Dispersion of a nanotube in vortex state

This notebook will calculate the dispersion relation of a nanotube with 60 nm outer and 40 nm inner diameters, permalloy material parameters. The obtained dispersion(s) must be equivalent to that(those) calculated analytically as well as with micromagnetic simulations published in Ref. 1.

In the last section we compute the same dispersion with the newly added geometry type, namely the Axial waveguides and compare the results.

First we import the TetraX package.

import tetrax as tx

Here, we create a sample, a waveguide with the cross section of a tube, with permalloy-like material parameters. Note, by default the material parameters are set to those of permalloy.

r_in = 20
r_out = 30
lc = 3

tube = tx.Sample(tx.geometries.waveguide.tube(r_in, r_out, lc),
                 name="Nanotube_20_30"
                )

tube.material["Msat"] = 800e3 # A/m
tube.material["Aex"] = 13e-12 # J/m

The tube (created sample) parameters, such as material and mesh, can be listed by typing tube.

tube

Sample

attribute	value
name	Nanotube_20_30
magnetic_order	FM
interactions	['exchange', 'dipole', 'zeeman', 'uniaxial_anisotropy', 'cubic_anisotropy', 'dmi_bulk', 'dmi_interfacial']
external_field (avrg.)	[0. 0. 0.] T
antenna	None
mag (avrg.)	[0. 0. 1.]

SampleMesh (Sample.mesh) of 'Nanotube_20_30'

attribute	value	description
geometry_type	waveguide	geometry type
nx	273	total nodes
nb	105	boundary nodes
scale	1e-09

SampleMaterial (Sample.material) of 'Nanotube_20_30'

name	average	description
Msat	800000.0 A/m	saturation magnetization
Aex	1.2999999999999999e-11 J/m	exchange stiffness
gamma	176085964399.99997 radHz/T	gyromagnetic ratio
alpha	0.008 (is_global)	Gilbert damping
Ku1	0.0 J/m^3	first-order unaxial-anistropy constant
e_u	[0. 0. 1.]	uniaxial anistropy direction
Dbulk	0 (is_global)	bulk DMI constant
Didmi	0 (is_global)	interfacial DMI constant
e_d	[0. 1. 0.]	interfacial DMI direction
Kc1	0.0 J/m^3	first-order cubic-anistropy constant
e_c1	[1. 0. 0.]	first cubic anistropy direction
e_c2	[0. 1. 0.]	second cubic anistropy direction
e_c3	[0. 0. 1.]	third cubic anistropy direction

Hint: Access each parameter using material['name'].
Hint: You can use Sample.get_field('interaction_name'). Same with get_energy and get_interaction.

Here we set an initial state and visualize it.

tube.mag = tx.vectorfields.helical(tube.mesh.xyz, 60, 1)
tube.show()

/Users/attilak/micromag/tetrax-tests/fresh_environement_test/venv/lib/python3.13/site-packages/traittypes/traittypes.py:97: UserWarning: Given trait value dtype "float32" does not match required type "float32". A coerced copy has been created.
  warnings.warn(

Here we set a circular field of 80 mT and visualize it together with the manetization.

Bphi = tx.vectorfields.helical(tube.mesh.xyz, 90, 1) * 0.08
tube.external_field = Bphi
tube.show([tube.mag,tube.external_field])

Computation and visualization of the equilibrium state.

success = False
while not success:
    relax = tx.experiments.relax(tube,tolerance=1e-13)
    success = relax.was_success
tube.show()

Minimizing energy in using 'L-BFGS-B' (tolerance 1e-13) ...
energy length density: -6.727228940911428e-11 J/m, <mag.x> = 0.00, <mag.y> = 0.00, <mag.z> = 0.00000
Success!

/Users/attilak/micromag/tetrax-tests/fresh_environement_test/venv/lib/python3.13/site-packages/traittypes/traittypes.py:97: UserWarning: Given trait value dtype "float32" does not match required type "float32". A coerced copy has been created.
  warnings.warn(

Now let us calculate the spin-wave spectra:

tube.spectrum = tx.experiments.eigenmodes(tube, num_cpus=-1, num_modes=10, kmin=-40e6, kmax=40e6, num_k=81)

100%|███████████████████| 41/41 [00:01<00:00, 28.28it/s]

We plot the dispersion for the first 5 modes (n=[0,4]), using the built-in plot() method.

tube.spectrum.plot(n=[0,4], renderer="notebook") # remove the renderer argument when running on your computer

The same dispersion can be computed using the waveguide-axial sample geometry, as we show in the following:

Here again we define the tube, set material parametes, magnetization, external field, relax the initial state and compute the spectrum.

tube_axial = tx.Sample(tx.geometries.waveguide_axial.tube(r_in, r_out, 1),
                      name="Nanotube_axial_20_30"
                      )

tube_axial.material["Msat"] = 800e3 # A/m
tube_axial.material["Aex"] = 13e-12 # J/m
tube_axial.mag = tx.vectorfields.helical(tube_axial.mesh.xyz,60,1)
tube_axial.external_field = 80e-3 * tx.vectorfields.helical(tube_axial.mesh.xyz,90,1)

Here we compute the equilibrium, that is expected to be in the \(\phi\) direction. (\(x\),\(y\),\(z\)) coordinates for the axial waveguide mean (\(\rho\),\(\phi\),\(z\)).

success = False
while not success:
    relax_axial = tx.experiments.relax(tube_axial, tolerance=1e-18)
    success = relax_axial.was_success

Minimizing energy in using 'L-BFGS-B' (tolerance 1e-18) ...
energy area density: -1.072744950755669e-02 J/(rad m), <mag.x> = -0.00, <mag.y> = 1.00, <mag.z> = -0.00
Failure. Returning initial state.
Minimizing energy in using 'L-BFGS-B' (tolerance 1e-18) ...
energy area density: -1.072744950755065e-02 J/(rad m), <mag.x> = 0.00, <mag.y> = 1.00, <mag.z> = -0.000
Success!

tube_axial.spectrum = tx.experiments.eigenmodes(tube_axial,num_cpus=-1, num_modes=10, kmin=-40e6, kmax=40e6, num_k=81)

100%|████████████████| 446/446 [00:01<00:00, 338.57it/s]

tube_axial.spectrum.plot(m=[-2,2], n=0, renderer="notebook") # remove the renderer argument when running on your computer