## 3 Nanomagnetic Switches  
Next, we consider a magnetic switch. We first point out that there are two types of switching errors: \"unintentional switching\" (i.e., switching that takes place when it is _not_ desired) and \"failure to switch\" (i.e., switching that does _not_ take place when it is desired). The previous example with the field-effect-transistor is of the first type and relates to errors caused by unintentional switching. Magnetic switches, fashioned out of nanomagnets (e.g., magnetic tunnel junctions), can also experience unintentional switching due to stray magnetic fields, thermal agitations, etc. The relationship between the energy dissipation  
Figure 1: Conduction band profile along the channel of a field effect transistor in the OFF state (solid line) and ON state (broken line). and error probability for unintentional switching was derived in ref. [7], which deduced that the minimum energy dissipation is related to the error probability according to:  
\[E_{d}^{\min}=2 kT\ln\left[-\frac{T_{c}}{\tau_{0}\ln(1-p)}\right] \tag{7}\]  
where \(T_{c}\) is the clock period and \(\tau_{0}\) is the inverse of the attempt frequency (which is in the range of 1 ps-1 ns) [8]. To maintain \(p=10^{-15}\) at a clock frequency of 1 GHz, the minimum energy dissipation would be 0.14-0.17 a) according to Equation (7). We caution that this is an overly optimistic estimate, since it assumes that the minimum energy dissipation needed to switch is the anisotropy energy barrier within the nanomagnet and no consideration has been made of additional energy losses due to Gilbert damping, etc. Next, we address \"failure to switch\" in a magnetic switch (i.e., switching not taking place when switching is desired and a switching stimulus is provided). Unlike the electronic switch, this case is not amenable to any analytical treatment and hence we will resort to simulations, using a specific example. A bistable nanomagnetic switch can be fashioned out of a ferromagnetic elliptical disk where, because of the elliptical shape, the magnetization can point only along the major axis, either pointing to the left or to the right (as shown in Figure 2a). This type of nanomagnet is said to possess in-plane magnetic anisotropy (IPA). In thinner nanomagnets, the surface anisotropy may be dominant and the magnetization can point perpendicular to the surface (either up or down). This type of nanomagnet is said to possess perpendicular magnetic anisotropy (PMA) (Figure 2b). Either type makes a binary switch if we encode the bit information in the magnetization orientation which can point in just two directions. In this paper, we will consider only the IPA nanomagnetic switch, although the results will apply equally to PMA nanomagnets. The IPA nanomagnet can be vertically integrated as the \"soft\" layer into a three-layer stack consisting of a \"hard\" ferromagnetic layer and an insulating (non-magnetic) spacer to form a magnetic tunnel junction (MTJ) (as shown in Figure 2c). The hard layer is permanently magnetized in one of its two stable directions. When the soft layer's magnetization is parallel to that of the hard layer, the MTJ resistance (measured between the two ferromagnetic layers) is low. If the two magnetizations are antiparallel, the resistance is high. Thus, the MTJ acts as a binary switch, much like the transistor, whose two resistance states (i.e., high and low) encode the binary bits 0 and 1. The difference between the transistor and the MTJ is that the former is _volatile_ (since charges leak out when the device is powered off), while the MTJ is _non-volatile_ (since the bit information is encoded in magnetization (spins) and not charge). In order to make the magnetizations of the hard and soft layers mutually parallel (ON state), we can employ spin-transfer-torque [2]. To do this, we apply a voltage across the MTJ with the negative polarity of the battery connected to the hard layer.