c ys als
A icle
La ge Uniaxial Magne ic Aniso opy o Hexagonal
Fe-H -Sb Alloys
Lukas Ky ala 1,2 , Maxim Tchaplianka 1, Alexande B. Shick 1,* and Se gii Khmele skyi 3
and Dominik Legu 2
1Ins i u e o Physics, Czech Academy o Sciences, Na Slo ance 2, CZ-18221 P ague, Czech Republic;
[email p o ec ed] (L.K.); [email p o ec ed] (M.T.)
2IT4Inno a ions and Nano echnology Cen e, VSB-Technical Uni e si y o Os a a, 17. lis opadu 2172/15,
CZ-70800 Os a a-Po uba, Czech Republic; [email p o ec ed]
3Cen e o Compu a ional Ma e ials Science, Vienna Uni e si y o Technology, Wiedne Haup s asse 8-10,
A-1040 Vienna, Aus ia; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 4 May 2020; Accep ed: 25 May 2020; Published: 27 May 2020


Abs ac :
We heo e ically in es iga e he elec onic and magne ic s uc u e o Fe
2
H . The densi y
unc ional heo y calcula ions a e shown o p oduce he nega i e, easy-plane, magne ic aniso opy
in he hexagonal Fe
2
H . An imony subs i u ion supp esses he plana magne iza ion di ec ion and
a o s he uniaxial magne ic aniso opy, in ag eemen wi h expe imen al obse a ions. Ou s udy
sugges s he possibili y o he chemical con ol o he magne ic aniso opy in Fe
2
H by Sb
subs i u ion, and illus a es he po en ial o (Fe,Sb)
2+x
H
1−x
La es phase alloys o he pe manen
magne applica ions.
Keywo ds: elec onic s uc u e; magne ism; magne ic aniso opy; pe manen magne s
1. In oduc ion
Pe manen magne s a e an indispensable pa o he mode n echnology [
1
]. They a e mainly
used o da a s o age and ene gy con e sion, and a e p esen in sma phones; lap ops; audio and
ideo de ices; and indus ial applica ions, such as elec ical mo o s and wind u bines. The e is a
high demand o pe manen magne s (PMs) wi h high pe o mance o e icien enewable ene gy
p oduc ion and con e sion [
2
]. Resea ch a ge ing new PMs is d i en by bo h echnological and
economical ac o s, o keep cos s low and ensu e supply s abili y.
A good PM has o ha e a high Cu ie empe a u e, la ge magne iza ion densi y, and coe ci i y.
The la ge coe ci i y is equi ed o esis demagne iza ion, and ad anced PM should achie e a coe ci i y
µ0Hc>
1.5 T [
3
]. I is de e mined by he magne o-c ys alline aniso opy ene gy (MAE), he ene gy o
o a e he magne iza ion om he easy o he ha d di ec ion. This la ge coe ci i y can be p o ided by
uniaxial magne ic phases wi h hexagonal, hombohed al, o e agonal c ys al s uc u es wi h he high
MAE >1 MJ/m3.
The bes known PMs a e based on Nd
2
Fe
14
B wi h he mos supe io PM p ope ies [
4
],
and SmCo
5
[
5
] which is used o he high empe a u e PM applica ions. These ma e ials belong
o he class o he a e-ea h (RE)- ansi ional me al (3
d
) in e me allics. In hese ma e ials, he RE
elemen p o ides high alue o he MAE, and he 3
d
me al componen yields a high magne iza ion
densi y and a high Cu ie empe a u e.
In ecen yea s, majo e o has been made o sea ch o he RE- ee PM ma e ials [
6
]. The eason
is a g owing conce n o he RE elemen s’ (Nd, Sm, Dy) sus ainabili y, oge he wi h g owing
demand om he indus y o cheap and s able high- empe a u e PMs [
2
]. So a , he esea ch
C ys als 2020,10, 430; doi:10.3390/c ys 10060430 www.mdpi.com/jou nal/c ys als
C ys als 2020,10, 430 2 o 10
has ollowed wo ou es: (i) o sea ch di ec ly o he ansi ion me al based in e me allics in hexagonal,
hombohed al, o e agonal c ys al s uc u es wi h he high MAE; (ii) o ind a modi ica ion o al eady
known e omagne s by subs i u ional and in e s i ial alloying o inc ease he MAE.
G ea p ospec s in he second di ec ion we e demons a ed by Coey and Sun [
7
] who ha e
obse ed s ong MAE inc ease in he i on-based RE in e me alics wi h subs i u ion o he in e s i ial
ni ogen. Fu he mo e, ecen ly i was p oposed expe imen ally and heo e ically ha alloying 6.25%
o Sb in o he hexagonal e omagne Fe
3
Sn changes he MAE and u ns he magne iza ion di ec ion
om he easy-plane o he easy axis [
8
]. Ano he success ul example o he chemical enginee ing o he
MAE is he ecen expe imen al obse a ion o a la ge uniaxial MAE in Fe
2
H wi h Sb subs i u ion [
9
].
The Fe-based La es phases wi h 1:2 s oichiome y we e ne e conside ed as candida es o
magne ically ha d ma e ials. They usually c ys allize in he cubic C15 s uc u e wi h he small MAE,
and posses supe io magne os ic i e p ope ies (e.g., gian magne os ic ion in a e-ea h-Fe
2
[
10
]).
Expe imen ally o he s oichiome ic H Fe
2
, he cubic C15 phase is sligh ly mo e s able han he
hexagonal C14 phase a he low empe a u es [
11
], and bo h phases we e epo ed o co-exis o high
empe a u es [12].
I was shown expe imen ally [
9
] ha he Sb subs i u ion s abilizes Fe-H -Sb alloy in hexagonal
C14 phase, and leads o he la ge posi i e MAE o
∼
1.5 MJ/m
3
o (Fe,Sb)
2+x
H
1−x
alloy wi h 13.5%
o Sb. Toge he wi h sa u a ion magne ic pola iza ion o
∼
0.8 MA/m (1 T), and he Cu ie empe a u e
o 470 K, hese p ope ies make he alloy a p omising candida e o he PM applica ions.
In his wo k, we aim o in es iga e he MAE in he hexagonal Fe
2
H compound, and he
Fe-H -Sb alloy making use o he densi y unc ional heo y (DFT) implemen ed in he p ojec o
augmen ed-wa e VASP [
13
], he ull-po en ial linea ized augmen ed plane wa e (FLAPW) [
14
],
and he Ko inga–Kohn–Ros okke (KKR) [15] me hods.
We show heo e ically ha he Sb subs i u ion in hexagonal Fe
2
H leads o he s ong uniaxial
MAE in good quan i a i e ag eemen wi h he expe imen al da a. We demons a e ha he ole o
Sb subs i u ion is no limi ed by s abiliza ion o he Fe
2
H in hexagonal C14 s uc u e. The la ge
uniaxial MAE occu s due o he p esence o he Sb a oms in he c ys alline ma ix. Apa om
he elec onic mechanism, he s uc u e elaxa ion plays an essen ial ole in he MAE inc ease.
In addi ion, we e alua e he in e -a omic exchange in e ac ions in Fe
2
H , and make an es ima e
o he Cu ie empe a u e.
2. Resul s
2.1. Hexagonal Fe2H
The Fe
2
H has hexagonal C14 c ys al s uc u e wi h P6
3
/mmc (numbe 194) space g oup and
con ains wel e a oms pe uni cell (see Figu e 1A). The e a e ou o mula uni s in he uni cell.
The Fe a oms occupy (2a) and (6h) Wycko posi ions, and H is in (4 ) Wycko posi ion (see Table 1).
Fi s , we pe o med he scala - ela i is ic VASP, KKR, and FLAPW calcula ions wi hou spin-o bi
coupling (SOC). The expe imen al la ice cons an s
a=
4.968 and
c=
8.098 , and in e nal posi ions [
16
]
shown in Table 1we e used in he calcula ions. The FLAPW calcula ed magne ic momen s inside
“mu in- in” (MT) sphe es o di e en Fe and H a oms in he uni cell oge he wi h he o al magne ic
momen pe cell a e shown in Table 1. The VASP and KKR calcula ions yield e y simila esul s.
No e ha he magne ic momen s a e aligned in he same di ec ion o he bo h Fe-subla ices, and he
H momen s a e poin ed in he opposi e di ec ion.
The spin-p ojec ed o al densi y o s a es (DOS), as a esul o FLAPW calcula ions wi hou
spin-o bi coupling (SOC), is shown in Figu e 2A, and he
d
-s a es p ojec ed DOS o Fe and H a oms
in di e en Wycko posi ions a e shown in Figu e 2B. Since he magne ic momen s o Fe and H a e
an i-aligned, he exchange spli ing o Fe and H is opposi e. Vi ually all he DOS cha ac e nea
EF
is om Fe
d
-s a es. Bo h Fe and H -p ojec ed DOS illus a e i ine an cha ac e o he
d
-s a es.
The calcula ed o al (spin-up plus spin-down) DOS(
EF
) o Fe
2
H o 1.075 1/eV pe Fe a om is close
C ys als 2020,10, 430 3 o 10
o he co esponding DOS alue o 1.11 1/eV o he
bcc
-Fe. Since he Fe-a om momen s in Fe
2
H
(see Table 1) a e smalle han 2.2 µBmomen o bcc-Fe, he momen o ma ion in Fe2H is weake .
Table 1.
The calcula ed magne ic momen s (in
µB
) inside he MT-sphe es o di e en a oms in
p is ine Fe
2
H and Fe58-H 25-Sb17 alloy uni cell, oge he wi h he o al magne ic momen pe
cell. The Wycko posi ions and in e nal coo dina es o di e en a oms in he uni cell a e shown.
Fe2H Fe58-H 25-Sb17
Wycko Elemen In e nal Magne ic Momen s Elemen Magne ic Momen s
Posi ions Posi ion
(2a) Fe (1) (0, 0, 0) 1.71 Sb −0.03
(2a) Fe (2) (0, 0, 1/2) 1.71 Fe 2.00
(6h) Fe (3) (0.17, 0.34, 1/4) 1.74 Fe 1.64
(6h) Fe (4) (-0.34, −0.17, 1/4) 1.74 Fe 1.64
(6h) Fe (5) (−0.17, −0.34, −1/4) 1.74 Fe 1.55
(6h) Fe (6) (0.34, 0.17, −1/4) 1.74 Fe 1.55
(6h) Fe (7) (0.17, −0.17, 1/4) 1.74 Fe 1.64
(6h) Fe (8) (−0.17, 0.17, −1/4) 1.74 Fe 1.55
(4 ) H (1) (1/3, 2/3, 0.563) −0.35 H −0.28
(4 ) H (2) (2/3, 1/3, 0.063) −0.35 H −0.30
(4 ) H (3) (1/3, 2/3, −0.063) −0.35 H −0.27
(4 ) H (4) (2/3, 1/3, −0.563) −0.35 Sb 0.01
To al magne ic momen 13.48 10.48
Figu e 1.
C ys al s uc u e (hexagonal C14) o p is ine Fe
2
H (
A
). I on a oms in (2a) Wycko
posi ions a e ep esen ed by ed-colo ed balls, i on a oms in (6h) Wycko posi ions a e ep esen ed by
blue-colo ed balls, and ha nium a oms in (4 ) Wycko posi ions a e ep esen ed by g ay balls. (
B
) The
uni cell o Fe58-H 25-Sb17 alloy in he lowes ene gy con igu a ion. An imony a oms a e shown by
da k-blue-colo ed balls.
C ys als 2020,10, 430 4 o 10
Figu e 2.
Densi y o s a es (DOS) o he Fe
2
H : (
A
) Spin-p ojec ed o al DOS. (
B
) Spin-p ojec ed DOS
o he d-s a es o Fe and H a oms in di e en Wycko posi ions. The Fe mi le el EFis se o EF=0.
In o de o es ima e he Cu ie empe a u e we calcula e he in e -si e pai exchange in e ac ions
Jij o he classical Heisenbe g Hamil onian,
ˆ
H=−∑
i,j
Ji,j~
ei~
ej,
whe e
~
ei
and
~
ej
ma k he uni di ec ional ec o s o he magne ic momen s a
i
and
j
di e en Fe
la ice si es o Fe
2
H . The calcula ions we e pe o med making use o he magne ic o ce heo em [
17
]
implemen ed in he bulk KKR o malism [
18
]. Since he magne ic o de ing happens ou o high
empe a u e pa amagne ic s a e, he exchange in e ac ions we e e alua ed in he diso de ed local
momen (DLM) [
19
] model, used o ea he he mal magne ic diso de . The impo ance o using he
DLM e e ence s a e o es ima ion o he Cu ie empe a u e is demons a ed in [20].
C ys als 2020,10, 430 5 o 10
The calcula ed exchange cons an s a e shown in he Figu e 3. The e a e h ee ypes o he
in e -si e exchange in e ac ions: in a-subla ice in e ac ions be ween Fe momen s on (2a)–(2a) si es
and (6h)–(6h) si es, and in e -subla ice (2a)–(6h) exchange coupling. I is seen om he Figu e 3
ha he exchange in e ac ions ha e a long ange, as expec ed o he me allic magne . The (6h)–(6h)
in a-subla ice exchange couplings a e he s onges , and p o ide main con ibu ion o he Cu ie
empe a u e. The in e -subla ice exchange coupling be ween he (2a) and (6h) i s nea es neighbo s is
also s ong. Thus, i is expec ed ha he magne iza ion o (2a) subla ice will ha e simila empe a u e
beha io o he magne iza ion o (6h) subla ice. The mean- ield heo y es ima e yields 445 K o he
Cu ie empe a u e. This alue is in ai ag eemen wi h he alue o 470 K expe imen ally de e mined
o Fe-H -Sb alloys [9].
The ene gy penal y o changing he o ien a ion o he magne iza ion om in-plane o ou -o -plane
is called he magne ic aniso opy ene gy (MAE). No e ha o a hexagonal c ys al symme y, he leading
e m in he MAE angula dependence is
K1sin2θ
, wi h
θ
he angle be ween he magne iza ion and
c ys allog aphic
c
-axis. The MAE is hen compu ed as he o al ene gy di e ence when magne iza ion
is o ien ed along [1100] and [0001] c ys al axes, MAE = E[1100]−E[0001].
S a ing om he scala - ela i is ic calcula ions, we pe o m he sel -consis en o al ene gy
calcula ions wi h he SOC included, making use o a ela i is ic e sion o FLAPW [
21
].
The con en ional local spin densi y app oxima ion ( on Ba h–Hedin) is adop ed, which we expec o
be alid o i ine an me allic sys ems. In hese calcula ions, he magne iza ion is ixed along [1100]
and [0001] axes. The special k-poin s me hod is used o he B illouin zone (BZ) in eg a ion wi h he
Gaussian smea ing o 1 mRy o k-poin s weigh ing, and he equi alen o abou 3200 k-poin s in he
ull BZ a e used. Impo an ly, he same se o k-poin s mus be used o di e en di ec ions o he
magne iza ion, in o de o achie e accu a e MAE esul s. The calcula ed MAE =
−
0.57 meV pe uni
cell (
−
0.54 MJ m
−3
) indica es he "in-plane" p e e en ial di ec ion o he magne iza ion in p is ine
Fe
2
H . Al hough hexagonal Fe
2
H has a sa u a ion magne ic momen and he Cu ie empe a u e
desi able o a pe manen magne applica ions, he magne iza ion lies in he hexagonal plane, which is
de imen al o PM ma e ial.
Figu e 3.
The in e -si e exchange in e ac ions in hexagonal Fe
2
H compound calcula ed in he
diso de ed local momen (DLM) s a e. Open symbols a e o he (2a)–(2a) Fe in a-subla ice in e ac ions,
closed symbols a e o (6h)–(6h) in a-subla ice in e ac ions, and semi-closed symbols deno e he
in e -subla ice (2a)–(6h) in e ac ions. The calcula ions a e p esen ed o up o 50 nea es -neighbo
shells.

C ys als 2020,10, 430 6 o 10
2.2. Fe–H –Sb Alloy
The MAE can be al e ed by alloying. Expe imen ally, Goll e al. [
9
] obse ed ha o -s oichiome ic
(Fe,Sb)
2+x
H
1−x
compounds ha e he uniaxial MAE o
∼
1.5 MJ m
−3
. In o de o examine he e ec
o Sb subs i u ional alloying as a possible uning o posi i e MAE, we conside subs i u ion o Fe
and H a oms by Sb a oms in he Fe
2
H uni cell. To s ay close o he expe imen al composi ion
Fe60.0-H 26.5-Sb13.5 [
9
], we subs i u e simul aneously one Fe and one H in he 12 a oms Fe
2
H uni
cell. This yields Fe58-H 25-Sb17 composi ion (Fe
7
H
3
Sb
2
uni cell) which we model conside ing 32
di e en o de ed con igu a ions o he Sb dopan .
We make use o he VASP code o e alua e he o al ene gy. The o al ene gies o di e en
con igu a ions o he Sb dopan a e shown in Appendix ATable A1. We iden i y he ou lowes ene gy
o de ed con igu a ions whe e he Sb dopan s subs i u e he pai s o [Fe (1)–H (4)] (see Table 1),
and [Fe (1)–H (1)],
[Fe (2)–H (2)], and [Fe (2)–H (3)]. All o hese con igu a ions in which Sb
subs i u es he Fe a oms in (2a) Wycko posi ions, a e ene ge ically equi alen . The alloy con igu a ion
whe e Sb dopan s subs i u e he pai [Fe (1)–H (4)]) is shown in Figu e 1B.
Addi ionally, we es ima e s abili y o he lowes ene gy con igu a ions calcula ing he en halpy o
o ma ion H.
H=E−∑
i
µixi
whe e
E
is he DFT o al ene gy o he alloy,
µi
is he chemical po en ial o elemen
i
, and
xi
is he
quan i y o elemen
i
in he compound. The s anda d con en ion is o ake he chemical po en ial
o each species o be he DFT o al ene gy o he elemen al Fe, H , and Sb g ound s a e. The
en halpy o o ma ion is ound
H
=
−
0.724 eV/ .u., which has somewha smalle magni ude han he
H=−1.165 eV/ .u.
o p is ine Fe
2
H . Nega i e sign o
H
suppo s he he modynamic s abili y o
Fe58-H 25-Sb17 alloy.
Mo eo e , we es ima e he en halpy o o ma ion o Fe58-H 25-Sb17 alloy o med om p is ine
Fe
2
H . In his case, he
∆H
is calcula ed om he alloy o al ene gy sub ac ing he o al ene gy o
Fe
2
H , and aking in o accoun excess o Fe and inco po a ion o wo Sb a oms in o he uni cell.
This esul s in
∆H
= 0.15 eV/ .u., and indica es he po en ial o Sb inco po a ion in al eady exis ing
Fe2H a ini e empe a u es.
Nex , we pe o m he FLAPW calcula ions o he alloy con igu a ion shown in Figu e 1B.
The calcula ed magne ic momen s inside MT-sphe es o di e en Fe and H a oms in he uni
cell oge he wi h he o al magne ic momen pe cell a e shown in Table 1. I is seen ha he
magne ic momen is inc eased o he Fe a om in (2a) posi ion, and is dec eased o he (6h) Fe
a oms. Addi ionally, magni ude o he momen on H a om is smalle han in he p is ine Fe
2
H .
The Sb dopan yields almos no con ibu ion o he magne ic momen , and he o al momen pe
Fe7H 3Sb2uni cell is dec eased by 3 µB.
S a ing om he scala - ela i is ic calcula ions, we pe o m he calcula ions wi h he SOC
included, making use o a ela i is ic e sion o FLAPW [
21
]. In hese calcula ions, he magne iza ion
is ixed along [1100] and [0001] axes, as i was done o Fe
2
H desc ibed abo e, wi h he same se o he
special k-poin s o he BZ-in eg a ion. The MAE o 0.88 meV pe uni cell (0.82 MJ m
−3
, see Table 2)
is calcula ed. This signi ican posi i e MAE indica es he "ou -o -plane" p e e en ial di ec ion o he
magne iza ion, and is c ucial o PM applica ions.
The calcula ions which a e p esen ed abo e we e done assuming he expe imen al uni cell
olume and in e nal posi ions o Fe
2
H [
16
]. The s uc u e op imiza ion can play impo an ole,
and we examined he e ec o s uc u al elaxa ion on he elec onic s uc u e, magne ism, and MAE
o Fe-H -Sb alloy. In hese calcula ions, he uni cell olume, c/a- a io, and in e nal coo dina es we e
op imized making use o he VASP me hod. In e nal posi ions we e elaxed un il esidual o ces we e
less han 0.1 meV/Å. The elaxed olume o 179 Å
3
and c/a- a io o 1.64 o Fe58-H 25-Sb17 alloy
a e bigge han he olume o 173 Å
3
and c/a- a io o 1.63 o un elaxed uni cell. The sa u a ed
magne iza ion o 10.93
µB
(0.57 MA m
−1
, see Table 2) sligh ly exceeds he un elaxed alue o
C ys als 2020,10, 430 7 o 10
10.48
µB
(0.55 MA m
−1
). This sligh inc ease o he magne iza ion is consis en wi h an inc ease
o equilib ium olume.
Impo an ly, he MAE is inc eased wi h elaxa ion om 0.88 meV pe uni cell (0.82 MJ m
−3
) o
1.37 meV pe uni cell (1.27 MJ m
−3
), as shown in Table 2. The MAE calcula ions equi e special ca e
o con e ge ene gy di e ences o o en less han 0.1 meV pe uni cell om he o al ene gy, which
may be en o de s o magni ude la ge . This is an in ol ed endea o and abou 5200 special k-poin s
in he ull BZ we e used in o de o achie e he accu acy be e han 0.05 meV pe uni cell in he
MAE calcula ions.
Table 2.
Magne ic momen (
Msa
, MA/m), and he uniaxial magne oc ys alline aniso opy
(MAE(K1), MJ/m3).
Composi ion Msa MAE
un elaxed Fe2H 0.72 −0.54
un elaxed Fe58-H 25-Sb17 0.55 +0.82
elaxed Fe58-H 25-Sb17 0.57 +1.27
exp. Fe60-H 26.5-Sb13.5 0.56–0.80 1.4–1.5
Compa ison be ween calcula ed elaxed Fe58-H 25-Sb17 and expe imen al
Fe60.0-H 26.5-Sb13.5 [
9
] alloy is shown in Table 2. The e is a ai ag eemen o he sa u a ion
magne iza ion
Msa
. The calcula ed alue o
Msa
is somewha smalle since he e is a de iciency o Fe
and H , and excess o Sb in he model supe cell as compa ed wi h he expe imen . Ag eemen be ween
he uniaxial MAE is good. Signi ican ly posi i e MAE and co esponding aniso opy ield
HA≈
2
K1/Msa
o 3.58 MA/m (
µ0HA
= 4.5 T), oge he wi h sa u a ion magne iza ion o 0.72 T
and easonably high Cu ie empe a u e o 470 K, allow us o sugges ha Fe-H -Sb alloys wi h he
op imal amoun o Sb dopan can be a sui able candida es o a magne ically ha d ma e ial.
3. Discussion
The MAE s eng h depends on he alue o he SOC cons an
ξ
[
22
]. In Fe-H -Sb alloy, he SOC
o
p
-s a es o Sb, and
d
-s a es o Fe and H play a ole o he MAE. The s onges
ξ
= 0.978 eV is
o he 5
p
-s a es o Sb, while he
ξ
alues o 3
d
-s a es o Fe and 5
d
-s a es H a e o 0.065 eV and
0.188 eV espec i ely. B uno [
22
] has shown ha he MAE is p opo ional o he aniso opy o he
o bi al momen ML,
MAE ≈ −ξ/4(ML||[1100]−ML||[001]) ,
when he exchange spli ing
ξ<∆ex
. Fo he Fe-H -Sb alloy, he exchange spli ing
∆ex
= 0.654 eV,
and we can use his app oach o es ima e con ibu ions o he MAE o Fe-H -Sb alloy om Fe and H
a oms. Making use o he calcula ed
ML
alues, we ob ain he nega i e Fe-a oms MAE o
−
1.14 meV
pe uni cell, and he posi i e H -a oms MAE o 0.28 meV pe uni cell. The o al con ibu ion o he
MAE =
−
0.88 meV due o he SOC o Fe and H
d
-s a es is semi-quan i a i ely consis en wi h he
nega i e MAE = −0.57 meV pe uni cell o p is ine Fe2H .
Since he Sb-
p
-s a es’ SOC is bigge han
∆ex
, no pe u ba ion heo y is expec ed o be alid
o hei con ibu ion o he MAE. A i s , we show in Figu e 4 he
p
-p ojec ed Sb-a oms’ DOS o
di e en di ec ions o he magne iza ion. While hey look alike, close examina ion shows small
p
-DOS
aniso opy o 0.02 s a es/eV a he Fe mi le el. This DOS aniso opy is usually ela ed o he MAE [
23
].
In o de o e alua e e ec o he Sb-
p
-s a es SOC on he MAE, we pe o med he o al ene gy
calcula ions, whe e we se
ξ
= 0 o he
p
-s a es o he Sb a oms; i.e., explici ly emo ed he Sb
p
-s a e’s
con ibu ion o he MAE. The calcula ed MAE = 0.13 meV pe uni cell (0.12 MJ/m
3
) is posi i e. I is
subs an ially smalle han he ull MAE o 1.37 meV pe uni cell (1.27 MJ/m
3
) shown in Table 2,
C ys als 2020,10, 430 8 o 10
whe e he Sb-
p
-s a es SOC is included. These calcula ions show explici ly and quan i a i ely ha he
p-s a es o Sb play a key ole in he posi i e MAE o Fe58-H 25-Sb17 alloy.
Figu e 4.
(
A
) Spin-p ojec ed
p
-DOS o
M||
[0001]; (
B
) spin-p ojec ed
p
-DOS o
M||
[1100] o Sb a oms
in (2a) and (4 ) Wycko posi ions. The Fe mi le el EFis se o EF=0.
4. Conclusions
To summa ize, we ha e in es iga ed he elec onic s uc u e and magne ic cha ac e o Fe
2
H
hexagonal C14 phase. The nega i e MAE was ound, and de e mined he "in-plane" p e e en ial
di ec ion o he magne iza ion in he p is ine ma e ial. The i s -p inciples calcula ions demons a e
ha Sb subs i u ion changes he MAE om plana o uniaxial, in ag eemen wi h expe imen al
obse a ions. We emphasize he essen ial con ibu ion o he s uc u al elaxa ion o he MAE o
Fe-H -Sb alloy. Ou esul s sugges ha he chemical subs i u ion o he
p
-elemen s o he Fe-based
hexagonal La es phases can be a p omising way o posi i e MAE con ol in he a e-ea h ee
pe manen magne s o echnological applica ions.
Au ho Con ibu ions:
L.K. and M.T. pe o med he o al ene gy VASP calcula ions o he elec onic s uc u e,
c ys al s uc u e op imiza ion, and en halpy o o ma ion; A.B.S. pe o med he o al ene gy FLAPW calcula ions
o he DOS, spin and o bi al magne ic momen s, and magne ic aniso opy ene gy; S.K. pe o med he KKR
calcula ions o exchange coupling cons an s and Cu ie- empe a u e; D.L. pe o med he p ojec adminis a ion,
supe ision, and analysis o he esul s. All au ho s con ibu ed in w i ing he manusc ip . All au ho s ha e ead
and ag eed o he published e sion o he manusc ip .
Funding:
Financial suppo was p o ided by Ope a ional P og am Resea ch, De elopmen and Educa ion
inanced by Eu opean S uc u al and In es men Funds and he Czech Minis y o Educa ion, You h and
Spo s (p ojec numbe SOLID21-CZ.02.1.01/0.0/0.0/16
−
019/0000760), Eu opean Regional De elopmen
Fund in he IT4Inno a ions na ional supe compu ing cen e —pa h o exascale p ojec , p ojec numbe
CZ.02.1.01/0.0/0.0/16
−
013/0001791 wi hin he Ope a ional P og am Resea ch, De elopmen and Educa ion,
mobili y g an numbe 8J18AT004, and by he Czech Science Founda ion g an numbe 18-06240S. D.L. was
suppo ed by he Czech Science Founda ion g an numbe 17-23964S.
C ys als 2020,10, 430 9 o 10
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A
The o al ene gies ela i e o he lowes ene gy con igu a ion o hi y wo di e en con igu a ions
whe e he Sb dopan s subs i u e he pai s o [Fe–H ] a e shown in Table A1. We iden i y ou
di e en g oups o he o de ed con igu a ions wi h he same ene gy. Inside o he each g oup,
di e en con igu a ions a e connec ed by he symme y ope a ions.
No e ha we made a compa ison be ween he en halpy o o ma ion o Fe58-H 25-Sb17 alloy
HFe58-H 25-Sb17 =−
0.724 eV/ .u. (one Fe and one H a oms a e subs i u ed by Sb), and wo alloys
whe e ei he Fe (Fe50-H 33-Sb17) o H (Fe66-H 17-Sb17) si es a e eplaced by Sb. We ound
HFe50-H 33-Sb17 =−
0.179 eV/ .u., and
HFe66-H 17-Sb17 =−
0.770 eV/ .u. Thus, he con igu a ion wi h
wo H si es subs i u ed by Sb may be ene ge ically p e e able o he 1:1 Fe58-H 25-Sb17 subs i u ion.
One has o ake his compa ison wi h some cau iousness since i is done o T = 0 K and wi hou
phonon con ibu ion.
Table A1.
The o al ene gy di e ence
∆E
(eV/ .u.) ela i e o he lowes ene gy con igu a ion o 32
o de ed con igu a ions o Fe7H 3Sb2.
G oup Sb Subs i u ion Si es ∆E(eV/ .u)
1 [Fe(1)–H (1)] 0
[Fe(2)–H (2)] 0
[Fe(2)–H (3)] 0
[Fe(1)–H (4)] 0
2 [Fe(3)–H (1)] 0.137
[Fe(4)–H (1)] 0.137
[Fe(7)–H (1)] 0.137
[Fe(5)–H (2)] 0.137
[Fe(6)–H (2)] 0.137
[Fe(8)–H (2)] 0.137
[Fe(3)–H (3)] 0.137
[Fe(4)–H (3)] 0.137
[Fe(7)–H (3)] 0.137
[Fe(5)–H (4)] 0.137
[Fe(6)–H (4)] 0.137
[Fe(8)–H (4)] 0.137
3 [Fe(5)–H (1)] 0.215
[Fe(6)–H (1)] 0.215
[Fe(8)–H (1)] 0.215
[Fe(3)–H (2)] 0.215
[Fe(4)–H (2)] 0.215
[Fe(7)–H (2)] 0.215
[Fe(5)–H (3)] 0.215
[Fe(6)–H (3)] 0.215
[Fe(8)–H (3)] 0.215
[Fe(3)–H (4)] 0.215
[Fe(4)–H (4)] 0.215
[Fe(7)–H (4)] 0.215
4 [Fe(2)–H (1)] 0.258
[Fe(1)–H (2)] 0.258
[Fe(1)–H (3)] 0.258
[Fe(2)–H (4)] 0.258