# E-Spin: A Stochastic Ising Spin Based on Electrically-Controlled MTJ for Constructing Large-Scale Ising Annealing Systems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Spin-Transfer Torque Magnetic Tunnel Junctions and E-Spin

#### 2.1. Mechanism of the Spin-Transfer Torque Magnetic Tunnel Junctions

#### 2.2. Schematic Diagram, Operation Paradigm, and Simulation of the E-Spin

#### 2.3. Advantages of the E-Spin

#### 2.4. Endurance of the E-Spin

^{15}, it can operate at the frequency of 10 MHz for 3.17 years. The endurance can be increased even more if the SOT-MRAM is included [27]. Because the current is not passed directly through the MTJ in SOT-MRAM, the MTJ is clearly protected, increasing the endurance to almost infinite levels.

## 3. Large-Scale Ising Annealing System for Solving CO Problems Using E-Spins

#### 3.1. Steps of Ising Annealing System Solving CO Problems

#### 3.2. Mapping Integer Factorization Problem to Ising Annealing System

#### 3.3. Stochastic Ising Annealing Algorithm Based on E-Spin

- Map $X$ and $Y$ to the corresponding spin ${x}_{i}$ and ${y}_{i}$ using Equations (7) and (8).
- If the calculated ${x}_{i}$ or ${y}_{i}$ equals value “1”, the MUX outputs ${V}_{H}$ to the corresponding ${I}_{i}$. Then, the E-spin provides a strong likelihood of an output value “1”. The ${V}_{H}$ is the parameter that can be adjusted. In this experiment, we set ${V}_{H}$ to be around 0.95 V and ${V}_{L}$ to be 0.55 V. The random flips of E-spin do not frequently occur, so the gradient descent procedure is dominant in most cases.
- Accordingly, if ${x}_{i}$ or ${y}_{i}$ equals ‘0’, the MUX outputs ${V}_{L}$ to the corresponding E-spin ${I}_{i}$. Then the E-spin provides a high probability of output value “0”.

#### 3.4. The Overall Diagram of the Proposed Ising Annealing System

## 4. Results

#### 4.1. Integer Factorization Results

#### 4.1.1. Examples of Factoring Integers

#### 4.1.2. Comparations for Simulated Annealing, Trial Division, and Ising Annealing Algorithm

#### 4.2. Analysis Results of the E-Spin

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Diagrammatic representation of the Double-MTJ employed in this investigation. (

**b**) Switching probability of Double-MTJ under various current and pulse duration conditions. (

**c**) Process of operation. Pulses for reset, excite, and read are applied in that order.

**Figure 2.**(

**a**) Schematic diagram of the E-spin. (

**b**) Operation paradigm of the E-spin. (

**c**) The layout of the MTJ’s film stack. The technology node of our logic circuit was 40 nm. The MTJ was fabricated on the top layer of the logic circuit. (

**d**) The applied voltage for the reset, excite, and read operations. (

**e**) The parameters of MTJ were measured at 300 K. The low resistance was 2.75$\mathrm{K}\mathsf{\Omega}$ and the high resistance was 7.425$\mathrm{K}\mathsf{\Omega}$. The TMR (Tunnel Magneto Resistance) was 170%. The switch error rate of the MTJ was correlated with the write voltage and pulse width. The minimum write voltage and pulse width were 100 mV and 2 ns, respectively. (

**f**) Time-averaged ${V}_{out}$, <${V}_{out}$>. <${V}_{out}$> is the function of the applied input voltage ${V}_{WL}$ and pulse width. The pulse width is 5 ns in this illustration. The function is fitted to the sigmoidal function. Each data point shown in this figure was averaged for 300 sampling points. (

**g**) Time snapshots of the MTJ’s resistance after each excite operation for various input voltage ${V}_{WL}$ (0.6 V, 0.75 V, and 0.9 V). The MTJ’s resistance was sampled 300 times for each ${V}_{WL}$.

**Figure 3.**MTJ’s lifetime at various operating frequencies. Under a clock frequency of 10 MHz, MTJ with $1\times {10}^{15}$ endurance can operate for 3.17 years.

**Figure 6.**The overall diagram of the proposed Ising annealing system (QFactor). It comprises an E-spin array with 64 E-spins, a digital circuit to implement the QFactor annealing algorithm, and analog MUXs.

**Figure 7.**Examples of factoring 24-bit integers 14019841 (

**a**) and 14166761 (

**b**) using Vivado. ${m}_{10~1}$ and ${m}_{22~11}$ represent {${x}_{10}$,…, ${x}_{1}$} and {${\mathrm{y}}_{10}$,…, ${\mathrm{y}}_{1}$}, respectively. The clock period in this front-end circuit simulation is 500 ns. (

**c**) Schematic view of CMOS-based stochastic Ising spin. It comprises a 32-bit LFSR, a piecewise linear approximation module, and a 16-bit comparator. The behavior of the CMOS-based stochastic Ising spin is the same as described in Equation (2). (

**d**) Schematic view (left) and circuitry (right) of a digital MUX.

**Figure 8.**The required cycles to factor $n$-bit integers. Nc in the Y axis represents the number of required cycles. The four algorithms are simulated using MATLAB software.

**Figure 9.**Layout view of a single CMOS-based stochastic Ising spin (left) and a single MTJ-based E-spin (right). The layout of the CMOS-based stochastic Ising spin is automatically generated by EDA (electronic design automation) software DC (Design Compiler) and ICC (IC Compiler). The layout of the MTJ-based E-spin is a customized design using Virtuoso.

Indicators | Spin Dice [26] | p-bit [16] | E-Spin (This Work) |
---|---|---|---|

probability | Fixed | adjustable | adjustable |

randomness | TRNG | TRNG | TRNG |

operation mode | electric | thermal | electric |

reliability | good | poor | good |

large-scale integration | easy | hard | easy |

**Table 2.**The relationship between the voltage drop of the MTJ and the probability of switching from a low-resistance state to a high-resistance state.

Voltage Drop (mV) ^{1} | Switching Probability |
---|---|

[0,144) | 0% |

[144,171) | 7% |

[171,212) | 20% |

[212,275) | 32% |

[275,342) | 48% |

[342,428) | 66% |

[428,584) | 81% |

[584,718) | 93% |

[718,731) | 98% |

[731,1100] | 100% |

^{1}In this table, the voltage pulse widths are 5 ns.

Indicators | CMOS-Based Stochastic Ising Spin | Thermal Disturbance MTJ-Based Spin (p-bit) [16] | E-Spin (This Work) |
---|---|---|---|

Area (μm^{2}) | 1600 | NA | 280 |

Speed (MHz) | 333 | <1 | 50 |

Power (μW) | 142.6 | 20 | 139.7 |

Technology node | 40 nm | NA (discrete) | 40 nm |

Randomness | PRNG | TRNG | TRNG |

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**MDPI and ACS Style**

Chen, W.; Tang, H.; Wang, Y.; Hu, X.; Lin, Y.; Min, T.; Xie, Y.
E-Spin: A Stochastic Ising Spin Based on Electrically-Controlled MTJ for Constructing Large-Scale Ising Annealing Systems. *Micromachines* **2023**, *14*, 258.
https://doi.org/10.3390/mi14020258

**AMA Style**

Chen W, Tang H, Wang Y, Hu X, Lin Y, Min T, Xie Y.
E-Spin: A Stochastic Ising Spin Based on Electrically-Controlled MTJ for Constructing Large-Scale Ising Annealing Systems. *Micromachines*. 2023; 14(2):258.
https://doi.org/10.3390/mi14020258

**Chicago/Turabian Style**

Chen, Wenhan, Haodi Tang, Yu Wang, Xianwu Hu, Yuming Lin, Tai Min, and Yufeng Xie.
2023. "E-Spin: A Stochastic Ising Spin Based on Electrically-Controlled MTJ for Constructing Large-Scale Ising Annealing Systems" *Micromachines* 14, no. 2: 258.
https://doi.org/10.3390/mi14020258