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davidcorteso
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Commented on the DMI convention we are using for micromagnetics. Added derivations in C libraries
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‎fidimag/micro/lib/dmi.c‎

Lines changed: 96 additions & 56 deletions
Original file line numberDiff line numberDiff line change
@@ -87,32 +87,51 @@ void dmi_field_bulk(double *m, double *field, double *energy, double *Ms_inv,
8787
* where j is the index of a NN neighbour and D_{ij} = D * r_{ij}, r_{ij}
8888
* being the position vector connectin the site i with the site j
8989
*
90-
* In the continuum, the field can be written as : - nabla X M , which can
91-
* be discretised with an expression similar to the atomic model one when
92-
* doing the finite difference approximation of the derivative, thus we
93-
* only need the DMI vector as: -D * r_{ij}
94-
*
95-
* So, in the loop through neighbours we compute:
96-
*
97-
* neighbour field sum this gives
98-
* -x: (D / dx) * (+x X M) --> Components in y, z
99-
* +x: (D / dx) * (-x X M) --> %
100-
* -y: (D / dy) * (+y X M) --> Components in x, z
101-
* +y: (D / dy) * (-y X M) --> %
102-
* -z: (D / dz) * (+z X M) --> Components in x, y
103-
* +z: (D / dz) * (-z X M) --> %
104-
*
105-
* which gives:
106-
* field_x = (D / dx) * (mz[-x] - mz[+x])
107-
* + (D / dz) * (mx[-z] - mx[+x])
90+
* In the continuum, the field can be written as (M = Ms m):
91+
* ->
92+
* - (2 D / mu0 Ms) nabla X m ,
93+
*
94+
* which can be discretised with an expression similar to the atomic model
95+
* one when doing the finite difference approximation of the derivative,
96+
* thus we only need the DMI vector as: -D * r_{ij}. Specifically, we have
97+
* that:
98+
*
99+
* ----> -> ^ ^
100+
* field = - 2 D nabla X m = -D [ (d m_z - d m_y) x + (...) y + ]
101+
* ---- ----- -----
102+
* mu0 Ms d y d z
103+
*
104+
* Discretising the derivatives, the field components are:
105+
*
106+
* field_x = - (2 D / mu0 Ms) [ (1 / 2 dy) * (mz[+y] - mz[-y]) -
107+
* (1 / 2 dz) * (my[+z] - my[-z])
108+
* ]
109+
*
110+
* field_y = - (2 D / mu0 Ms) [ (1 / 2 dz) * (mx[+z] - mx[-z]) -
111+
* (1 / 2 dx) * (mz[+x] - mz[-x])
112+
* ]
108113
* ...
109114
*
110-
* which are the first order derivatives.
115+
* where +x, -x, etc are the neighbours at position +x, -x, etc
116+
* We now can collect the terms involving the neighbours at
117+
* +x, -x, +y, -y, +z and -z.
118+
* For example the field H where the +x neighbour is involved is:
119+
*
120+
* -> ^ ^
121+
* H(+x) = (-2 D / mu0 Ms) * ( my(+x) z - mz(+x) y ) * (1 / 2 dx)
122+
*
123+
* ^ ->
124+
* = (- D / mu0 Ms dx) * ( x X m )
125+
*
126+
* In general, the field for every component will be the cross product of
127+
* r_ij with m, where r_ij is the vector towards the j-th neighbour, as in
128+
* the discrete case. This term is divided by the mesh discretisation
129+
* (dx, dy or dz) in the corresponding direction
111130
*
112131
* The DMI vector norms are given by the *D array. If our simulation has
113132
* n mesh nodes, then the D array is (6 * n) long, i.e. the DMI vector norm
114133
* per every neighbour per every mesh node. The order is the same than the NNs
115-
* array, i.e.
134+
* array, i.e.
116135
*
117136
* D = [D(x)_0, D(-x)_0, D(y)_0, D(-y)_0, D(z)_0, D(-z)_0, D(x)_1, ...]
118137
*
@@ -130,13 +149,14 @@ void dmi_field_bulk(double *m, double *field, double *energy, double *Ms_inv,
130149
/* The DMI vector directions are the same according to the neighbours
131150
* positions. Thus we set them here to avoid compute them
132151
* every time in the loop . So, if we have the j-th NN,
133-
* the DMI vector will be dmivector[3 * j] */
134-
double dmivector[18] = {-1, 0, 0,
135-
1, 0, 0,
136-
0, -1, 0,
137-
0, 1, 0,
138-
0, 0, -1,
139-
0, 0, 1
152+
* the DMI vector will be dmivector[3 * j]
153+
* */
154+
double dmivector[18] = {-1, 0, 0, // -x
155+
1, 0, 0, // +x
156+
0, -1, 0, // -y
157+
0, 1, 0, // +y
158+
0, 0, -1, // -z
159+
0, 0, 1 // +z
140160
};
141161

142162
/* These are for the DMI prefactor or coefficient */
@@ -175,7 +195,7 @@ void dmi_field_bulk(double *m, double *field, double *energy, double *Ms_inv,
175195
* is larger than zero */
176196
if (Ms_inv[ngbs[idn + j]] > 0){
177197

178-
/* We do here: (D / dx_i) * ( r_{ij} X M_{j} )
198+
/* We do here: -(D / dx_i) * ( r_{ij} X M_{j} )
179199
* The cross_i function gives the i component of the
180200
* cross product. The coefficient is computed according
181201
* to the DMI strength of the current lattice site.
@@ -232,44 +252,64 @@ void dmi_field_interfacial(double *m, double *field, double *energy, double *Ms_
232252
/* In the atomic model, the effective field in the i-th spin site has the
233253
* summation: D_{ij} x S_{ij}
234254
*
235-
* For the Interfacial DMI, D_{ij} has the structure: D_{ij} = r_{ij} X z
236-
* See Rohart et al. Phys. Rev. B 88, 184422)
237-
* We will use R&T's convention for the DMI sign.
255+
* For the Interfacial DMI, we use the following energy density structure
256+
* with Lifshitz invariants (see Rohart et al. Phys. Rev. B 88, 184422))
257+
*
258+
* w_DM = D ( L_{xz}^{(x)} + L_{yz}^{(y)} )
259+
*
260+
* Hence, using a variational derivative, the field can be computed as
261+
*
262+
* / -> -> \
263+
* field = - 2 D | ^ dm ^ dm |
264+
* ---- | y X -- - x X -- |
265+
* mu0 Ms \ dx dy /
266+
*
267+
* Using finite differences for the derivatives, we can get the components
268+
* contributed to the field by the neighbours in the +x, -x, +y and -y
269+
* directions. For instance,
238270
*
239-
* Notice that in [Yang et al. Phys. Rev. Lett. 115, 267210] they use the
240-
* opposite sign.
271+
* / \
272+
* field(+x) = -2 D | mz(+x) ^ mx(+x) ^ |
273+
* ---- | ----- x - ----- z |
274+
* mu0 Ms \ 2 dx 2 dx /
275+
*
276+
* - D 1 ^ ->
277+
* = ---- -- ( y X M(+x) )
278+
* mu0 Ms dx
241279
*
242-
* The Dzyaloshinskii vectors are IN plane. This function only works
243-
* along the XY plane since we assume there is a non magnetic material below
244-
* with a different SOC which gives the DMI for neighbouring in plane spins
245280
*
246-
* The computation is similar than before, only that we no longer have
247-
* a z component. In the continuum the H_dmi field is:
281+
* In general, the derivative will be given as in the discrete spin model
282+
* by using the Dzyaloshinskii vector as
248283
*
249-
* H_dmi = (2 * D / (Ms a a_z)) * ( x X dM/dy - y X dM/dx )
284+
* ^ ->
285+
* D_ij = z X r_ij
250286
*
251-
* where "a" is the lattice spacing in the plane and "a_z" the system
252-
* thickness along z
287+
* and dividing by the mesh discretisation in the direction of the
288+
* neighbour.
289+
* This DMI vector convention is given by [Yang et al. PRL 115, 267210]
253290
*
254-
* This derivative can be obtained doing the cross product
255-
* as in the atomic model, hence when going through the neighbours
291+
* The Dzyaloshinskii vectors are IN plane. This function only works along
292+
* the XY plane since we assume there is a non magnetic material below with
293+
* a different SOC which gives the DMI for neighbouring in plane spins
294+
*
295+
* As in the atomic model, when going through the neighbours
256296
* loop, we can obtain the field doing the following cross products:
257297
*
258298
* neighbour field sum
259-
* -x: (D / dx) * (+y X M)
260-
* +x: (D / dx) * (-y X M)
261-
* -y: (D / dy) * (-x X M)
262-
* +y (D / dy) * (+x X M)
299+
* -x: (D / dx) * (-y X M)
300+
* +x: (D / dx) * (+y X M)
301+
* -y: (D / dy) * (+x X M)
302+
* +y (D / dy) * (-x X M)
263303
*
264304
* So, our Dzyaloshinskii vectors can be depicted in the cuboid mesh as
265305
*
266306
* o +y
267307
* |
268-
* --> D
269-
* ^ |
308+
* <-- D
309+
* | ^
270310
* -x o __ | __ o __ | _ o +x
271-
* | v
272-
* <--
311+
* v |
312+
* -->
273313
* |
274314
* o -y
275315
*
@@ -279,7 +319,7 @@ void dmi_field_interfacial(double *m, double *field, double *energy, double *Ms_
279319
* The DMI vector norms are given by the *D array. If our simulation has
280320
* n mesh nodes, then the D array is (4 * n) long, i.e. the DMI vector norm
281321
* per every neighbour per every mesh node. The order is the same than the NNs
282-
* array, i.e.
322+
* array, i.e.
283323
*
284324
* D = [D(x)_0, D(-x)_0, D(y)_0, D(-y)_0, D(x)_1, ...]
285325
*
@@ -294,10 +334,10 @@ void dmi_field_interfacial(double *m, double *field, double *energy, double *Ms_
294334
* (NNs are in the order: -x, +x, -y, +y)
295335
* For interfacial DMI we only compute the DMI in 2D
296336
* */
297-
double dmivector[12] = { 0, 1, 0,
298-
0, -1, 0,
299-
-1, 0, 0,
337+
double dmivector[12] = { 0, -1, 0,
338+
0, 1, 0,
300339
1, 0, 0,
340+
-1, 0, 0,
301341
};
302342
double dxs[4] = {dx, dx, dy, dy};
303343

@@ -327,7 +367,7 @@ void dmi_field_interfacial(double *m, double *field, double *energy, double *Ms_
327367
if (ngbs[idn + j] >= 0) {
328368

329369
/* DMI coefficient according to the neighbour position */
330-
DMIc = D[idn_DMI + j] / dxs[j];
370+
DMIc = -D[idn_DMI + j] / dxs[j];
331371

332372
/* Magnetisation array index of the neighbouring spin
333373
* since ngbs gives the neighbour's index */

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