PHY.K02UF Molecular and Solid State Physics

Phonon dispersion of Zincblende Crystals

Zincblende has an fcc Bravais lattice and there are two atoms in the basis. Examples of crystals with the zincblende structure are GaAs, InP, and ZnS. The diamond structure is closely related to zincblende. While zincblende has two different atoms in the basis, for diamond, the two atoms in the basis are the same. Otherwise diamond and zincblende are the same. The fcc primitive lattice vectors are,

\begin{equation} \vec{a}_1=\frac{a}{2}(\hat{x}+\hat{z}),\quad \vec{a}_2=\frac{a}{2}(\hat{x}+\hat{y}),\quad\vec{a}_3=\frac{a}{2}(\hat{y}+\hat{z}), \end{equation}

and the basis is S (0,0,0) and Zn (0.25,0.25,0.25). Each S atom has Zn nearest neighbor As atoms and each Zn atom has four nearest neighbor S atoms. Let $u^x_{lmn}$ the displacement of S atom in the unit cell at position $\vec{r}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ in the $x$-direction from its equilibrium position and $v^x_{lmn}$ the displacement of Zn atom in the unit cell at position $\vec{r}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ in the $x$-direction from its equilibrium position. If the restoring force that pushes the atoms back to their equilibrium positions is modelled with linear springs with spring constant $C$, the equations of motion are,

\begin{equation} \begin{split} M_{\text{S}} \frac {d^{2}u^{x}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (v^{x}_{lmn} - u^{x}_{lmn})+ (v^{x}_{(l-1)mn} - u^{x}_{lmn})+ (v^{x}_{l(m-1)n} - u^{x}_{lmn}) + (v^{x}_{lm(n-1)} - u^{x}_{lmn})\\ &+ (v^{y}_{l(m-1)n} - u^{y}_{lmn}) + (v^{y}_{lmn} - u^{y}_{lmn}) - (v^{y}_{(l-1)mn} - u^{y}_{lmn}) - (v^{y}_{lm(n-1)} - u^{y}_{lmn})\\ &+ (v^{z}_{(l-1)mn}- u^{z}_{lmn})+ (v^{z}_{l(m-1)n} - u^{z}_{lmn})-(v^{z}_{lmn} - u^{z}_{lmn})- (v^{z}_{lm(n-1)} - u^{z}_{lmn})\Big] \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{S}} \frac {d^{2}u^{y}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (v^{y}_{lmn} - u^{y}_{lmn})+ (v^{y}_{(l-1)mn} - u^{y}_{lmn})+ (v^{y}_{l(m-1)n} - u^{y}_{lmn}) + (v^{y}_{lm(n-1)} - u^{y}_{lmn})\\ &+ (v^{x}_{l(m-1)n} - u^{x}_{lmn}) + (v^{x}_{lmn} - u^{x}_{lmn}) - (v^{x}_{(l-1)mn} - u^{x}_{lmn}) - (v^{x}_{lm(n-1)} - u^{x}_{lmn})\\ &+ (v^{z}_{lmn} - u^{z}_{lmn}) + (v^{z}_{lm(n-1)} - u^{z}_{lmn}) - (v^{z}_{(l-1)mn}- u^{z}_{lmn}) - (v^{z}_{l(m-1)n} - u^{z}_{lmn})\Big] \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{S}} \frac {d^{2}u^{z}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (v^{z}_{lmn} - u^{z}_{lmn})+ (v^{z}_{(l-1)mn} - u^{z}_{lmn})+ (v^{z}_{l(m-1)n} - u^{z}_{lmn}) + (v^{z}_{lm(n-1)} - u^{z}_{lmn})\\ &+ (v^{x}_{(l-1)mn}- u^{x}_{lmn})+ (v^{x}_{l(m-1)n} - u^{x}_{lmn})-(v^{x}_{lmn} - u^{x}_{lmn})- (v^{x}_{lm(n-1)} - u^{x}_{lmn})\\ &+ (v^{y}_{lm(n-1)} - u^{y}_{lmn}) + (v^{y}_{lmn} - u^{y}_{lmn}) - (v^{y}_{l(m-1)n} - u^{y}_{lmn})- (v^{y}_{(l-1)mn} - u^{y}_{lmn})\Big] \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Zn}} \frac {d^{2}v^{x}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (u^{x}_{lmn} - v^{x}_{lmn})+ (u^{x}_{(l+1)mn} - v^{x}_{lmn})+ (u^{x}_{l(m+1)n} - v^{x}_{lmn}) + (u^{x}_{lm(n+1)} - v^{x}_{lmn})\\ &+ (u^{y}_{l(m+1)n} - v^{y}_{lmn}) + (u^{y}_{lmn} - v^{y}_{lmn}) - (u^{y}_{(l+1)mn} - v^{y}_{lmn}) - (u^{y}_{lm(n+1)} - v^{y}_{lmn})\\ &+ (u^{z}_{(l+1)mn}- v^{z}_{lmn})+ (u^{z}_{lmn} - v^{z}_{lmn})-(u^{z}_{l(m+1)n} - v^{z}_{lmn})- (u^{z}_{lm(n+1)} - v^{z}_{lmn})\Big] \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Zn}} \frac {d^{2}v^{y}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (u^{y}_{lmn} - v^{y}_{lmn})+ (u^{y}_{(l+1)mn} - v^{y}_{lmn})+ (u^{y}_{l(m+1)n} - v^{y}_{lmn}) + (u^{y}_{lm(n+1)} - v^{y}_{lmn})\\ &+ (u^{x}_{l(m+1)n} - v^{x}_{lmn}) + (u^{x}_{lmn} - v^{x}_{lmn}) - (u^{x}_{(l+1)mn} - v^{x}_{lmn}) - (u^{x}_{lm(n+1)} - v^{x}_{lmn})\\ &+ (u^{z}_{lmn} - v^{z}_{lmn}) + (u^{z}_{lm(n+1)} - v^{z}_{lmn}) - (u^{z}_{(l+1)mn}- v^{z}_{lmn}) - (u^{z}_{l(m+1)n} - v^{z}_{lmn})\Big] \end{split} \end{equation} \begin{equation} \begin{split} M_{\text{Zn}} \frac {d^{2}v^{z}_{lmn}}{dt^{2}} & = \frac{C}{3}\Big[ (u^{z}_{lmn} - v^{z}_{lmn})+ (u^{z}_{(l+1)mn} - v^{z}_{lmn})+ (u^{z}_{l(m+1)n} - v^{z}_{lmn}) + (u^{z}_{lm(n+1)} - v^{z}_{lmn})\\ &+ (u^{x}_{(l+1)mn}- v^{x}_{lmn})+ (u^{x}_{l(m+1)n} - v^{x}_{lmn})-(u^{x}_{lmn} - v^{x}_{lmn})- (u^{x}_{lm(n+1)} - v^{x}_{lmn})\\ &+ (u^{y}_{lm(n+1)} - v^{y}_{lmn}) + (u^{y}_{lmn} - v^{y}_{lmn}) - (u^{y}_{l(m+1)n} - v^{y}_{lmn})- (u^{y}_{(l+1)mn} - v^{y}_{lmn})\Big] \end{split} \end{equation}

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