Properties You Can Calculate with MuMax3

MuMax3 is best suited for mesoscale simulations, meaning it can model magnetization behavior at the micro- and nano-scale, which is crucial for understanding domain structures, coercivity, and thermal stability.

1. Coercivity ($H_c$)

• What it is: The field strength required to reverse magnetization in a material.

• Why it matters: High coercivity prevents unwanted demagnetization.

How to compute in MuMax3:

1. Set up a magnetic system (e.g., NdFeB grains).

2. Apply an external field sweep (e.g., a decreasing field in the opposite direction).

3. Track magnetization reversal ($M$) as a function of the field.

4. Find the field strength at which magnetization crosses zero.

Example Code (lua):

B_ext = vector(1, 0, 0) * 2e-1  -- Apply an external field (T)
relax()                         -- Let the system reach equilibrium
tablesave()                     -- Save initial state

-- Sweep field to determine coercivity
for B = 0, -1, -0.01 do
    B_ext = vector(1, 0, 0) * B
    minimize()  -- Find new equilibrium
    tablesave()  -- Save data
end

Result: Plot $M(H)$ to extract $H_c$.

2. Hysteresis Loop & Energy Product ($BH_{\max}$)

• What it is: The full magnetization curve showing how a magnet responds to an external field.

• Why it matters: The area enclosed in the hysteresis loop determines the maximum energy stored in a magnet.

How to compute in MuMax3:

1. Run a full field loop (+H to -H and back).

2. Extract remanence ($M_r$) and coercivity ($H_c$).

3. Calculate energy product from $B$ and $H$.

Example Code:

for B = -1, 1, 0.01 do
    B_ext = vector(B, 0, 0)
    relax()
    tablesave()
end
for B = 1, -1, -0.01 do
    B_ext = vector(B, 0, 0)
    relax()
    tablesave()
end

Result: Use $B H$ plot to find $BH_{\max}$.

3. Domain Structures & Magnetization Dynamics

• What it is: The formation of magnetic domains and their evolution under applied fields.

• Why it matters: Domain structure impacts coercivity and magnetization reversal.

How to compute in MuMax3:

1. Define grain sizes and magnetic parameters.

2. Simulate relaxation to find equilibrium states.

3. Apply a small perturbation and track domain wall motion.

Example Code:

Aex = 1e-11          -- Exchange stiffness
Ku1 = 1e6           -- Anisotropy constant
Msat = 1.2e6        -- Saturation magnetization
alpha = 0.02        -- Damping parameter

-- Initial magnetization state
m = randomMag()

relax()  -- Find domain equilibrium

-- Apply small external field pulse
B_ext = vector(0.01, 0, 0)
run(1e-9)

Result: Observe domain wall motion in MuMax3’s visualization.

4. Thermal Stability & Curie Temperature Estimation

• What it is: The ability of a magnet to retain its properties at high temperatures.

• Why it matters: High $T_C$ prevents thermal demagnetization.

How to compute in MuMax3:

1. Introduce a random thermal noise field.

2. Measure how magnetization decreases over time.

3. Extract an effective Curie temperature based on changes in $M_s(T)$.

Example Code:

Temp = 300          -- Set temperature (K)
alpha = 0.02        -- Damping
thermalnoise = true -- Enable thermal fluctuations

run(1e-9)  -- Simulate magnetization decay

tablesave()  -- Save final magnetization state

Result: Fit the $M_s(T)$ curve to estimate $T_C$.

5. Anisotropy Energy & Hard-Axis Switching

• What it is: The energy required to rotate magnetization out of its easy axis.

• Why it matters: Strong anisotropy prevents unwanted switching.

How to compute in MuMax3:

1. Set uniaxial anisotropy with varying field angles.

2. Measure energy barriers for magnetization switching.

Example Code:

Ku1 = 5e5       -- High anisotropy constant
m = uniform(1, 0, 0)  -- Start along easy axis

-- Apply external field along different angles
B_ext = vector(0.1, 0, 0)
relax()
tablesave()

Result: Extract energy barrier values to confirm high coercivity.

What MuMax3 CANNOT Directly Compute

MuMax3 works at the micromagnetic scale (~nm to µm). However, some properties require atomic-scale calculations, such as:

• Electronic structure (e.g., Density Functional Theory for Curie temperature)

• Intrinsic anisotropy (DFT calculations needed for Ku1 values)

• Magneto-elastic effects (Requires coupled micromagnetic and structural modeling)

Solution: Combine MuMax3 (micromagnetic) with DFT calculations (quantum-scale) from tools like VASP, Quantum ESPRESSO, or Material Project data.

Summary: What You Can Calculate in MuMax3

Final Thoughts

MuMax3 is great for micromagnetic modeling, but not for atomic-scale properties. We should use:

• MuMax3 → Micromagnetic simulations for coercivity, domain walls, hysteresis.

• DFT (Quantum ESPRESSO, VASP, Materials Project) → Intrinsic material properties like $T_C$, $K_u$, and $M_s$.