Sensitivity of the Threshold Current for Switching of a Magnetic Tunnel Junction to Fabrication Defects and Its Application in Physical Unclonable Functions

Abstract

A physical unclonable function (PUF) leverages the unclonable random variations in device behavior due to defects incurred during manufacturing to produce a unique “biometric” that can be used for authentication. Here, we show that the threshold current for the switching of a magnetic tunnel junction via spin transfer torque is sensitive to the nature of structural defects introduced during manufacturing and hence can be the basis of a PUF. We use micromagnetic simulations to study the threshold currents for six different defect morphologies at two different temperatures to establish the viability of a PUF. We also derive the challenge–response set at the two different temperatures to calculate the inter- and intra-Hamming distances for a given challenge.

1. Introduction

Mobile and embedded devices are increasingly called upon to authenticate or be authenticated by another party for trust. These devices include cell phones that are used routinely for bank transactions and medically implanted devices that form an integral part of acute care and patient monitoring. The usual approach to providing protection against unauthorized use, data interception, etc., is to place a secret key in a nonvolatile electrically erasable programmable read-only memory (EEPROM) or battery-backed static random access memory (SRAM) and use hardware cryptographic operations such as digital signatures or encryption for authentication [1]. This approach is expensive both in terms of design area and power consumption [1]. Furthermore, nonvolatile memory is often vulnerable to invasive attacks that can be thwarted only using active tamper detection/prevention circuitry, which is continually powered [1] and hence inappropriate for edge devices or Internet of Things, where energy is a premium and continual powering is prohibitive. Physical unclonable functions (PUFs), on the other hand, are a promising alternative that enables authentication and secret key storage via the physical characteristics of a device that are unpredictable and impossible to anticipate, duplicate, or hack.

2. PUFs Based on Magnetic Tunnel Junctions

A popular implementation of PUFs involves magnetic tunnel junctions (MTJs) [2,3,4,5,6,7,8], which are essentially binary switches with two resistance states—“high” and “low.” An MTJ is a three-layered structure with the two outer layers ferromagnetic and the middle layer a non-magnetic insulator through which spin-dependent tunneling of electrons takes place between the two ferromagnetic layers. Each layer is shaped like an elliptical disk, and the magnetization of each will align along the so-called “easy axis”, which is along the major axis of the ellipse, either pointing to the left or to the right [see Figure 1] (this makes the magnetization “bistable”). One layer is permanently magnetized in one of the two directions along the major axis and is called the hard layer, while the other is the soft layer, whose magnetization can be switched from right to left, or vice versa, by passing a spin-polarized current pulse of the correct polarity between the two layers. The MTJ resistance is high when the magnetizations of the two ferromagnetic layers are antiparallel and low when they are parallel. Thus, flipping the magnetization of the soft layer results in switching the resistance of the MTJ from high to low, or vice versa.

Threshold Current-Based PUF

The minimum (spin-polarized) charge current (threshold current) needed for switching depends on the current pulse width, degree of spin polarization, and various MTJ soft layer parameters such as size, ellipticity, material composition, etc. Even if we keep all these parameters fixed, the threshold current will vary because of unintentional defects introduced in the soft layer during fabrication. Since these defects are unclonable and unpredictable, the threshold current is also unclonable and unpredictable. Thus, if we pass a current of fixed amplitude and duration through a number of nominally identical MTJs that have different (random) defect morphologies, as shown in Figure 2, some will switch and some will not, and which ones will switch is unpredictable and unclonable. This can be the basis of a PUF. Such a PUF is usually superior to many other types of PUFs, such as memristor-based PUFs [9], because the defects are extremely stable.

Let us say that we have 8 MTJs, which are all initialized to the low resistance state with a global magnetic field. We then pass a current of amplitude $I_0$ (of the correct polarity) for a duration $\Delta t$ through all of them, as shown in Figure 2. Let us say that because of the defects in the soft layers of the MTJs, devices 3, 5, and 8 switch to the high-resistance state because their threshold currents (which depend on the defect morphology) are below $I_0$. We will denote the resistance of the MTJ by a single bit—0 for low resistance and 1 for high resistance. In this specific case, the output string representing the resistance states of the 8 MTJs will be 00101001, where the n-th bit represents the resistance state of the n-th MTJ after current injection.

Let us now say that we increase the current pulse amplitude to 1.2$I_0$. This time, MTJs 1, 2, 3, 5, and 8 switch because their threshold currents are below 1.2$I_0$. The corresponding output bit stream representing the final resistance states will be 11101001. Thus, each current value produces a different output bit stream. This forms a set of “challenge” (current pulse amplitude) and the corresponding “response” (output bit streams). This set is specific to this particular unit and is a fingerprint or biometric of that unit, which is unpredictable and unclonable. A different unit with different unpredictable manufacturing defects will produce a different challenge–response set. It is impossible to predict what the response to a specific challenge will be for a given unit, and it is impossible to intentionally fabricate two units with the exact same challenge–response set since the defects are uncontrollable. This is the basis of a PUF.

Instead of changing the current pulse amplitude, one can change the current pulse width or duration while keeping the amplitude fixed. This will also have a similar effect and can be the basis of a PUF. Here, we study the amplitude challenge rather than the pulse width challenge.

To proceed further, we have to evaluate the threshold currents in nominally identical soft layers, which are all made of cobalt and all shaped like elliptical disks of major axis 100 nm, minor axis 90 nm, and average thickness 3 nm. However, they have different defect morphologies associated with voids, random thickness variations, etc. We will consider one pristine (defect-free) soft layer and five defective soft layers, each having a different type of defect. These are shown in Figure 3.

To calculate the threshold current at room temperature, we have to calculate the probability of the magnetization flipping as a function of the injected current amplitude when the current pulse width is fixed at 3 ns. For this purpose, the magnetization of the soft layer is initially aligned along one direction, say the –y-direction, as shown in Figure 3, with an external magnetic field. Next, spin-polarized current, with the majority of the spins polarized along the +y-direction, is injected perpendicular to the surface. Flipping takes place when the magnetization aligns close to the +y-direction and the normalized y-component of the magnetization exceeds 0.9, regardless of what the x- and z-components are. We simulate the time evolution of the magnetization, which is governed by the Landau–Lifshitz–Gilbert equation, and track the temporal evolution of the magnetization for a duration of 3 ns (current pulse width) in the presence of thermal noise. This is accomplished with the micromagnetic simulator MuMax3 [10]. The ambient temperature is assumed to be 300 K. The thermal noise makes the switching trajectories random. Hence, we generate 100 trajectories, and the probability of switching is defined as the fraction of the trajectories that, at the end of the 3 ns, show a y-component of normalized magnetization greater than or equal to 0.9. We calculate these probabilities for different current pulse amplitudes and for a fixed pulse width of 3 ns. They are calculated for the six different defect morphologies shown in Figure 3, and the resulting plots are shown in Figure 4. These morphologies are very similar to the ones studied in ref. [11] and are frequently found in fabricated nanomagnets. In fact, ref. [11] showed that if the nanomagnets are fabricated with electron beam lithography followed by metal evaporation and lift-off, then the defects that are formed can indeed be mimicked by the types we consider here.

Figure 4 shows that the probabilities do not change abruptly from near zero to near 100% as we increase the current amplitude. That makes it difficult to define a “threshold current”. Absent any better definition, we define the threshold current as the current that results in an 80% or more likelihood of flipping. Clearly, the threshold current depends on the defect morphology.

3. Materials and Methods

For the purpose of simulation, we map the different defect morphologies onto the MuMax3 software package using a uniform mesh size of 2.8 nm × 3.1 nm × 1 nm. There are 32 × 32 × 4 cells, giving a total volume of 90 nm × 100 nm × 4 nm, and we use a time step of 0.09 ps. An effective magnetic field due to noise is introduced as [12]

$$h_i^{noise}\left(t\right)=\sqrt{{\displaystyle \frac{2\alpha kT}{\gamma \left(1+{\alpha }^2\right){\mu }_0{M}_s\Omega \Delta t}}}{G}_{(0,1)}^i\left(t\right),$$

where $\alpha$ is the Gilbert damping constant, $\gamma$ is the universal gyromagnetic factor, $G_{(0,1)}^i\left(t\right)$ ($i=x,y,z$) are three uncorrelated Gaussians of zero mean and unit standard deviation, $\Omega$ is the nanomagnet volume, k is the Boltzmann constant, T is the absolute temperature, and $\Delta t$ is the time step of the simulation (0.09 ps). The magnet material is assumed to be cobalt, and we use its material parameters. The current is spin-polarized, and we assume the spin polarization percentage to be 30%.

We track the magnetization components (normalized to the saturation magnetization of cobalt) $m_x\left(t\right)$, $m_y\left(t\right)$, and $m_x\left(t\right)$ as a function of time, starting with the initial condition $m_x\left(0\right)=m_z\left(0\right)$ = 0 and $m_y\left(0\right)$ = −1. The simulation is continued for 3 ns, and the final state of $m_y\left(t\right)$ is noted. If it is greater than or equal to 0.9, we conclude that switching has taken place.

We repeat the simulation 100 times to generate 100 switching trajectories. Because of the presence of thermal noise, the final state of $m_y\left(t\right)$ and hence the switching outcome is different for different trajectories. The probability of switching is the fraction of trajectories that result in successful switching after 3 ns of current injection, i.e., the final $m_y\left(t\right)\ge 0.9$. Since we have 100 trajectories, the accuracy in calculating the switching probability is 1%. This is adequate since, as we will see later, the difference between the switching probabilities at the threshold current for any given defect morphology is more than 1%.

We calculate the switching probability as a function of current amplitude for all six morphologies shown in Figure 3 and plot them in Figure 4. This allows us to determine the threshold currents, which are tabulated in

Table 1. Threshold currents for different defect morphologies at 300 K for 3 ns current pulse duration.

Defect	Threshold Current (mA)	Threshold Current Density (A/m2)
${\mathrm{C}}_0$	2.77	3.91 $\times {10}^{11}$
${\mathrm{C}}_1$	2.79	3.94 $\times {10}^{11}$
${\mathrm{C}}_{2–3}$	3.26	4.61 $\times {10}^{11}$
${\mathrm{C}}_4$	2.77	3.91 $\times {10}^{11}$
${\mathrm{C}}_5$	2.84	4.00 $\times {10}^{11}$
${\mathrm{C}}_6$	2.74	3.87 $\times {10}^{11}$

. Note that the plots in Figure 4 are not monotonic because of the presence of thermal noise.

As already mentioned, we define the threshold current as that which ensures that switching occurs with an 80% probability. If we increase this to a higher percentage, we will just have a higher threshold current in all cases. What we are looking for, however, is not so much the exact value of the threshold current but whether the threshold is different for different defect morphologies, as that enables a PUF.

4. Results and Discussion

The reason why the threshold current is sensitive to the defect morphology can be understood as follows. There is an approximate analytical expression for the threshold current $I_{th}$ given by [13,14]

$$I_{th}={\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{2q}{\hslash }}\right)\alpha \Omega H_k{M}_s\left[1+{\displaystyle \frac{2\pi M_s}{H_k}}\right],$$

(1)

where $\eta$ is the spin polarization fraction of the current ($0\le \eta \le 1$), $\alpha$ is the Gilbert damping constant in the soft layer material, $\Omega$ is the volume of the soft layer, $M_s$ is the saturation magnetization of the nanomagnet material, and $H_k$ is the uniaxial (shape) anisotropy field. The defects effectively change the “shape” of the nanomagnet and hence $H_k$. This is why the defect morphology affects the threshold current.

Table 1 shows some interesting features. The threshold current for defect type $C_{2–3}$ is markedly different from those of the others. This was also observed in ref. [11], which studied strain-induced switching as opposed to current-induced switching. It is obviously due to the fact that $C_{2–3}$ increases $H_k$ much more than the other defects and hence increases the threshold current dramatically.

To implement PUFs, we need the threshold current distribution among the MTJs in a PUF to be unpredictable (random), unclonable, and different in each PUF so that the threshold current distribution becomes a fingerprint or biometric. We note that the threshold currents for defect types $C_0$ and $C_4$ are the same up to two decimal places. Therefore, the differences between them may not be resolvable. The threshold currents for others are different from those of $C_0$ and $C_4$ by at least 20 $\mu$A, which is easily measurable, and hence those other types of defects can be resolved from $C_0$ and $C_4$. It is obvious that for PUF realization, not all defects can be of the type $C_0$ and $C_4$ alone. There must be some $C_1$, $C_{2–3}$, $C_5$, and $C_6$ as well. To ensure that, one can increase the number of MTJs in a PUF so that not all the defects are of types $C_0$ and $C_4$ alone. Since not all defects occur with the same probability during manufacturing and their relative probability of occurrence is unknown, we cannot calculate the minimum number of MTJs that will be required to implement a PUF, but this can be established experimentally by trial and error.

4.1. PUF Behavior

Let us consider a unit consisting of just three MTJs instead of the eight shown in Figure 2. We assume that MTJ1 is defect-free (threshold current = 2.75 mA), MTJ2 has a defect of type $C_1$ (threshold current = 2.78 mA), and MTJ3 has a defect of type $C_{2–3}$ (threshold current = 3.26 mA), all introduced during fabrication. Let us say that we pass a current of 2.9 mA, which exceeds the thresholds of MTJ 1 and 2 significantly, to the point that their switching probabilities are ∼90%, but it falls short of the threshold current of MTJ3, whose switching probability at that current is only ∼55%. This current input can be considered the “challenge”, and the “response” bit stream in this case will be 110, indicating that the first two MTJs switch with high probability, while the last fails to switch almost half of the time.

Let us take another unit where the defect morphologies of MTJ2 and MTJ3 are switched. In this case, for the same current injection, the response bit stream will be 101.

We can generate the response bit streams for all possible combinations of defect morphologies in a 3-MTJ unit (each MTJ has a different defect $\in (C_0,C_1,C_{2–3})$), following the convention that the first bit in the response bit stream corresponds to the response of the first MTJ, the second to the second MTJ, and the third to the third MTJ. Using this convention, we can generate the response bit streams for six different 3-MTJ units, each having a different defect distribution. This is shown in

Table 2. Response bit streams under a current injection challenge of 2.9 mA for six different 3-MTJ units with different distributions of the defect morphologies at 300 K.

Defect Distribution	Response Bit Stream
Unit 1: ${\mathrm{C}}_0$  ${\mathrm{C}}_1$  ${\mathrm{C}}_{2–3}$	110
Unit 2: ${\mathrm{C}}_0$  ${\mathrm{C}}_{2–3}$  ${\mathrm{C}}_1$	101
Unit 3: ${\mathrm{C}}_1$  ${\mathrm{C}}_0$  ${\mathrm{C}}_{2–3}$	110
Unit 4: ${\mathrm{C}}_1$  ${\mathrm{C}}_{2–3}$  ${\mathrm{C}}_0$	101
Unit 5: ${\mathrm{C}}_{2–3}$  ${\mathrm{C}}_1$  ${\mathrm{C}}_0$	011
Unit 6: ${\mathrm{C}}_{2–3}$  ${\mathrm{C}}_0$  ${\mathrm{C}}_1$	011

.

It is easy to see that if the challenge current had been 2.76 mA, then the response bit stream would have been [100], [100], [010], [001], [001], [010]. Thus, by using different challenges and the corresponding responses, we can generate a challenge-response table that will depend on the defect morphologies and hence will be a fingerprint of the specific unit under consideration. Since the defect morphologies are unpredictable, unknown, and unclonable, the challenge-response table is also unpredictable, unknown, and unclonable. One will have to measure the response to each challenge and establish the unique challenge–response characteristic of any unit, which then becomes a biometric of that unit and enables a PUF.

4.2. Inter-Hamming Distance

The inter-Hamming distance (IHD) is defined as the number of positions where the bits in the response bit stream are different for two different units, divided by the number of bits, averaged over all possible pairs. We can see from Table 2 that the Hamming distance between units 1 and 2 is 2/3, between units 1 and 3 is 0, between units 1 and 4 is 2/3, between units 1 and 5 is 2/3, and between 1 and 6 is also 2/3.

Similarly, between units 2 and 3, it is 2/3; between 2 and 4, it is 0; between 2 and 5, it is 2/3; and between 2 and 6, it is 2/3.

We can continue with the other combinations and then average over all combinations to find that the IHD is 8/15, or 0.533. This is only true for the challenge current of 2.9 mA and will change for a different challenge. The ideal IHD for a PUF is 0.5 [2].

In this example, we arbitrarily assumed that no two MTJs in a unit have the same defect morphology. In reality, this restriction should not exist, and the defect morphology can be repeated in two or even all three MTJs. In that case, we will have $3^3$ = 27 possible combinations (27 different units), and then the IHD will have to be computed by comparing the response bit streams of $(n/2)(n+1)$ [n = 26], i.e., 351 pairs. This process leads to an IHD = 0.479 for the challenge current of 2.9 mA. A different challenge current will, of course, produce a different IHD.

The IHDs we calculate are very similar to those reported in the experiments of refs. [2,3], which reported values of 0.498 and 0.53, respectively.

4.3. Sensitivity to Temperature and Intra-Hamming Distance

We recalculated the switching probability as a function of current at an increased temperature of 330 K to examine whether temperature has an effect on the challenge–response behavior. A strong effect would be undesirable since then the challenge–response set would no longer be a unique fingerprint of the defect morphology. The measure of uniqueness is the intra-Hamming distance, which is the number of positions where the response bits to the same challenge are different for the same unit under two different conditions, such as different temperatures, divided by the number of bits in the response stream. Ideally, the intra-Hamming distance should be zero, indicating that the challenge–response behavior of the PUF is invariant and therefore can act as a true fingerprint or biometric.

The switching probability versus current plot at 330 K is shown in Figure 5. The threshold currents do change somewhat for a 30 Kelvin increase in temperature. We tabulate the threshold currents at 330 K in

Table 3. Threshold currents for different defect morphologies at 330 K for 3 ns current pulse duration.

Defect	Threshold Current (mA)	Threshold Current Density (A/m2)
${\mathrm{C}}_0$	2.75	3.89 $\times {10}^{11}$
${\mathrm{C}}_1$	2.72	3.84 $\times {10}^{11}$
${\mathrm{C}}_{2–3}$	3.12	4.41 $\times {10}^{11}$
${\mathrm{C}}_4$	2.82	3.99 $\times {10}^{11}$
${\mathrm{C}}_5$	2.80	3.95 $\times {10}^{11}$
${\mathrm{C}}_6$	2.75	3.88 $\times {10}^{11}$

. Note that the increase in temperature can either increase or decrease the threshold current depending on the defect morphology.

If we now use the same 3-MTJ unit as before and use the same challenge current of 2.9 mA, then the response bit streams will be exactly the same as in Table 2. This is because the sensitivity of the threshold current (of any defect morphology) to reasonable temperature variations, albeit nonzero, is too weak to change the challenge-response set for the 2.9 mA challenge. Hence, the intra-Hammin distance in this case is zero, attesting to the uniqueness of the challenge–response set for a 2.9 mA current challenge. This may not hold for a different challenge current.

Finally, we wished to study whether the sensitivity of the threshold current to defect morphology persists if a longer pulse duration is used. A longer pulse duration will reduce the absolute value of the threshold current since the integrated spin momentum transfer will increase with increasing pulse duration. Therefore, one would logically expect the variation in the threshold current to decrease if the pulse duration is increased. Hence, we calculated the switching probabilities when the pulse duration is increased from 3 ns to 5 ns. The results are shown in Figure 6.

The threshold currents for different defect morphologies are tabulated in

Table 4. Threshold currents for different defect morphologies at 300 K for 5 ns current pulse duration.

Defect	Threshold Current (mA)	Threshold Current Density (A/m2)
${\mathrm{C}}_0$	0.65	8.98 $\times {10}^{10}$
${\mathrm{C}}_1$	0.66	9.12 $\times {10}^{10}$
${\mathrm{C}}_{2–3}$	0.78	1.07 $\times {10}^{11}$
${\mathrm{C}}_4$	0.68	9.40 $\times {10}^{10}$
${\mathrm{C}}_5$	0.66	9.12 $\times {10}^{10}$
${\mathrm{C}}_6$	0.65	8.98 $\times {10}^{10}$

for a 5 ns pulse duration. One can see that both the average and the standard deviation of the threshold current decrease, but the sensitivity to defect morphology, although weakened, is still visibly present. This underscores the need to optimize the pulse duration in order to obtain the highest sensitivity of the threshold current to defect morphology.

One question remains: what if each MTJ has every type of defect in each manufacturing run? This is a possibility only if the MTJ area is so large that it can accommodate every type of defect. That is why one uses small-area MTJs to eliminate this possibility. Moreover, even if every type of defect is present, it is unlikely that they will be present in exactly the same ratios. Hence, it is not likely that all distinctions between the MTJs will be erased to make the challenge–response set not unique for each PUF.

5. Conclusions

We showed that the threshold current for switching the magnetization of the soft layer of a magnetic tunnel junction via spin transfer torque depends sensitively on the nature of defect(s) introduced during manufacturing. This feature can be exploited to implement a physical unclonable function (PUF). We also calculated the inter- and intra-Hamming distances for a given challenge.

If the soft layer of the MTJ is made magnetostrictive, then its magnetization can be flipped with electrically generated strain [15]. This is more energy efficient than switching with spin transfer torque, which is the method discussed here. More importantly, the threshold strain for strain-mediated switching is even more sensitive to defect morphologies [11] and hence may enable an even stronger PUF. This is left for future study.