PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Zincblende is a cubic crystal structure that is a similar to diamond. The basis consists of one Zn atom at (0,0,0) and one S atom at (¼,¼,¼). These two atoms are repeated on an fcc bravais lattice to form the crystal structure. If all of the atoms were of the same type, it would be the diamond crystal structure. Many semiconductors such as GaAs, InAs, BN, AlAs, AlP, AlSb, CdS, CdSe, CdTe, GaP, GaSb, HgTe, InP, and InSb form the zincblende structure. Often the bonds between the atoms are partly ionic and there is some charge transfer between the atoms. The Madelung constant for zincblende is $\alpha=1.638$.
Crystal structure: Zincblende
Bravais lattice: Face-centered cubic (cF)
Space group: 216 ($F\overline{4}3m$),
Point group: $\overline{4}3m = T_d$ three 2-fold rotations, four 3-fold rotations, six mirror planes, no inversion
The lattice parameters are:
\[ \begin{equation} a = b = c, \hspace{0.5cm} \alpha = 90^{\circ} , \,\beta = 90^{\circ} ,\,\gamma = 90^{\circ} . \end{equation} \]The primitive lattice vectors are,
$\vec{a}_1=\frac{a}{2}\hat{x}+\frac{a}{2}\hat{y}$, $\vec{a}_2=\frac{a}{2}\hat{x}+\frac{a}{2}\hat{z}$, $\vec{a}_3=\frac{a}{2}\hat{y}+\frac{a}{2}\hat{z}$.
There are two atoms in the basis. In fractional coordinates of the conventional unit, the positions of the two atoms are,
$\vec{B}_{\text{Zn}}=(0,0,0)$, $\vec{B}_{\text{S}}=(0.25,0.25,0.25)$.
The asymmetric unit is the same as the basis.
The conventional unit cell is cubic and the lattice vectors are,
\begin{equation} \vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z}. \end{equation}Space group: 216 ($F\overline{4}3m$)
96 symmetry operations:
1 x,y,z identity 2 x,-y,-z C2 axis 3 -x,y,-z C2 axis 4 -x,-y,z C2 axis 5 y,z,x C3 axis 6 y,-z,-x C3 axis 7 -y,z,-x C3 axis 8 -y,-z,x C3 axis 9 z,x,y C3 axis 10 z,-x,-y C3 axis 11 -z,x,-y C3 axis 12 -z,-x,y C3 axis 13 y,x,z mirror plane 14 y,-x,-z 4-bar axis|inversion center at 0 0 0 15 -y,x,-z 4-bar axis|inversion center at 0 0 0 16 -y,-x,z mirror plane 17 x,z,y mirror plane 18 x,-z,-y mirror plane 19 -x,z,-y 4-bar axis|inversion center at 0 0 0 20 -x,-z,y 4-bar axis|inversion center at 0 0 0 21 z,y,x mirror plane 22 z,-y,-x 4-bar axis|inversion center at 0 0 0 23 -z,y,-x mirror plane 24 -z,-y,x 4-bar axis|inversion center at 0 0 0 25 x,y+1/2,z+1/2 translation: 0 1/2 1/2 26 x,-y+1/2,-z+1/2 C2 axis 27 -x,y+1/2,-z+1/2 2-fold screw axis|translation: 0 1/2 0 28 -x,-y+1/2,z+1/2 2-fold screw axis|translation: 0 0 1/2 29 y,z+1/2,x+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 30 y,-z+1/2,-x+1/2 C3 axis 31 -y,z+1/2,-x+1/2 3-fold screw axis|translation: -1/3 1/3 1/3 32 -y,-z+1/2,x+1/2 C3 axis 33 z,x+1/2,y+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 34 z,-x+1/2,-y+1/2 C3 axis 35 -z,x+1/2,-y+1/2 C3 axis 36 -z,-x+1/2,y+1/2 3-fold screw axis|translation: -1/3 1/3 1/3 37 y,x+1/2,z+1/2 g-glide plane|translation: 1/4 1/4 1/2 38 y,-x+1/2,-z+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 39 -y,x+1/2,-z+1/2 4-bar axis|inversion center at -1/4 1/4 1/4 40 -y,-x+1/2,z+1/2 g-glide plane|translation: -1/4 1/4 1/2 41 x,z+1/2,y+1/2 g-glide plane|translation: 0 1/2 1/2 42 x,-z+1/2,-y+1/2 mirror plane 43 -x,z+1/2,-y+1/2 4-bar axis|inversion center at 0 1/2 0 44 -x,-z+1/2,y+1/2 4-bar axis|inversion center at 0 0 1/2 45 z,y+1/2,x+1/2 g-glide plane|translation: 1/4 1/2 1/4 46 z,-y+1/2,-x+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 47 -z,y+1/2,-x+1/2 g-glide plane|translation: -1/4 1/2 1/4 48 -z,-y+1/2,x+1/2 4-bar axis|inversion center at -1/4 1/4 1/4 49 x+1/2,y,z+1/2 translation: 1/2 0 1/2 50 x+1/2,-y,-z+1/2 2-fold screw axis|translation: 1/2 0 0 51 -x+1/2,y,-z+1/2 C2 axis 52 -x+1/2,-y,z+1/2 2-fold screw axis|translation: 0 0 1/2 53 y+1/2,z,x+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 54 y+1/2,-z,-x+1/2 C3 axis 55 -y+1/2,z,-x+1/2 C3 axis 56 -y+1/2,-z,x+1/2 3-fold screw axis|translation: 1/3 -1/3 1/3 57 z+1/2,x,y+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 58 z+1/2,-x,-y+1/2 3-fold screw axis|translation: 1/3 -1/3 1/3 59 -z+1/2,x,-y+1/2 C3 axis 60 -z+1/2,-x,y+1/2 C3 axis 61 y+1/2,x,z+1/2 g-glide plane|translation: 1/4 1/4 1/2 62 y+1/2,-x,-z+1/2 4-bar axis|inversion center at 1/4 -1/4 1/4 63 -y+1/2,x,-z+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 64 -y+1/2,-x,z+1/2 g-glide plane|translation: 1/4 -1/4 1/2 65 x+1/2,z,y+1/2 g-glide plane|translation: 1/2 1/4 1/4 66 x+1/2,-z,-y+1/2 g-glide plane|translation: 1/2 -1/4 1/4 67 -x+1/2,z,-y+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 68 -x+1/2,-z,y+1/2 4-bar axis|inversion center at 1/4 -1/4 1/4 69 z+1/2,y,x+1/2 g-glide plane|translation: 1/2 0 1/2 70 z+1/2,-y,-x+1/2 4-bar axis|inversion center at 1/2 0 0 71 -z+1/2,y,-x+1/2 mirror plane 72 -z+1/2,-y,x+1/2 4-bar axis|inversion center at 0 0 1/2 73 x+1/2,y+1/2,z translation: 1/2 1/2 0 74 x+1/2,-y+1/2,-z 2-fold screw axis|translation: 1/2 0 0 75 -x+1/2,y+1/2,-z 2-fold screw axis|translation: 0 1/2 0 76 -x+1/2,-y+1/2,z C2 axis 77 y+1/2,z+1/2,x 3-fold screw axis|translation: 1/3 1/3 1/3 78 y+1/2,-z+1/2,-x 3-fold screw axis|translation: 1/3 1/3 -1/3 79 -y+1/2,z+1/2,-x C3 axis 80 -y+1/2,-z+1/2,x C3 axis 81 z+1/2,x+1/2,y 3-fold screw axis|translation: 1/3 1/3 1/3 82 z+1/2,-x+1/2,-y C3 axis 83 -z+1/2,x+1/2,-y 3-fold screw axis|translation: 1/3 1/3 -1/3 84 -z+1/2,-x+1/2,y C3 axis 85 y+1/2,x+1/2,z g-glide plane|translation: 1/2 1/2 0 86 y+1/2,-x+1/2,-z 4-bar axis|inversion center at 1/2 0 0 87 -y+1/2,x+1/2,-z 4-bar axis|inversion center at 0 1/2 0 88 -y+1/2,-x+1/2,z mirror plane 89 x+1/2,z+1/2,y g-glide plane|translation: 1/2 1/4 1/4 90 x+1/2,-z+1/2,-y g-glide plane|translation: 1/2 1/4 -1/4 91 -x+1/2,z+1/2,-y 4-bar axis|inversion center at 1/4 1/4 -1/4 92 -x+1/2,-z+1/2,y 4-bar axis|inversion center at 1/4 1/4 1/4 93 z+1/2,y+1/2,x g-glide plane|translation: 1/4 1/2 1/4 94 z+1/2,-y+1/2,-x 4-bar axis|inversion center at 1/4 1/4 -1/4 95 -z+1/2,y+1/2,-x g-glide plane|translation: 1/4 1/2 -1/4 96 -z+1/2,-y+1/2,x 4-bar axis|inversion center at 1/4 1/4 1/4