Auxiliary Material for On the magnetocrystalline

Auxiliary Material for On the magnetocrystalline anisotropy of greigite (Fe3S4) Michael Winklhofer1, Liao Chang2,3, and Stephan H. K. Eder1 1

Department of Earth and Environmental Sciences, Ludwig-Maximilians-University Munich, Theresienstr. 41, D-80333 Munich, Germany

2

Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia 3

Paleomagnetic Laboratory ‘Fort Hoofddijk’, Department of Earth Sciences, Utrecht University, Budapestlaan 17, 3584 CD Utrecht, The Netherlands

Geochemistry, Geophysics, Geosystems, 2014 Introduction The auxiliary material consists of an Appendix (Sections Ax) and supplementary slides. Figures A1, A2 and Tables A1-A3 are included in this README. Structure of Appendix: A1. Higher order cubic magnetocrystalline anisotropy terms a) Series representations of 𝐹!" and definitions of 𝐾! , 𝐾! , 𝐾!   b) Easy axes orientation along symmetry axes and non-major cubic axes (Figure A1a,b) c) Temperature dependence of the 𝐾!

A2. Computational details of the fitting procedure a) Calculation of resonance fields b) Forward calculation of powder spectra c) Fitting of FMR powder spectra d) Performance

A3. Results from spectral fitting a) Sensitivity study: Role of uncertainty in g-factor (Table A1a,b) b) Fits for X-band data of cement-embedded samples recorded at 290 K (Table A2a,b) c) Comparison between X- and Q-band FMR data for sample SYN519 (Figure A2 and Table A3)

Auxiliary slides “TEM_ED“ (Fig_SI_Revisiting_ED_YamaguchiWada.pdf) The supplement shows our re-analysis of the electron diffraction pattern that Yamaguchi and Wada [1970] considered as evidence of a easy axes orientation in greigite. We suggest that the reflections in question are not due to (400), but (440). Either way, the incompleteness of the original diffractograms does not allow for a texture analysis.

Appendix A1 Higher order cubic magnetocrystalline anisotropy terms a) Series representations of 𝑭𝐚𝐧 and definitions of 𝑲𝟏 , 𝑲𝟐 , 𝑲𝟑   The cubic magnetocrystalline anisotropy energy (MAE) density is usually expanded in even powers of directional cosines 𝛼! of the magnetization (i=1,2,3), 𝐹!" 𝛼! , 𝛼! , 𝛼! ; 𝐾! , 𝐾! , 𝐾! , … = 𝐾! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + +𝐾! 𝛼!! 𝛼!! 𝛼!! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + ⋯ ,

(A1)  

where 𝐾! , 𝐾! , and  𝐾! are referred to as first-, second-, and third-order magnetocrystalline ansiotropy constants, respectively. Importantly, 𝐹!" is the Helmholtz free energy, which is measured at constant temperature and volume (strain free), which should hold reasonably well for particles embedded in a mechanically strong matrix. The following representation for the MAE is also found in the literature, 𝐹!" ′(𝛼! , 𝛼! , 𝛼! ; 𝐾! , 𝐾!! , 𝐾! ) = 𝐾! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + +𝐾!! 𝛼!! 𝛼!! 𝛼!! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! ! ,

(A2)  

which because of 𝛼!! + 𝛼!! + 𝛼!! = 1  is equal to 𝐹!" (𝛼! , 𝛼! , 𝛼! ; 𝐾! , 𝐾!! , 𝐾! ) = 𝐾! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + + 𝐾!! + 2𝐾! 𝛼!! 𝛼!! 𝛼!! + 𝐾! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! ,

(A3)

so that 𝐾! of (A1) corresponds to 𝐾!! + 2𝐾! of (A2). This mixing of coefficients for the 𝛼!! 𝛼!! 𝛼!! term in Eq. (A3) is confusing and can be avoided by using (A1). b) Easy axes orientation along symmetry axes and non-major cubic axes Depending on the relative magnitude of 𝐾! , 𝐾! , 𝐾! , any of the crystallographic axis of high symmetry ( – four fold, - two fold, and three fold) can be an easy, intermediate, or hard axis (see Fig. A1a for 𝐾! >0 and Fig. S1b for 𝐾! 0 is the minimum of the MAE, while it is the maximum for 𝐾! 0), then there is a region in the 𝐾! /𝐾! , 𝐾! /𝐾! plane (above the red line in Fig. A1b), where the easy axis lies between a and an adjacent axis (or between a and axis, black field). Such a more general (or ) orientation of the easy axis was indeed observed in some cubic Laves compounds RFe2 (R=Ce, Lu) [Atzmony & Dariel, 1974].

Figure A1a “Phase diagram” showing the energetic ordering of symmetry axes (easy axis < intermediate axis < hard axis) for K 1 > 0 with the magnetocrystalline anisotropy energy given by (A1). Each one of the symmetry axes can be an easy, intermediate, or hard axis.

Figure A1b As in Fig A1a, now for K 1 < 0. As opposed to the case K1 > 0, the easy axis now need not coincide with a major symmetry axis. Above the red line, the easy axis lies between a axis and axis and has a [uuw] orientation. In the black field, the easy axis is between and . The relative ordering of the symmetry axes in terms of energy is not affected by the tilt. Below the red line, the (left) or axes define the easy axes of magnetization.

c) Temperature dependence of the 𝑲𝒊 Although representation (A1) of the MAE is compact and useful for computational purposes, it is not the most appropriate representation for analysis of the temperature dependence of the MAE in terms of the Akulov-Zener-law [see Callen and Callen, 1966, J. Phys. Chem. Solids, 27, 1271-85] which states that 𝑀! (𝑇) ℓ𝓁(ℓ𝓁!!)/! 𝜅ℓ𝓁 (𝑇)   = , (A4) 𝜅ℓ𝓁 (0)   𝑀! (0)   where the 𝜅ℓ𝓁 are magnetocrystalline anisotropy coefficients, which are related to an expansion of the MAE in terms of spherical harmonics. Including 𝜅ℓ𝓁 coefficients up to order ℓ𝓁 = 6, Zener [1954, Phys. Rev. 96, 1335-37;] gave the following expression for a “zero-mean” cubic MAE ! 𝐹!" in terms of directional cosines ! 𝐹!" 𝛼! , 𝛼! , 𝛼! ; 𝜅! , 𝜅! = 𝜅! 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! −

+𝜅! 𝛼!! 𝛼!! 𝛼!! − where 𝜅! = 𝐾! +

! !!

1 2 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + 𝛼!! 𝛼!! + , 11 231

1 +⋯ 5 (A5)

𝐾! and 𝜅! = 𝐾! . The numerical constants in (A3) (1/5 and -2/231) are

! chosen such that the average of 𝐹!" 𝛼! , 𝛼! , 𝛼! ; 𝜅! , 𝜅! over the surface of the unit sphere ! ! ! ! vanishes, while the average of 𝐹!" 𝛼! , 𝛼! , 𝛼! ; 𝐾! , 𝐾! is 𝐾! + 𝐾! . Note that − ∙

−

! !"#

!

!"#

!"#

!

! !!

=

.

When 𝐾! cannot be neglected in (A1), then 𝐾! will contribute to 𝜅! and 𝜅! and equation (A5) needs to be expanded to a higher order. The relationships between the 𝜅ℓ𝓁 and 𝐾! can be obtained with the procedure detailed in Callen and Callen [1960, J. Phys. Chem. Solids, 16, 310-328, Appendix B therein], which they carried out to order ℓ𝓁 = 6. When including 𝐾! (i.e., going to order ℓ𝓁 = 8), we obtain the following relationships: 1 24 1 18 𝐾! + 𝐾! ,    𝐾! = 𝜅! − 𝜅! − 𝜅 , 11 143 11 65 ! 6 6  (A6)         𝜅! = 𝐾! − 𝐾! , 𝐾! = 𝜅! + 𝜅! , 5 5 𝜅! = 𝐾! , 𝐾! = 𝜅! ,

𝜅! = 𝐾! +

!

and 𝜅! = 𝐾! + 21𝐾! + 𝐾! + 3𝐾! . Therefore, if 𝐾! is similar in magnitude to 𝐾! , it has a !"# stronger influence on the anisotropy than 𝐾! has. Hence the temperature dependence of 𝐾! and 𝐾! predicted by (A4) then is: 1 24 𝑀! 𝑇 𝐾! 𝑇! + 𝐾! 𝑇! 𝑀! 𝑇! 11 143 !" 6 18 𝑀! 𝑇 𝑀! 𝑇 − 𝐾! 𝑇! −  𝐾! T! 𝑀! 𝑇! 𝑀! 𝑇! 5 65

!"

!"

!"

𝐾! (𝑇)   = 𝐾! 𝑇! + −

1   𝐾 𝑇 11 ! !

and 6 𝐾! 𝑇 = 𝐾! 𝑇! − 𝐾! 𝑇! 5

𝑀! 𝑇 𝑀! 𝑇!

6 +  𝐾! T! 5

𝑀! 𝑇 𝑀! 𝑇!

+ !"

, (A7)

,

(A8)

For greigite, Chang et al. [2008] reported a 7-8% decrease of M! from 𝑇! =5 K to room temperature.

A2: Computational details of the fitting procedure a) Calculation of resonance fields: The resonance field 𝐵!"# (𝜙! , 𝜃! ) for a given set of anisotropy parameters and for a given orientation of the external field vector with respect to the [001] axis of the crystal was obtained by simultaneously solving (numerically) the resonance condition (Eq. 1 in paper) and the equilibrium condition (𝑑𝐹/𝑑𝜙 = 0, 𝑑𝐹/𝑑𝜃 = 0) for 𝜙 → 𝜙!" , 𝜃 → 𝜃!" , and 𝐵 → 𝐵!"# . The search was performed with the function FindRoot in Mathematica (Wolfram Research). The initial guesses for 𝜙!" and 𝜃!" were 𝜙! and 𝜃! , respectively, and the search range for 𝐵!"#  was restricted to the range defined by the minima and !!!!! !!!"! !!""! , 𝐵!"# , 𝐵!"# } (see Eq. 3 in paper), except for maxima of the set SB = {𝐵!"# or easy axis orientations. As initial guess for 𝐵!"# we chose the value from the analytical first-order approximation derived by Schlöman [1958], provided that value was inside the search range. In case it was not, the intermediate value of SB was used. b) Forward calculation of FMR powder spectra: 𝐵!"# (𝜙! , 𝜃! ) was first computed according to A2.a) for 100 points in the field ! ! orientation range 𝜙! = 0, , 𝜃! = [0.01, ]. The 𝐵!"# (𝜙! , 𝜃! ) surface was then !

!

interpolated on a finer grid (100 x 100 points) equidistant in 𝜙! and cos 𝜃! , using an interpolation order of 3. The resulting 𝐵!"# 𝜙! , cos 𝜃! dataset was histogrammed to obtain the pdf(𝐵!"# ), which then was convolved with the derivative of a Gaussian in order to obtain a derivative absorption spectrum. The convolution was performed in Fourier space, using the discrete Fourier transform. c) Fitting of FMR powder spectra The best-fit model parameters K1, K2, K3, g, Δ𝐵 were obtained by minimizing the sum of squared differences between experimental and model spectra, using the NMinimize function in Mathematica with the NelderMead method (a simplex scheme). Both experimental and simulated spectra were normalized in the same way [ (max-min)/2 norm] so as to avoid another fit parameter. Control minimizations were performed with the computation-time intensive SimulatedAnnealing method to check if there were other parameter combinations in the global search space which had not been found with the simplex method under the given initial guesses.

d) Performance: The simulation of a derivative absorption spectrum for a given anisotropy model takes about a second on a MacBook Air. The fitting of an experimental spectrum may take several hours (for the full set of five fitting parameters) depending on the distance between initial guesses and final values, but can be reduced to about 10 - 40 minutes when two of the five model parameters have prescribed values (i.e., K3=0, g=2.08). Of course, the run time depends on the requested precision and on how close the initial guesses are to the final guess.

A3. Results from spectral fitting a) Sensitivity study: Role of uncertainties in g-factor The intrinsic g value (“g-factor”) of greigite is not known, but likely to be in the range 2.05 to 2.1 (see paper). To find out how uncertainties in g affect the fitted value of K1, we forward calculated FMR absorption derivative spectra for a given set of anisotropy parameters K1, K2, g and effective intrinsic Gaussian linewidth Δ𝐵 and then fitted the spectra with prescribed values of g. The results in Table A1a,b show that overestimation of the g value leads to an underestimation of the magnitude of K1 and vice versa. The fit error parameter s depends quadratically on (gest – gtrue), where gtrue and gest are the true and estimated values of g. g 𝐾!!"#

2.04

2.05

2.06

2.07

2.08

2.09

2.1

2.12

2.14

   

-2.987

-2.916

-2.847

-2.773

-2.701

-2.630

-2.560

-2.405

-2.249

   

-0.578

-0.586

-0.582

-0.586

-0.583

-0.582

-0.583

-0.595

-0.590

Δ𝐵 [mT]

57.6

58.5

58.8

59.4

60.0

60.7

61.3

62.9

64.4

δ𝐵!" [mT]

4.65

5.28

6.01

6.84

7.67

8.48

9.25

11.17

13.37

s

0.020

0.01

0.005

0.001

0. δ𝐵!" = 𝐵!"# − 𝐵!"

ii) for K1 0  models for sample SYN519 is twice as large as that for sample SYN702 (compare Table 1 in paper with Table A3 here), which is readily explained by the different 𝑔 factors (1.93 vs. 2.08). As a consequence, the anisotropy field is twice as large, too. Even so, the energy barriers Δ𝐹!" remain at about 2.5 kJ/m3, irrespective of the g factor. As far as the 𝐾! < 0 models are concerned, the magnitude of 𝐾! in the models for sample SYN519 is between four and five times larger compared to the models for sample SYN702 (compare Table 1 and Table A3), and most strikingly, 𝐾! now has large positive values instead of small negative values. Yet, as with the 𝐾! > 0 models for SYN519, the energy barriers  ∆𝐹!" calculated for the 𝐾! < 0 models do not differ much from those listed in Table 1. The large negative 𝐾! values obtained are related to the high field side of the FMR derivative spectrum (Figure A2), which at Q-band frequencies has a long negative tail that extends beyond the maximum field of 2 T. This tail is even more pronounced for sister sample SYN706 from Chang et al. [2011].

Model

𝐾!   [kJ/m3]

𝐾! 𝐾!

𝐾! 𝐾!

g

a519

−68.9

−2.2

−1.3

2.08

b519

−72.8

−2.4

−1.2

2.05

c519

−83.3

−2.2

−0.3

1.89

d519

−74.6

−1.5

−0.3

1.90

e519

−69.4

−1.8

0

1.90

f519

−33.3

−0.4

−0.2

2.04

519

−80.4

−5.7

−2.4

1.92

a519

+16.3

+3.5

0

2.08

b519

+16.8

+4.1

−0.8

2.04

c519

+16.7

+5.6

+0.7

2.04

d519

+18.5

+4.5

−0.3

2.04

f519

+20.6

+9.8

−2.1

1.93

f [GHz] 9.4 (34.0) (9.4) 34.0 9.4 34.0 9.4 34.0 9.4 34.0 (9.4) 34.0 9.4 34.0 (9.4) 34.0 (9.4) 34.0 (9.4) 34.0 (9.4) 34.0 9.4 34.0

Δ𝐵 [mT] 73 (95) (71) 112 77 91 66 105 81 103 (70) 100 61 92 (68) 80 (63) 88 (75) 90 (67) 89 65 88

fit categ. 1 3 2 2 3 2 2 1 2 2 3 3 1 2 3 2 4 3 4 2 3 2 2 2

𝐵!" [mT]

Δ𝐹!" [kJ/m3]

104

2.5

82

1.9

116

0.6

205

2.6

150

1.1

160

2.5

208

1.0

135

4.0

140

3.3

241

4.9

154

4.2

171

2.4

Table A3 Overview of anisotropy models A! = 𝐾! , 𝐾! , 𝐾! ; 𝑔 ! found for greigite powder SYN519 dispersed in eicosane. Symbols as in Table 1 of paper. The column ‘fit category’ (1 = very good) is a qualitative rank for the goodness of fit for a given solution at X-band (9.40 GHz) and Q-band (34.0 GHz). For illustration, see Figure A2. For most of the cases, experimental data were only fitted in one band, so that the values for ΔB in the other band indicate approximate values for ΔB and are therefore set in parentheses.

Figure A2 (Following three pages) X- and Q-band data for greigite powder SYN519 dispersed in eicosane (solid line), with simulated spectra (red dots) for the models listed in Table A3. All spectra are normalized.

aS519 û Q-band

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

aS519 û X-band

0.0

-0.5

-0.5 -1.0 0

0.0

200

B @mTD 400

600

-1.0 600

800

bS519 û Q-band

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

bS519 û X-band

0.0

0.0

-0.5

-0.5

-1.0 0

-1.0 600

200

B @mTD 400

600

800

1.0

1.0

0.5

0.5

0.0

0.0 -0.5

-0.5 200

B @mTD 400

600

-1.0 600

800

dS519 û X-band 1.0

0.5

dc''êdB

dc''êdB

B @mTD

800 1000 1200 1400 1600 1800 2000

dS519 û Q-band

1.0

0.0

0.5 0.0 -0.5

-0.5 -1.0 0

B @mTD

800 1000 1200 1400 1600 1800 2000

cS519 û Q-band

dc''êdB

dc''êdB

cS519 û X-band

-1.0 0

B @mTD

800 1000 1200 1400 1600 1800 2000

200

B @mTD 400

600

800

-1.0 600 800 1000 1200 1400 1600 1800 2000

B @mTD

eS519 û Q-band

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

eS519 û X-band

0.0

-0.5

-0.5

B @mTD fS519 û X-band

200

400

600

-1.0 600 800 1000 1200 1400 1600 1800 2000

B @mTD fS519 û Q-band

800

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

-1.0 0

0.0

0.0

0.0 -0.5

-1.0 0

-1.0 600 800 1000 1200 1400 1600 1800 2000

B @mTD S519 û X-band

200

400

600

B @mTD S519 û Q-band

800

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

-0.5

0.0

-0.5

-0.5

B @mTD aS519 û X-band

200

400

600

-1.0 600 800 1000 1200 1400 1600 1800 2000

B @mTD aS519 û Q-band

800

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

-1.0 0

0.0

0.0

0.0

-0.5

-0.5

-1.0 0

-1.0 600 800 1000 1200 1400 1600 1800 2000

200

B @mTD 400

600

800

B @mTD

bS519 û Q-band

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

bS519 û X-band

0.0

0.0 -0.5

-0.5 -1.0 0

200

B @mTD 400

600

-1.0 600

800

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

cS519 û X-band

0.0

0.0 -0.5

-0.5 -1.0 0

200

B @mTD 400

600

800

-1.0 600

1.0

1.0

0.5

0.5

dc''êdB

dc''êdB

dS519 û X-band

0.0

-0.5

-1.0 0

-1.0 600

B @mTD eS519 û X-band

200

400

600

800

0.5

0.5

dc''êdB

1.0

0.0

0.0

-0.5

-0.5

-1.0 0

-1.0 600

B @mTD 400

600

B @mTD

800 1000 1200 1400 1600 1800 2000

eS519 û Q-band

1.0

200

B @mTD dS519 û Q-band

800 1000 1200 1400 1600 1800 2000

0.0

-0.5

dc''êdB

B @mTD cS519 û Q-band

800 1000 1200 1400 1600 1800 2000

800

B @mTD

800 1000 1200 1400 1600 1800 2000