times 1$ Surface :

原子配置(単位胞内の原子位置) $1\times 1$表面 :
Slab Model with 2 Si Layers：X-Si(111)$_{2}$-H (For Test Calculation)
Slab Model with 8 Si Layers：X-Si(111)$_{8}$-H
Slab Model with 16 Si Layers：X-Si(111)$_{16}$-H
Slab Model with 24 Si layers：X-Si(111)$_{24}$-H
Slab Model with 48 Si layers：X-Si(111)$_{48}$-H

Si 2 Layers, X=Tl (Thallium)
:

シリコン2層、X=Tl(タリウム)のとき

cellmd0
4 7.25562268 0.0000 3.262045268 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81 1 13.0 1.0 204.37 2.5 2.5 NZA NA ZV RCMAX PMASS RATS RATS1
     0.000000     2.216741     6.926633 1 0.0 0.0 0.0
14 2 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1 
     0.000000     0.000000     4.656633 1 0.0 0.0 0.0
     0.000000     2.216741     3.918682 0 0.0 0.0 0.0
1 1 1.00 0.800 1.000 0.80 0.80  NZA NA ZV RCMAX PMASS RATS RATS1
     0.000000     2.216741     2.401982 0 0.0 0.0 0.0
esm.dat
0  8.10                   jtresm tresm
11.83408481 -11.83408481  z1esm z2esm
0.00  0.00                vesmp vesmm
0.00  0.00                eesmp eesmm

Si 8 Layers, X=Tl (Thalium), without Symmetry, Optimization with LDA
:

シリコン8層、X=Tl(タリウム)のとき(対称性を考慮していない、LDAでの最適化)

celldm0
4 7.25562268 0.0000 6.12372436 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81 1 13.0 1.0 204.37 2.5 2.5 NZA NA ZV RCMAX PMASS RATS RATS1
     0.002210     2.221417    16.211970 1 0.0 0.0 0.0
14 8 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1
    -0.005615     0.006414    14.051625 1 0.0 0.0 0.0
    -0.004728     2.222474    13.182383 1 0.0 0.0 0.0
    -0.000425     2.219121    10.850667 1 0.0 0.0 0.0
     1.919991     1.110267    10.087031 1 0.0 0.0 0.0
     1.919225     1.109942     7.750835 1 0.0 0.0 0.0
     0.000098     0.001101     6.987061 1 0.0 0.0 0.0
    -0.000584     0.000447     4.656690 1 0.0 0.0 0.0
     0.000000     2.216741     3.918682 0 0.0 0.0 0.0
1 1 1.00 0.800 1.000 0.80 0.80  NZ NA ZV RCMAX PMASS RATS RATS1
     0.000000     2.216741     2.401982 0 0.0 0.0 0.0
esm.dat
0  20.0            jtresm tresm
22.2157  -22.2157  z1esm z2esm
0.00 -0.00         vesmp vesmm
0.00 -0.00         eesmp eesmm

Si 16 Layers, X=Tl (Thalium) without Symmetry
:

シリコン16層、X=Tl(タリウム)のとき(ただし対称性を考慮しないもの)

celldm0
4 7.25562268 0.0000 9.38971068 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81 1 13.0 1.0 204.37 2.5 2.5 NZA NA ZV RCMAX PMASS RATS RATS1
     0.001034     2.218402    28.892452 1 0.0 0.0 0.0
14 16 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1
     0.000180     0.000093    26.654440 1 0.0 0.0 0.0
     0.000133     2.216747    25.778368 1 0.0 0.0 0.0
     0.000063     2.217026    23.428419 1 0.0 0.0 0.0
     1.919839     1.108659    22.654277 1 0.0 0.0 0.0
     1.919870     1.108655    20.301335 1 0.0 0.0 0.0
     0.000129     0.000284    19.516598 1 0.0 0.0 0.0
     0.000232     0.000388    17.166724 1 0.0 0.0 0.0
     0.000235     2.217116    16.388417 1 0.0 0.0 0.0
     0.000152     2.216986    14.041783 1 0.0 0.0 0.0
     1.919908     1.108594    13.263860 1 0.0 0.0 0.0
     1.919869     1.108568    10.918207 1 0.0 0.0 0.0
     0.000129     0.000191    10.141660 1 0.0 0.0 0.0
     0.000057     0.000115     7.795737 1 0.0 0.0 0.0
     0.000062     2.216838     7.013195 1 0.0 0.0 0.0
    -0.000003     2.216762     4.665858 1 0.0 0.0 0.0
     1.919754     1.108371     3.918682 0 0.0 0.0 0.0
1 1 1.00 0.800 1.000 0.80 0.80  NZA NA ZV RCMAX PMASS RATS RATS1
     1.919754     1.108371     2.401982 0 0.0 0.0 0.0
esm.dat
0  31.0            jtresm tresm
34.0640  -34.0640  z1esm z2esm
0.00 -0.00         vesmp vesmm
0.00 -0.00         eesmp eesmm

Si 24 Layers, X=Tl (Thalium)
:

シリコン24層、X=Tl(タリウム)のとき

celldm0
4 7.25562268 0.0000 12.655697 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81 1 13.0 1.0 204.37 2.5 2.5 NZA NA ZV RCMAX PMASS RATS RATS1
     0.000000     2.216741    41.448226 1  0.0 0.0 0.0
14 24 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1
     0.000000     0.000000    39.179795 1  0.0 0.0 0.0
     0.000000     2.216741    38.304007 1  0.0 0.0 0.0
     0.000000     2.216741    35.944652 1  0.0 0.0 0.0
     1.919754     1.108371    35.162001 1  0.0 0.0 0.0
     1.919754     1.108371    32.805119 1  0.0 0.0 0.0
     0.000000     0.000000    32.021616 1  0.0 0.0 0.0
     0.000000     0.000000    29.669066 1  0.0 0.0 0.0
     0.000000     2.216741    28.885720 1  0.0 0.0 0.0
     0.000000     2.216741    26.534154 1  0.0 0.0 0.0
     1.919754     1.108371    25.752551 1  0.0 0.0 0.0
     1.919754     1.108371    23.402703 1  0.0 0.0 0.0
     0.000000     0.000000    22.621308 1  0.0 0.0 0.0
     0.000000     0.000000    20.271191 1  0.0 0.0 0.0
     0.000000     2.216741    19.490612 1  0.0 0.0 0.0
     0.000000     2.216741    17.141383 1  0.0 0.0 0.0
     1.919754     1.108371    16.361210 1  0.0 0.0 0.0
     1.919754     1.108371    14.012111 1  0.0 0.0 0.0
     0.000000     0.000000    13.232278 1  0.0 0.0 0.0
     0.000000     0.000000    10.885493 1  0.0 0.0 0.0
     0.000000     2.216741    10.109400 1  0.0 0.0 0.0
     0.000000     2.216741     7.766571 1  0.0 0.0 0.0
     1.919754     1.108371     6.993588 1  0.0 0.0 0.0
     1.919754     1.108371     4.656982 1  0.0 0.0 0.0
     0.000000     0.000000     3.918682 0  0.0 0.0 0.0
1 1 1.00 0.800 1.000 0.80 0.80  NZA NA ZV RCMAX PMASS RATS RATS1
     0.000000     0.000000     2.401982 0  0.0 0.0 0.0
esm.dat
0  43.0            jtresm tresm
45.9124  -45.9124  z1esm z2esm
0.00 -0.00         vesmp vesmm
0.00 -0.00         eesmp eesmm

Si 48 Layers, X=Tl (Thalium)
:

シリコン48層、X=Tl(タリウム)のとき

celldm0
4 7.25562268 0.0000 22.39726382 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81 1 13.0 1.0 204.37 2.5 2.5 NZA NA ZV RCMAX PMASS RATS RATS1
       0.000000000      2.216741349     78.851104499 1 0.0 0.0 0.0
14 48 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1
       0.000000000      0.000000000     76.581167456 1 0.0 0.0 0.0
       0.000000000      2.216741349     75.709655439 1 0.0 0.0 0.0
       0.000000000      2.216741349     73.347939394 1 0.0 0.0 0.0
       1.919754321      1.108370674     72.562208379 1 0.0 0.0 0.0
       1.919754321      1.108370674     70.208787335 1 0.0 0.0 0.0
       0.000000000      0.000000000     69.423731320 1 0.0 0.0 0.0
       0.000000000      0.000000000     67.071444275 1 0.0 0.0 0.0
       0.000000000      2.216741349     66.289127260 1 0.0 0.0 0.0
       0.000000000      2.216741349     63.934613215 1 0.0 0.0 0.0
       1.919754321      1.108370674     63.156533201 1 0.0 0.0 0.0
       1.919754321      1.108370674     60.805350156 1 0.0 0.0 0.0
       0.000000000      0.000000000     60.022928141 1 0.0 0.0 0.0
       0.000000000      0.000000000     57.675122096 1 0.0 0.0 0.0
       0.000000000      2.216741349     56.891510081 1 0.0 0.0 0.0
       0.000000000      2.216741349     54.544154037 1 0.0 0.0 0.0
       1.919754321      1.108370674     53.765585022 1 0.0 0.0 0.0
       1.919754321      1.108370674     51.414474977 1 0.0 0.0 0.0
       0.000000000      0.000000000     50.633779963 1 0.0 0.0 0.0
       0.000000000      0.000000000     48.287721918 1 0.0 0.0 0.0
       0.000000000      2.216741349     47.512390903 1 0.0 0.0 0.0
       0.000000000      2.216741349     45.169375859 1 0.0 0.0 0.0
       1.919754321      1.108370674     44.396014844 1 0.0 0.0 0.0
       1.919754321      1.108370674     42.059463800 1 0.0 0.0 0.0
       0.000000000      0.000000000     41.321512785 1 0.0 0.0 0.0
       0.000000000      0.000000000     38.973706771 1 0.0 0.0 0.0
       0.000000000      2.216741349     38.190094741 1 0.0 0.0 0.0
       0.000000000      2.216741349     35.842738651 1 0.0 0.0 0.0
       1.919754321      1.108370674     35.064169623 1 0.0 0.0 0.0
       1.919754321      1.108370674     32.713059533 1 0.0 0.0 0.0
       0.000000000      0.000000000     31.932364503 1 0.0 0.0 0.0
       0.000000000      0.000000000     29.586306413 1 0.0 0.0 0.0
       0.000000000      2.216741349     28.810975385 1 0.0 0.0 0.0
       0.000000000      2.216741349     26.467960295 1 0.0 0.0 0.0
       1.919754321      1.108370674     25.694599265 1 0.0 0.0 0.0
       1.919754321      1.108370674     23.358048177 1 0.0 0.0 0.0
       0.000000000      0.000000000     22.620097149 1 0.0 0.0 0.0
       0.000000000      0.000000000     20.272291089 1 0.0 0.0 0.0
       0.000000000      2.216741349     19.488679044 1 0.0 0.0 0.0
       0.000000000      2.216741349     17.141322911 1 0.0 0.0 0.0
       1.919754321      1.108370674     16.362753867 1 0.0 0.0 0.0
       1.919754321      1.108370674     14.011643732 1 0.0 0.0 0.0
       0.000000000      0.000000000     22.620097149 1 0.0 0.0 0.0
       0.000000000      0.000000000     20.272291089 1 0.0 0.0 0.0
       0.000000000      2.216741349     19.488679044 1 0.0 0.0 0.0
       0.000000000      2.216741349     17.141322911 1 0.0 0.0 0.0
       1.919754321      1.108370674     16.362753867 1 0.0 0.0 0.0
       1.919754321      1.108370674     14.011643732 1 0.0 0.0 0.0
       0.000000000      0.000000000     13.230948689 1 0.0 0.0 0.0
       0.000000000      0.000000000     10.884890554 1 0.0 0.0 0.0
       0.000000000      2.216741349     10.109559510 1 0.0 0.0 0.0
       0.000000000      2.216741349      7.766544377 1 0.0 0.0 0.0
       1.919754321      1.108370674      6.993183332 1 0.0 0.0 0.0
       1.919754321      1.108370674      4.656632200 1 0.0 0.0 0.0
       0.000000000      0.000000000      3.918681156 0 0.0 0.0 0.0
1 1 1.00 0.800 1.000 0.80 0.80  NZA NA ZV RCMAX PMASS RATS RATS1
       0.000000000      0.000000000         2.401982 0 0.0 0.0 0.0
esm.dat
0  77.0                jtresm tresm
81.253047  -81.253047  z1esm z2esm
0.00 -0.00             vesmp vesmm
0.00 -0.00             eesmp eesmm

Atomic Positions in the Unit Cell $\sqrt{3}\times \sqrt{3}$ Surface
Slab Model with 2 Si Layers：X-Si(111)$_{2}$-H (No Test)
Slab Model with 8 Si Layers：X-Si(111)$_{8}$-H

原子配置(単位胞内の原子位置) $\sqrt{3}\times \sqrt{3}$表面
シリコン2層スラブモデル：X-Si(111)$_{2}$-H (テスト計算用なし)
シリコン8層スラブモデル：X-Si(111)$_{8}$-H

Si 8 Layers, X=Tl (Thalium)

シリコン8層、X=Tl(タリウム)のとき

celldm0
4 12.56710713 0.0000 3.53541985 0.0000 0.0000 0.0000
atomic coordinate in sample.dat.sh
81  1 13.0 1.0 204.37 2.5  2.5    NZA NA ZV RCMAX PMASS RATS RATS1
       0.000000000      0.000000000     16.480658773  1  0.0 0.0 0.0
14 24 4.00 1.0 28.086 1.75 1.75   NZ NA ZV RCMAX PMASS RATS RATS1
       1.108465436     -1.919354453     14.210722475  1  0.0 0.0 0.0
       4.433577585      0.000399941     14.210722475  1  0.0 0.0 0.0
       1.108465436      1.920154337     14.210722475  1  0.0 0.0 0.0
       0.000101380      0.000224405     13.339210442  1  0.0 0.0 0.0
       3.325213529      1.919978799     13.339210442  1  0.0 0.0 0.0
       0.000101380      3.839733195     13.339210442  1  0.0 0.0 0.0
      -0.000157749      0.000451229     10.977494352  1  0.0 0.0 0.0
       3.324954400      1.920205624     10.977494352  1  0.0 0.0 0.0
      -0.000157749      3.839960019     10.977494352  1  0.0 0.0 0.0
       2.216605451      0.000466885     10.191763322  1  0.0 0.0 0.0
       5.541717600      1.920221280     10.191763322  1  0.0 0.0 0.0
       2.216605451      3.839975675     10.191763322  1  0.0 0.0 0.0
       2.216651207      0.000525633      7.838342233  1  0.0 0.0 0.0
       5.541763356      1.920280028      7.838342233  1  0.0 0.0 0.0
       2.216651207      3.840034423      7.838342233  1  0.0 0.0 0.0
       1.108310850     -1.919234702      7.053286203  1  0.0 0.0 0.0
       4.433422999      0.000519691      7.053286203  1  0.0 0.0 0.0
       1.108310850      1.920274088      7.053286203  1  0.0 0.0 0.0
       1.108340154     -1.919297458      4.700999113  1  0.0 0.0 0.0
       4.433452303      0.000456936      4.700999113  1  0.0 0.0 0.0
       1.108340154      1.920211332      4.700999113  1  0.0 0.0 0.0
       0.000000000      0.000000000      3.918682083  0  0.0 0.0 0.0
       3.325108510      1.920220699      3.918682083  0  0.0 0.0 0.0
       0.000000000      3.839975094      3.918682083  0  0.0 0.0 0.0
1 3 1.00 0.800 1.000 0.80 0.80  NZA NA ZV RCMAX PMASS RATS RATS1
       0.000000000      0.000000000      2.401982056  0  0.0 0.0 0.0
       3.325108510      1.920220699      2.401982056  0  0.0 0.0 0.0
       0.000000000      3.839975094      2.401982056  0  0.0 0.0 0.0
esm.dat
0  18.0          jtresm tresm
22.215  -22.215  z1esm z2esm
0.00 -0.00       vesmp vesmm
0.00 -0.00       eesmp eesmm

Convergence of the Electron System :

電子系の収束 :

ex_opte.sh

Optimization of Atomic Positions :

原子位置の最適化 :

ex_opts.sh

Calculation of Dispersion Relations :

分散関係の計算 :

ex_ek.sh
Dispersion Relations :
分散関係 :

(ekdispup.xy, ekdisp)
Direction of Spin-Polarization of Band Electrons :
バンド電子のスピン分極方向 :

(eksurfspn.xy, eksurfchg)
Expected Value of Orbital Angular Momentum at Each Atomic Position of Band Electrons
バンド電子の各原子位置での軌道角運動量の期待値(軌道分局方向) :

(eklave.xy, lave.ek)

Output of lave.ek(lave.ekの出力)

The magnetic momoment and orbital angular momentum are output with respect to each k-point, each orbital and each atom.

各k点各軌道各原子毎に磁気モーメントおよび軌道角運動量を出力する

n,mx,my,mz
lx, ly, lz

They are output in the order as shown below.

の順に出力されている。

Physical Quantities Related to Rashba Effect :

ラシュバ効果に関する物理量 :

ex_bchg.sh

(1)

Electron Density Distribution and Spin Density Distribution in Depth Direction of Surface

(表面深さ方向の電子密度分布、スピン密度分布)

$$\rho_{\alpha}(z) = \frac{1}{S_{xy}} \int \rho_{\alpha}(x,y,z) dx dy$$

Here, there are four types of $\alpha$, in other words, $n$, $\hat{z}$, $\vec{k}$ and $\hat{z}\times\vec{k}$. $n$ represents the electron density, $\hat{z}$ represents the direction which is outward perpendicular to the surface, $\hat{k}$ represents the direction of two-dimensional wave vector and $\hat{z}\times\vec{k}$ represents the direction of Rashba spin, however counterclockwise rotation in $xy$-plane is positice. By setting JRHOZ in ex_bchg.sh to 1, this calculation is performed. After finishing this calculatio, the data for $gnuplot$ is output.

ここで$\alpha$には $n, \hat{z}, \vec{k}, \hat{z}\times\vec{k}$の4種類がある。$n$は電子密度を表し、$\hat{z}$は 表面外向き面直、$\hat{k}$は2次元波数ベクトル方向、 $\hat{z}\times\vec{k}$ラシュバスピン方向($xy$面内反時計まわりが正) を表す。ex_bchg.sh内のオプションJRHOZ=1として計算することができる。$gnuplot$用の データが出力される。

(2)

Distribution of Electron Density or Three-Dimensional Spin Density

(電子密度あるいはスピン密度の3次元分布)

By setting JRHOHEX to 1 in ex_bchg.sh, you can perform these calculation. $\rho_{\alpha}(x,y,z)$ are output in the form of xsf for XCrySDen. You can calculate four types, in other words, $n$, $m_{x}$, $m_{y}$ and $m_{z}$.

ex_bchg.sh内のオプションJRHOHEX=1として計算することができる。 $\rho_{\alpha}(x,y,z)$をXcrysdenの xsf形式で出力される。このとき$\alpha$は、 $n, m_{x}, m_{y}, m_{z}$の4種類を計算することができる。

(3)

Spin directions of band electron in constant energy are displayed with two-dimentional wave number space. To output these data, you have to calculate ex_eksp.sh rather than ex_ek.sh. The output of data is performed in ekspplot.x. The output data is in the form of xsf for XCrysDen. The data used for describing a constant energy curve by $gnuplot$ is also output.

エネルギー一定でのバンド電子のスピン方向を2次元波数空間で表示する。これを出力するためには、あらかじめ ex_ek.shではなくex_eksp.shを計算する。(データの出力はekspplot.x内で行われる)出力データは、 Xcrysdenのxsf形式のデータである。2次元面内のエネルギー一定の曲線を、gnuplot等で描く曲線も出力される。