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Sometimes plasmons are defined as collective plasma oscillations of the free electron gas in a metal. Therefore, plasmons must be a periodic modulation of electron charge density in the metal. But sometimes, plasmons are defined as electromagnetic waves that exist on the surface of metals and decays inside. What is plasmon really? Is it a charge density wave of electron gas or an EM wave that exists across the metal surface?

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A plasmon is both of these things! With the charge-density wave comes an associated EM wave across the metal surface (or vice-versa). These are two sides of the same coin of a plasmon. You can't have one without the other.

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