We used 2D coupled sine-Gordon equations combined with 3D heat diffusion equations to numerically investigate the thermal and electromagnetic properties of a $250 times 70,mumathrm{m}^2$ intrinsic Josephson junction stack. The 700 junctions are grouped to 20 segments; we assume that in a segment all junctions behave identically. At large input power a hot spot forms in the stack. Resonant electromagnetic modes, oscillating either along the length ((0, $n$) modes) or the width (($m$, 0) modes) of the stack or having a more complex structure, can be excited both with and without a hot spot. At fixed bath temperature and bias current several cavity modes can coexist in the absence of a magnetic field. The (1, 0) mode, considered to be the most favorable mode for THz emission, can be stabilized by applying a small magnetic field along the length of the stack. A strong field-induced enhancement of the emission power is also found in experiment, for an applied field around 5.9 mT.
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