In 1971, Leon Chua theorized the existence of the fourth fundamental passive circuit element, which he called the "memristor" (memory + resistor). The memristor completes the symmetry between the four fundamental circuit variables: current (i), voltage (v), charge (q), and magnetic flux (φ).
The memristor is defined by a relationship between charge and magnetic flux:
\[M(q) = \frac{d\phi}{dq}\]
Where \(M(q)\) is the memristance (measured in ohms, Ω), which depends on the charge \(q\) that has flowed through the device.
Since flux linkage is related to voltage by:
\[\phi(t) = \int_{-\infty}^{t} v(\tau) \, d\tau\]
And charge is related to current by:
\[q(t) = \int_{-\infty}^{t} i(\tau) \, d\tau\]
We can also express the memristor relation as:
\[v(t) = M(q(t)) \cdot i(t)\]
This shows that the memristor acts like a resistor whose resistance varies according to the history of current that has flowed through it.
In 2008, HP Labs announced the first physical implementation of a memristor. Their model can be described by:
\[M(w) = R_{OFF}(1-\frac{w}{D}) + R_{ON}\frac{w}{D}\]
Where \(w\) is the state variable representing the width of the doped region, \(D\) is the total thickness, and \(R_{ON}\) and \(R_{OFF}\) are the resistance values when the device is fully ON or OFF.
The state variable evolves according to:
\[\frac{dw}{dt} = \mu_v \frac{R_{ON}}{D} i(t) \cdot f(w)\]
Where \(\mu_v\) is the mobility of oxygen vacancies and \(f(w)\) is a window function that ensures \(w\) stays within physical boundaries.