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I saw an experiment where red laser diode was submerged into liquid Nitrogen. After a couple of second, the laser color shifted to yellow.

Why does the laser color changes from red to yellow when the diode is cooled to liquid nitrogen's temperature?

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Because the bandgap energy of materials as an active layer is changed by temperature, since the lattice constant is changed by the temperature. The bandgap dependence on the temperature can be calculated by Vegard's law and Varshni's empirical expression.

For example, for red colored laser diodes, it is considered that the materials are AlGaInP, GaAsP and AlGaAs.

In the case of AlGaInP, the composition ratio of (Al$_x$Ga$_{1-x}$)$_{0.5}$In$_{0.5}$P is used to use GaAs substrate. In this case, the bandgap energy at 77 K has more than 1.92 eV (630 nm) for the range of $x$ from 0 to 1.

In the case of GaAs$_x$P$_{1-x}$, the bandgap energy for the range of $x$ from 0.3 to 0.4 has 2.09 eV (592 nm) - 1.93 eV (617 nm) at 77 K and 2.02 eV (614 nm) - 1.93 eV (641 nm) at 300 K.

In the case of Al$_x$Ga$_{1-x}$As, the bandgap energy for the range of $x$ from 0.7 to 0.8 has 2.09 eV (593 nm) - 2.02 eV (615 nm) at 77 K and 2.02 eV (615 nm) - 1.94 eV (638 nm) at 300 K.

So it is estimated that the laser diode you saw is made of GaAsP or AlGaAs.

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Diodes produce light from an election falling in energy across a semiconductor's band gap, emitting a photon in the process. Note that yellow light is higher in energy than red light.

Band gaps in semiconductors tend to increase at lower temperatures. This happens occurs because atoms vibrate with smaller displacements from their equilibrium positions as temperature decreases, which causes the effective "distance" between adjacent atoms to increase.

As can be shown with the "hydrogen molecule" quantum mechanics problem (which isn't directly applicable, but is analogous), closer interatomic spacing causes a larger difference in energy states - or on the case of a semiconductor, a larger band gap.

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