We calculate the terahertz absorption spectrum of the high-electron-mobility transistor with a short gate and long ungated channel regions and show that the main contribution to the linewidth of the gated plasmon resonance can be attributed to the plasmon-plasmon intermode scattering. The results allow interpreting recent experimental results on resonant terahertz detection by InGaAs nanotransistors. The physics of the plasmon-plasmon intermode scattering is discussed and efficiency of this process is quantitatively compared to the other possible plasmon damping mechanisms.
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