Super-Earths have solid cores

The melting curve of iron at extreme pressures: implications for planetary cores

Looks like terrestrial super-Earths more massive than about 2 times the mass of Earth will have entirely solid cores, preventing the formation of a magnetic field. Ocean planets might be able to maintain a magnetic field generated by salt water convection.

STAGNANT INTERIORS SUPPRESS CHANCES OF SUPER-EARTHS SUPPORTING LIFE

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The high melting temperatures and the large viscosities that we've calculated for super-Earths suggest either a slow core formation or no core formation at all. This raises doubts about whether super-Earths could generate magnetic fields," said Stamenkovic.

Even if those super-Earths are differentiated, the research indicates that convection would be sluggish or that stagnant layers could form deep in the mantle, putting an effective lid on heat flow from the core. Strongly depending on the initial conditions, conduction may become the dominant form of heat transport. This would reduce the cooling rate of the core, again potentially quashing dynamo action.

The team has found the propensity of plate tectonics to rather decline with planetary mass. But they also find that water in the lithosphere can easily buffer these effects. Hence plate tectonics on super-Earths is not inevitable, but rather depends on a set of unknown planetary characteristics, which can’t be observed on exoplanets in the near future.

A new study support the opposite theory

http://www.space.com/18604-super-earth-planets-liquid-metal.html

Apparently the idea of un-differentiated interiors of Super-Earths holds on, but this study point out that the rocky material there could turn in a metallic liquid state, allowing the creation of magnetic fields.

That's interesting. Maybe yet another parallel between the "super-Earths" and Uranus and Neptune which also are thought to have magnetic fields generated nearer the surface, leading to a magnetic axis that is offset from the centre of the planet.

The Effect of Lower Mantle Metallization on Magnetic Field Generation in Rocky Exoplanets
http://arxiv.org/abs/1304.6953

A conducting mantle can lead to weaker surface magnetic fields as it screens the dynamo.