LED conversion efficiency exceed 100%

The device is apparently working as a heat pump, for which I give a brief theoretical analysis here.

In the example given, the $P_h=69{\rm pW}$ light output comprises the $W=30{\rm pW}$ input by the researchers together with $P_c=39{\rm pW}$ of heat that was formerly in the chip.

We can model the process by ideal heat pumping as follows. Heat drawn from the chip will lead to a drop in the chip's entropy of $\Delta S_c = -\frac{P_c}{T_c}$, and the light and output to the ambient World increases the entropy of the latter by $\Delta S_w = \frac{W + P_c}{T_h}$, where $T_h$ is the effective temperature of the light (measuring the latter's degree of thermalization together with its optical grasp). Since the light ends up in the environment, its effective temperature is ambient or greater.

The total entropy change of the World is then

$$\frac{W}{T_h} + P_c\,\left(\frac{1}{T_h} - \frac{1}{T_c}\right)$$

We know that $T_h>T_c$ because the effective "exhaust" temperature is at least ambient and $T_c$ must wind up less than this, because heat is being pumped out. So the second term with the brackets is negative: this means we must supply enough work $W$ to at least make the quantity positive. So the device can very plausibly (and probably does) work as claimed and comply with both the first and second laws of thermodynamics.

Given that the device is extracting heat (vibrational energy) from the lattice that the LED is embedded in, the conservation of energy issue can be understood. The question you really should be worried about is, "does this violate the second law of thermodynamis?" Is the overall free energy increasing?

You can use thermoelectric coooling, for example, exploiting the the Peltier effect to make a device which cools one side and transfers the heat to the opposite side of a plate. When you place this in an environment of initially uniform temperature, the free energy apparently gained (you could after all exploit the new temerature gradient by a Carnot engine to extract work) is less than or equal to the free energy supplied by the current driving the cooler. You can certainly use this to heat the "other side" of a plate by more than you would by just using the current in a resistor.

Similarly, the LED device is converting heat on one side to heat-converted-to-light on the other and is exploiting the free energy associated with the current to do this (so that the entropy is not decreasing.