**The Extension of Circuit Theory and Device Physics for a Memristor**

*Supervisor*: Dr Gerard Edwards

*BEng Project Student*: Mr Abou Sangary

In 1971 Chua (Joglekar & Wolf, 2009) proposed the existence, from the point of view of symmetry, of the memristor, which supplements the well-established (R, C, L) passive circuit elements. Looking at the relationships between the fundamental circuit quantities (charge *q*, current *i*, voltage *v*, magnetic flux f), there ought to be a memristor defined by *dϕ=M**q**dq* (Joglekar & Wolf, 2009).

*Figure 1 In circuit theory there are four basic variables (charge q, current i, voltage v, flux f) and the relations between them are indicated by the black lines define (R, C, L). The upper horizontal line represents Lenz’s law while the lower horizontal line indicates that current is the rate of change of charge. Symmetry suggest that there should be a memristor element M relating flux and charge.*

The memristor was discovered by (Strukov, et al., 2008) in a nanoscale thin film device (~ 5 nm thick) consisting of one layer of insulating titanium dioxide (Ti02) and oxygen poor titanium dioxide (Ti02 - *x*), both sandwiched between platinum contacts. The oxygen vacancies act as an effective charge *2|**e|* where *e* is the charge on the electron (Joglekar & Wolf, 2009) (Strukov, et al., 2008). The doped Ti02 – *x *region has a significantly lower resistance than the undoped Ti02 region. The overall device can be viewed as two resistors in series where the width w of the doped region can vary with time i.e. *w=w(**t)*. The simplest treatment for the movement of *w* is the linear drift model where the doped region/undoped region boundary drifts at constant speed (Joglekar & Wolf, 2009). The polarity of the memristor is given by h = +1 when the doped region is expanding and is given by h = -1 when the doped region is contracting.

In this study, the transient response for the ML and MC circuits are investigated theoretically, within the linear drift model (Joglekar & Wolf, 2009). A semi-analytic approach is taken and the results are presented for the time dependence of *q=q(**t)*, *i=i(**t)* and *w=w(**t)* for both polarities of the MC circuit in Figure 2 and the ML circuit in Figure 3. Ongoing work is to employ more realistic nonlinear drift models for MCL circuits, using a fully numerical approach based on the MATLAB implemented Berkeley Model and Algorithm Prototyping Platform (MAPP) simulator (Wang, et al., 2015).

*Figure 2 The transient response of an MC circuit with parameters w0 = 0.5D, ROFF = 20 RON , C = 1 C0, qInitial = 0.45 Q0 (Joglekar & Wolf, 2009)*

*Figure 3 The transient response of an ML circuit with parameters w0 = 0.5D, ROFF = 30 RON , L = 30 L0, iInitial = 0.135 I0 (Joglekar & Wolf, 2009)*

## References

Joglekar, Y. N. & Wolf, S. J., 2009. The elusive memristor: properties of basic electrical circuits. *European Journal of Physics, *Volume 30, p. 661.

Strukov, D. B., Snider, G. S., Stewart, D. R. & Williams, S. R., 2008. The missing memristor found. *Nature, *Volume 453, p. 80.

Wang, T. et al., 2015. MAPP: The Berkeley Model and Algorithm Prototyping Platform. *IEEE Custom Integrated Circuits Conference (CICC), *Volume doi: 10.1109/CICC.2015.7338431, p. 1.