Numerical method on the heterodyne terahertz detection characteristics of field effect transistors is studied in this paper which is based on the hydrodynamic equations which govern the terahertz signal transport in field effect transistors (FETs). A modification is made in an existed numerical tool established by our group by coupling the heterodyne characteristics. This modified numerical tool work well in all operation regions of FETs from sub-threshold to strong inversion and from linear to saturation. And the results are used to demonstrate the potential for using MOS transistors as THz detectors and investigate the optimization of the device structure.
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In recent years, terahertz frequency radiation has received a lot of attention due to its potential for non-invasive imaging. Dyakonov and Shur had proposed a theory for using field effect transistors as detectors of terahertz radiation [1,2]. The operations of these detectors are based on plasma wave excited in GaAs, Si, and GaN FETs [3–6] under high modulation frequency. Plasma waves excited in un-gated two dimensional electron layers  and the influence of DC current on THz detection had been studied [7,8] in earlier works. FET to be used as an ultra fast detector of modulated terahertz radiation was predicted in 2007 . Imaging of field-effect transistors by focused terahertz radiation was developed in 2009 .
In 2008, our group has developed a numerical tool for simulating plasma wave transmission in FETs based on the structure shown in Fig. 1 [11,12]. In the figure, the THz radiation induced AC voltage between gate and source is assumed to be, and the frequency of the incoming THz signal is. The radiation induced voltage dropbetween source and drain is considered to be the detector response.
A theory of nonlinear response of a field effect transistor subjected to two terahertz beams with close frequencies has been proposed in 2008 . It predicts that the heterodyne efficiency can be significantly increased by the electric current flowing in the transistor channel. In addition, it has been demonstrated that such a heterodyne detector is capable to operate at very high intermediate frequencies up to 50~100GHz. However, existed simulation programs developed so far only shows the relationship between the amplitude of heterodyne signal and drain-to-source current in the strong inversion region rather than the entire region including the sub-threshold region. In the sub-threshold region, since the existence of gate leakage current in the transistor decreases the magnitude of detection response remarkably and the enhancement of magnitude emerges close to the sub-threshold voltage, developing the numerical tool covering the sub-threshold region plays a significant role in the further investigation on heterodyne terahertz detection. Thus, in this work, numerical study on heterodyne terahertz detection is made, and a numerical simulation tool for calculating the terahertz response under different incoming signal amplitudes and frequencies at all operation regions from sub-threshold to strong inversion is established. By using our developed numerical tool, not only can the validity of the theory of nonlinear response of the FET channel subjected to two terahertz beams with the close frequencies be demonstrated, but also the characteristics on heterodyne detection in sub-threshold region can be simulated.
2. Numerical model
The basic principle of the mechanism of heterodyne detection is described as follows. An incoming signal expressed with the amplitude and a frequencyis mixed with a strong signal from a local oscillator with the amplitude and a frequency. We assume that the voltage at the source is the sum of the fixed gate-to-channel swing and an oscillating part induced by the two beams, so that1,2].
In the equations, is the electron density, is the average velocity, is the elementary charge, is the electron momentum relaxation, is the electron effective mass and is the quasi Fermi potential. is the density of the gate leakage current which is a constant. The general equation for the electron density in the FET’s channel is given by 
The following expressions are obtained
Using the forward-difference method in the time domain and central-difference method in the spatial domain, one gains the difference equation:Eq. (5)
By fixing the DC current of drain at, the left boundary is obtained. Similarly, the right boundary condition is described asEqs. (7), (8) and (9). The implicit numerical of this numerical model is stable for a wide range of conditions.
Here we have chosen to use the backward Euler method to carry out iterative calculation. In analyzing the error of the numerical method, it is found that the oscillation tends toward stability and eventually forms the stable oscillation, and it could attain dozens of the order of magnitudes at a determinate time step. Thus the analysis on the astringency of algorithm is crucial. According to the LAX equivalence theory, the stability is the necessary and sufficient condition of astringency. By keeping the constant, the compatibility of the difference and discrete form of two basic questions can be guaranteed. The stability of algorithm can be ensured with the advancing of time steps. Similarly, the methods for discretion of fundamental equation set-up basing on Eq. (4) are compatible as well. Following the same analysis methods, it is drawn that the solutions of the differential equations can always be close to their genuine solutions under an adequate precision.
When it comes to the question of adding boundary conditions, since there are only two differential equations, and two boundary conditions were needed. However, due to the limits of the difference algorithm itself, the current density should be given on the left margin, and the electron concentration given on the right sector. So the situation of adding boundary conditions will also affect the accuracy of the algorithm. Here we use a stable implicit algorithm, and this algorithm is used in the current boundary layer grid points, the next boundary layer grid points and the boundary adjacent grid points. The stability of such boundary conditions is much higher than that of the explicit algorithm as well as the leap-frog scheme.
3.Result and discussions
First, we simulated a field effect transistor with gate oxide thickness, substrate doping concentration and gate length. The temperature is fixed at 300K in all simulations. The threshold voltage is set at and by applying an average gate bias, the mean swing voltage is. Parameters of the fixed gate-to-channel swing and the oscillating signal induced by two beams are chosen as following: , . Figure 2 shows (a) the drain voltage and (b) the amplitude-frequency characteristic of the photoresponse corresponding to the spectrum of the drain voltage near the frequenciesand. The two components at frequency of andare called the harmonic responses induced by two incoming signals. Then a low-pass filter is used to separate the photoresponse from the other frequency components and the corresponding results are shown in Fig. 3(a) . According to the theory introduced in , the phase of the intermediate signal is equal to the phase difference of the signal and the phase of the local oscillation with a small correction, here is the time delay and . Both and are fixed and the THz detection spectrum is picked up by the FET spectrometer. The predicted time delay is not included in the red curve so as to show the time resulting from the heterodyne detection clearly. The amplitude of the photoresponse is shown in Fig. 3(b). The component at the frequency is the signal generated at the intermediate frequency (IF) .
In the preceding example, we showed the typical detection spectra at a gate bias . Now, we change the gate bias from 0.8V to 1.15V. Figure 4(a) shows that the photoresponse is modulated by the gate bias. Figure 4(b) shows the dependence of time delay and the amplitude of the response as a function of the gate bias. It is observed that both the time delay of the response and the dc component decrease as gate biasincreases. The black and green dotted lines are extracted from the simulation and the red and blue dotted lines are calculated from the theory. It is observed that the numerical simulations have excellent agreement with the analytical prediction.
To estimate the influence of the electron mobilityon the response of the MOSFET detector, we will use the output characteristics of the transistors as shown in Fig. 5 (a) . In Fig. 5(b), the numerical time delays at different relaxation times, are compared with the theoretical curves derived from the output characteristics of transistor. Comparison between the theoretical calculation and numerical simulation in Fig. 5(b) indicates that the time delay decreases while increases, and the simulation results agree with the theory. Besides, the amplitude of the response from the numerical simulation agrees with the theoretical results in the case of small electron mobility. However, for high electron mobility, the numerical results are a little larger than the theoretical results. It is because the theory introduced in  is restricted to the condition. With the increase in electron mobility, such condition will not be satisfied. The numerical simulation reflects the true THz detection in FET-based terahertz detectors more consistently.
All the above simulations are restricted in the strong inversion region. Our simulator can also be used in the sub-threshold region while . In the experiment of THz detection in Ref. , for smaller than the threshold voltage (negative values of ), the experimentally observed response decreases, so that the response always reaches a maximum value close to the FET threshold. Due to the fact that no systematic numerical tool for the investigation on heterodyne terahertz detection has been established till now, developing the numerical study on heterodyne terahertz detection in the sub-threshold region is necessary. Electron concentration in the FET channel becomes very small in such situation. The leakage current plays an important role and should be taken into account . In Fig. 6(a) , four different gate leakage current densities (0A/m2, 1A/m2, 10A/m2 and 100A/m2) are chosen to study their influences on the response, here is fixed at 3. When the gate leakage current density is neglected, the amplitude of the response in sub-threshold is independent on the gate bias. According to the theory , the amplitude of the response is just inversely proportional to the temperature. The analytical theory cannot reflect the actual situation when there is a gate leakage current. And the previous theory does not include the situation that gate voltage is near the threshold voltage, while the numerical tool constructed by us not only considers the influence of the gate current to response in the sub-threshold areas, but also has a good simulation capability in the situation near the threshold voltage. In Fig. 6(a), it is clear that when the gate bias decreases, the amplitude of the response rapidly decreases in the sub-threshold region, and the peak value of the amplitude appears when. A maximum value near zero swing voltage can be obtained by a decreasing leakage current, and the maximum broadens and shifts toward negative swing biases. A highland is gained at low values of the gate current. Gate leakage current does not influence the amplitude of response in strong inversion region. Well above threshold, the gate leakage is small compared to the current in the channel and does not affect the response of the detector. It may be interesting to study how the photoresponse of the MOSFET detector is influenced by the temperature. Three different temperatures (300K, 200K and 100K) are chosen for investigating their influences on the response. In Fig. 6(b), the peak amplitude of the photoresponse increases with the decrease of temperature. With decreasing temperature, the peak value shifts to smaller gate bias. That is because the threshold voltage decreases when the temperature decreases.
The numerical description, presented above, shows that parameters necessary to describe the heterodyne terahertz detection of field effect transistor are: (i) electron mobility (ii) the threshold voltage (iii) the leakage current. Therefore, the development of numerical tools is instructive to the study of the device structure about changing parameters on the heterodyne detection.
In this paper, a numerical method on heterodyne detection has been investigated which is used for calculating the terahertz signal transport in field effect MOS transistors basing on the hydrodynamic equations. By implementing the heterodyne characteristics into the fundamental framework of a numerical tool established by our group in 2008, a comprehensive numerical method particularly for heterodyne detection is formed. By comparing the numerical results with the existing theory, the validity of the proposed numerical method has been proved, and the developed numerical tool has been demonstrated to work well in all operation regions of field a MOS transistor with continuous transit from the sub-threshold to the strong inversion and from the linear to the saturation regions. Thus the proposed numerical method has potential to be a useful tool for studying the nonresonant detection response by changing certain parameters of field effect MOS transistors.
This work is supported by the National Natural Science Funds of China (60976066), the Shenzhen Science & Technology Foundation (JSA200903160146A) and the Open Project of State Key Laboratory of Functional Materials for Informatics.
References and links
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