Table 1 Spin arrangements investigated for GdFeO3

From: Improper electric polarization in simple perovskite oxides with two magnetic sublattices

Gd spins

Fe spins

 

G a

G b

G c

G a

Pn′a′21 Fe (Ga, Ab, Fc) Gd (Ga, Ab, Fc) Pc=−0.42

P212121 Fe (Aa, Gb, Cc) Gd (Ga, Ab, Cc) P=0

P21n′m′ Fe (Fa, Cb, Gc) Gd (Ga, Cb, 0) Pa<0.01

G b

P21 Fe (Ga, Ab, Fc) Gd (Aa, Gb, Fc) P=0

Pna21 Fe (Aa, Gb, Cc) Gd (Aa, Gb, Cc) Pc=−0.41

Pbm′ Fe (Fa, Cb, Gc) Gd (Fa, Gb, 0) Pb<0.01

G c

Pn′m Fe (Ga, Ab, Fc) Gd (0, 0, Gc) Pa<0.01

Pb21m Fe (Aa, Gb, Cc) Gd (0, 0, Gc) Pb<0.01

Pn′a Fe (Fa, Cb, Gc) Gd (Fa, Cb, Gc) Pc=−0.41

A a

P21 Fe (Ga, Ab, Fc) Gd (Aa, Gb, Fc) P=0

Pna21 Fe (Aa, Gb, Cc) Gd (Aa, Gb, Cc) Pc<0.01

Pbm′ Fe (Fa, Cb, Gc) Gd (Aa, Cb, 0) Pb<0.01

A b

Pn′a′21 Fe (Ga, Ab, Fc) Gd (Ga, Ab, Fc) Pc<0.01

P212121 Fe (Aa, Gb, Cc) Gd (Ga, Ab, Cc) P=0

P21n′m′ Fe (Fa, Cb, Gc) Gd (Fa, Ab, 0) Pa<0.01

A c

Pb′m Fe (Ga, Ab, Fc) Gd (0, 0, Ac)

P21nm Fe (Aa, Gb, Cc) Gd (0, 0, Ac) Pa<0.01

P21 Fe (Fa, Cb, Gc) Gd (Fa, Cb, Ac) P=0

C a

P/c′

P21/c

P/m′

C b

P/c′

P21/c

Pbn′m′

C c

P21/m

Pbnm

P21/c

F a

P/c′

P21/c

Pbn′m′

F b

P/c′

P21/c

P/m′

F c

Pb′n′m

P21/m

  1. The G, A, C and F letters indicate the presence of the spin order thus denoted (see text). The a, b and c subindexes indicate the direction of the corresponding spin order in the orthorhombic setting (see sketch in Fig. 1b). The spin arrangement used to initialize the simulation is indicated in bold font; the other (canted) orders appear in our calculations as a result of the symmetry breaking. (We only report magnetic-order components for which we obtain a sizeable result from our numerical calculations; other symmetry-allowed orders may in principle exist in some of the cases considered.) Magnetic space groups are given. In the cases in which the symmetry is polar, the polarization (computed using the Berry-phase formalism; see Methods) is indicated in μC cm−2. Note that in some instances the polarization is very small and falls below the accuracy of our numerical method (see text).
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