## Band engineering and heterointerface ### Possible band engineering techniques Some techniques may adjust the band structures: - By stacking layers of materials (up to 15 $\mu$m), one may achieve a strained layer without creating dislocations. Such a strained layer is called "pseudomorphic". - Alloying ([[Doping and remote doping|doping]]) also shifts band gap (and band structure). - Of course, unstrained layers by MBE may be achieved for materials with the same lattice constant and crystal structure, like GaAs/AlAs material systems. ### Heterointerface Band structure will change at the interface. This may be caused by different electron affinity, or say work function, of materials — $\chi$, also different band gaps. These give the band offset. ![[Drawing 2024-09-01 16.30.07.excalidraw.svg]] The band offset given by experiment could be greatly different from the theoretical ones. The following plots show 3 types of heterointerfaces. ![[Drawing 2024-09-01 16.37.28.excalidraw.svg]] GaAs and AlAs interface belongs to type I. When crossing the interface, WF, or say, the envelope function (if we consider band extrema at $\Gamma$) $F$ should satisfy some condition: $F_c^{(A)}(\mathbf{r}) = F_c^{(B)}(\mathbf{r})$ namely the continuity of $F$, and $\mathbf{j}_A(\mathbf{r}) = \mathbf{j}_B(\mathbf{r}) $ namely $\frac{1}{m_A^*} \nabla F_c^{(A)}(\mathbf{r}) = \frac{1}{m_B^*} \nabla F_c^{(B)}(\mathbf{r})$ continuity of the probability current. >[!Note] >The continuity at interface, or say boundary conditions, can also be noticed in transport problems and electrodynamics problems. >For Maxwell equation, see [[Maxwell equation recap#Boundary conditions for waves across an interface]] >[!Example] >Now apply this in a quantum well. Consider the structure below: >![[Drawing 2024-09-01 16.52.44.excalidraw.svg]] > - B: 10 nm GaAs; > - A: Al0.3Ga0.7As; > > Schrödinger equations are written as: > $-\frac{\hbar^2}{2m_A^*} \Delta F(\mathbf{r}) = (E - {E}_c^{(A)}) F(\mathbf{r}), \quad |z| > \frac{w}{2}$ > $-\frac{\hbar^2}{2m_B^*} \Delta F(\mathbf{r}) = (E - {E}_c^{(B)}) F(\mathbf{r}), \quad |z| < \frac{w}{2}$ > Write $F(\mathbf{r})$ in the form of 2 plane waves times $\chi(z)$, > $F(\mathbf{r}) = \xi(x) \eta(y) \chi(z) = e^{ik_x x} \cdot e^{ik_y y} \chi(z)$ > $\begin{cases} \frac{\hbar^2}{2m_A^*} \left( -\frac{d^2}{dz^2} + k_{||}^2 \right) \chi(z) = (E - E_c^{(A)}) \chi(z), & \text{for } |z| > \frac{w}{2}, \\ \frac{\hbar^2}{2m_B^*} \left( -\frac{d^2}{dz^2} + k_{||}^2 \right) \chi(z) = (E - E_c^{(B)}) \chi(z), & \text{for } |z| < \frac{w}{2}. \end{cases}$ > Here, $k_{\parallel} = \sqrt{k_x^2 + k_y^2}$, defined similarly to the standard QW (quantum well) problems. > $\chi(z) = B \begin{cases} \sin(k_z z), \\ \cos(k_z z) \end{cases} \quad \text{for } |z| < \frac{w}{2}$ > And > $X(z) = A e^{-\kappa |z|} \text{ for } |z| > \frac{w}{2}$ > where, > $k_z = \sqrt{\frac{2m_B^*(E - E_c^{(B)})}{\hbar^2} - k_{||}^2}, \quad \kappa = \sqrt{\frac{2m_A^*[E_c^{(A)} - E]}{\hbar^2} + k_{||}^2}.$ > Here we assumed our interested energy: ${E}_c^{(B)} < E < {E}_c^{(A)}$. > If we solve this with the boundary condition, we eventually get the subband dispersion relation. > $E = E_n(k_{||}) + \frac{\hbar^2 k_{||}^2}{2m_B^*}$ > $E(k_{\parallel})$ is approximately parabolic with negative curvature, but dependence of $k_{\parallel}$ is quite weak (see Figure 5.6 in the book [[Semiconductor Nanostructures]]). > ![[Drawing 2024-09-01 17.17.38.excalidraw.svg]] > And the overall dependence is parabolic, for both $k_x$ and $k_y$ (so, also $k_{\parallel}$). This gives the constant energy surface $k_x, k_{\parallel}$, as shown in the figure, also the DOS (Density of States) as a step-like function, if multiple sub-bands are taken into account. > If only one sub-band is occupied (i.e., the energetically lowest one), we say it's in the quantum limit. In this case, the DOS is a constant, > $D(E)=\frac{m^{*}}{\pi \hbar^{2}}$ > This expression assumes spin degeneracy, otherwise one should add $g$ factors, namely $D(E)=\frac{g_{s}g_{v}m^{*}}{\pi \hbar^{2}}$. > 2DEG will be extensively discussed in later chapters. For 2D hole gas, more complicated phenomena exist, like mass inversion and state mix. Check the book chapter for more information.