# Monolithically Integrated THz Detectors Based on High-Electron-Mobility Transistors
## 2 Resistive Self-Mixing  
With resistive mixing, the channel of an HEMT is used as a variable resistor. It is like using the same signal for the local-oscillator (LO) and radio-frequency (RF) signal in an FET-based mixer. The HEMT device works around the origin of the output characteristics [20] at zero drain-source voltage. To achieve this operating state in the THz detector, the THz radiation is fed between the gate and source connection, representing the LO signal. Through a broadband short circuit between the gate and drain connection, the same signal is present between the drain and source connection, which represents the RF signal. Figure 1 shows the equivalent circuit for resistive self-mixing. The drain-source current \(i_{ds}\) can be determined using a two-dimensional Taylor series expansion around an operating point with an arbitrary gate-source voltage and zero drain-source voltage. The transconductance of an HEMT \(g_{m}\) as a function of gate-source voltage and the drain-source channel conductance \(g_{ds}\) as a function of drain-source voltage  
\[\frac{\partial\dot{i}_{ds}}{\partial u_{gs}}=g_{m}\quad;\quad\frac{\partial\dot{ i}_{ds}}{\partial u_{ds}}=g_{ds} \tag{1}\]  
lead to a THz detection current at the drain terminals. \[ \begin{split} i_{ds}(u_{gs},u_{ds})&=g_{ds}(U_{GS}, 0)\Delta u_{ds}\\ &+\frac{\partial}{\partial u_{gs}}g_{ds}(U_{GS},0)\Delta u_{gs} \Delta u_{ds}\\ &+\frac{1}{2}\bigg{\{}\frac{\partial}{\partial u_{ds}}g_{ds}(U_{ GS},0)\Delta u_{ds}^{2}\bigg{\}}\end{split}  \tag{2}\]  
The formula of the drain current in Equation (2) describes the drain current at a direct-current (DC) operating point where the drain-source voltage is zero and the gate-source voltage is a chosen parameter. Because of the RF short circuit, \(\Delta u_{gs}\approx\Delta u_{ds}\), which results in  
\[ \begin{split} i_{ds}(u_{sg},u_{ds})&=g_{ds}(U_{GS}, 0)\Delta u_{gs}\\ &+\bigg{\{}\frac{\partial}{\partial u_{gs}}+\frac{1}{2}\frac{ \partial}{\partial u_{ds}}\bigg{\}}g_{ds}(U_{GS},0)\Delta u_{gs}^{2}\end{split}  \tag{3}\]  
With a sinusoidal small-signal modulation between the drain and gate terminal, the THz detected drain current is  
\[\Delta u_{gs}=A_{0}\sin(wt) \tag{4}\]  
\[i_{ds}(u_{gs},u_{ds})=\frac{1}{2}A_{0}^{2}\bigg{(}\frac{\partial}{\partial u_{ gs}}+\frac{1}{2}\frac{\partial}{\partial u_{ds}}\bigg{)}g_{ds}(U_{GS},0)+\ldots \tag{5}\]  
Figure 1: Schematic representation of the HEMT wiring for resistance mixing. The green arrows illustrate the THz radiation, which generates a voltage across the gate–source connection. The quantities represented through red arrows are the RF voltages and currents used for the THz detection. In Equation (5), only the DC component has been retained, and all higher-frequency components were discarded. In this relation, the differential output conductance at the operating point is highlighted in blue, the differentiation according to the drain voltage in green, and the differentiation according to the gate voltage in red. The DC component depends quadratically on the THz voltage amplitude, \(A_{0}\), which is determined by the source power and the antenna coupling. The second term in the parentheses describes the differential change in the output conductance as a function of the drain-source voltage (green). This represents the linearity of the differential output conductance. Since the drain-source bias voltage is zero at the selected operating point, this is the linearity of the differential conductance around the origin at the transition from the first to the third quadrant of the output IV characteristics. In a first approximation, this part can be neglected compared to the first term since, under small-signal conditions, the differential conductance at the origin represents a linear function of the drain-source voltage. The first term in the parenthesis is the change in the output conductance depending on the gate-source voltage (red), i.e., the ability of the HEMT to control the charge carriers in the channel.