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MTJ Matlab Model

A Physics Based Approach

The Landau–Lifshitz–Gilbert Equation

Describes the precessional motion of magnetization in a solid. (Magnetization is related to magnetic field by µ0)

The equation can be modified to include Spin Torque Transfer via Slonczewski's STT term:

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t

mmMhmM

t

mSeffS

pj mmmT xS

j tMe

JT

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The Full LLG + STT Model

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Transfer TorqueSpin

Effect Damping :Gilbert

Lifshitz–Landau

pjSeffS mmmTt

mmMhmM

t

m

xSj tMe

JT

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Material Magnetization Reduced Planck constant

Magnetization Saturation Torque Factor of Current

Effective Magnetic Field Current Density

Damping Constant Elementary Charge

Gyromagnetic Ratio Permeability of Free Space

Vector Direction of the Spin Torque Thickness of the Magnetic Layer

SM

effh

m

pm xt0eJ

Matlab Simulation Results

Implemented in 3 Matlab files:– instantiateMTJ.m: creates an MTJ object– MTJtransient.m: simulates a single time step given an

input current to the MTJ– getMTJresistance.m: calculates the resistance across the

MTJ. (Assumed to be linear between Rp and Rap)

Observed DC and AC response of the model appears to correlate very well to published MTJ behavior.

Simulation results use the following MTJ parameters (extracted from Ilya’s MTJ cell moding email):

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RP 640Ω Length 140nm

RAP 1550Ω Width 90nm

α 0.01 Thickness 1nm

Simulated DC Response (1)

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

1400

1600

1800

2000

Applied Current (mA)

Mea

sure

d R

esis

tanc

e (

)MTJ Resistance vs. Current

Switching Current:~432 μA

Simulated DC Response (2)

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

1400

1600

1800

2000

Applied Voltage (V)

Mea

sure

d R

esis

tanc

e (

)MTJ Resistance vs. Voltage

Switching Voltage:~620 mV

Simulated AC Response (1)

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0 2 4 6 8 10 12 14 16 18 20500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

Time (ns)

Mea

sure

d R

esis

tanc

e (

)

Sample Transient Response of MTJ:Above Switching Threshold

Rp

RapMTJ Resistance

Simulated AC Response (2)

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0 2 4 6 8 10 12 14 16 18 20500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

Time (ns)

Mea

sure

d R

esis

tanc

e (

)

Sample Transient Response of MTJ:Well Below Switching Threshold

Rp

RapMTJ Resistance

Simulated AC Response (3)

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0 2 4 6 8 10 12 14 16 18 20500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

Time (ns)

Mea

sure

d R

esis

tanc

e (

)

Sample Transient Response of MTJ:Very Near, But Not Above Switching Threshold

Rp

RapMTJ Resistance

High Frequency Resistance Oscillations

Simulated AC Response (4)

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0 2 4 6 8 10 12 14 16 18 20500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

Time (ns)

Mea

sure

d R

esis

tanc

e (

)Varying the Layer Thickness by 5%

Nominal

-5%+5%

~1.5 ns difference in switching times

Summary of Results

DC Response– Behaves very similar to literature– Predicted switching threshold of 432 μA, very close to

Ilya’s prediction.

AC Response– Switching is delayed, resistance change is rapid– Resistance displays damped oscillations during switching– Resistance oscillates when near switching current (does

this actually happen?) resulting in a lower “DC” resistance– Small variations in the MTJ can have a large impact on

switching delay

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Future Work

Current Matlab model does not take into account temperature dependences.– Model could be easily adapted to capture temperature

effects

Develop model into VerilogA and SPICE– SPICE implementation potentially problematic

Fit model to experimental MTJ data from Ilya

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