## Metal electrodes on semiconductor surfaces In this section, we'll have two contacts, Schottky contacts and Ohmic contacts. ### Schottky contacts Schottky contacts can be made by depositing/evaporating a thin film of metal (aluminum, for example) onto a semiconductor layer, in UHV (ultra-high vacuum) environments. For some materials, the energy difference between the Fermi level and the conduction band edge at the surface depends only weakly on the metal, e.g., GaAs, due to the Fermi pinning of the surface state. But typically, it's dependent, like in Si. For GaAs, this potential barrier $\Phi_{B}=0.8\ \text{eV}$. This value is called built-in potential. ![[Drawing 2024-09-03 15.01.02.excalidraw.svg]] And because in contact, $e^-$ moves to the metal, and we have the ionized donors left, forming a depletion region with width $d$. (bottom plot) Assume constant charge density, for volume doping, we'll have $\frac{\mathrm{d}^2 \phi(z)}{\mathrm{d}z^2} = -\frac{|e| N_D}{\varepsilon \varepsilon_0}$ And cause the minimum of $\phi$ is at $z=-d$ (namely the end of depletion region), $\phi(z) = -\frac{|e| N_D}{2 \varepsilon \varepsilon_0} |z + d|^2$and $\Phi_{B}=-|e|\phi(0)=\frac{|e|^{2} N_D}{\varepsilon \varepsilon_0}d^{2}$. This gives the thickness of the space charge layer, or say, the depletion region, $d = \sqrt{\frac{2 \varepsilon \varepsilon_0 \Phi_B}{e^2 N_D}}$And if we apply an external field, i.e., gate voltage, we'll have $d$ being a function of $V_G$, i.e., $d(V_G) = \sqrt{\frac{2 \varepsilon \varepsilon_0 (\Phi_b - |e| V_G)}{e^2 N_D}}$ This is called the field effect. For $N_D = 3 \times 10^{17} \, \text{cm}^{-3}$, $V_G = 0$, $d$ would be 61 nm, which is a pretty large value. Combining with remote doping and a gate, the electron density of the sheet is adjustable by a linear relation, $n_s = n_s^{(0)} + \frac{\varepsilon \varepsilon_0}{e\ d } V_G$ With [[Drude model with magnetic field|Hall effect]], one may measure this electron density experimentally. ### Ohmic contacts Unlike Schottky contacts, which aim to provide a potential barrier, Ohmic contacts aim to provide "wires" to connect to our circuit device. This may be done by very high concentration doping to make the tunnel barrier become sufficiently thin. So the contact shows an ohmic behavior, i.e., $V = IR$. By diffusing Ge and Au into GaAs, by heating at 450 $^\circ$C, it is possible to achieve a concentration of $n$-doping with $10^{19} \, \text{cm}^{-3}$. This allows us to bury contacts into 2DEG. The quality of these contacts is characterized by the contact resistance, $R_c$. Since the contact resistance depends on the contact area, we can define the specific contact resistance as $\rho_c = R_c A$. The unit of $\rho_c$ is $\Omega \cdot \text{cm}^2$. For AlGaAs/GaAs heterostructures, the typical value of $\rho_c$ ranges from $10^{-6}$ to $10^{-5} \, \Omega \cdot \text{cm}^2$, with the exact value depending on the doping level of the AlGaAs.