The SSDs that we're using today are built on floating gate transistor technology, illustrated in Figure 1 below from Anandtech.

Between the floating gate and the substrate is the tunnel oxide the barrier to the floating gate through which the electrons "tunnel" into the floating gate. The transistor either has electrons tunneled into the floating gate (indicating a logical 0) or does not have any electrons tunneled into the floating gate (indicating a logical 1). The process of forcing electrons into or out of the floating gate, called Fowler-Nordheim Tunneling (F-N Tunneling), is achieved by applying a voltage between the control gate and the source or drain. When the charge is removed, the floating gate either retains the electrons (if they were tunneled into the gate) or has no extra electrons if they were removed. This allows Flash memory to retain values after power is removed.
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