$d^0$-$d$ half-Heusler alloys: A class of future spintronic materials

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by  S. Davatolhagh

d0-d half-Heusler alloys: A class of future spintronic materials
∗
S. Davatolhagh and A. Dehghan
Department of Physics, College of Sciences, Shiraz University, Shiraz 71946, Iran
(Dated: February 28, 2017)
It is shown by rigorous ab initio calculations that half-Heusler alloys of transition metals and d0
metals,definedbythevalenceelectronicconfigurationns1,2,(n−1)d0,canproduceallkindsofhalf-
metallicbehaviorincludingtheelusiveDirachalf-semimetallicitythatisreportedforthefirsttimein
thereal3DmaterialCoKSb. Togetherwiththepredictedmagneticandchemicalstability,thispaves
7
1 theway for massless and dissipationless spintronics of thefuture. Furthermore, theintroduction of
0 d0 atoms is shown to stabilize the otherwise instable chemical structure of zinc-blende transition
2 metalpnictidesandchalcogeneides without alteringthep-dexchangethatismainlyresponsible for
theirhalf-metallicity, therefore, making their application in spintronic devices feasible.
b
e
PACSnumbers:
F

6
2 The half-metallic (HM) materials with characteristic The spin degeneracyof electronic bands in solidstate,
100% spin polarization at the Fermi level, have been re- En(k,↑)=En(k,↓),originatesfromthesimultaneousef-

] gardedas the ideal materials for spintronics due to their fect of time-reversal and spatial-inversion symmetry. As
i
c potential application as the source of spin-polarizedcur- pointed out by de Groot etal. [3], the lack of spatial-
s rent [1, 2]. Following the seminal work of de Groot etal. inversion symmetry in the HH structure in addition to
-
l [3],predicting forthe firsttime the HMpropertyinhalf- the broken time-reversal symmetry as in ordinary ferro-
r
t Heusler (HH) alloy NiMnSb, much work has been de- magnets, further removes the spin degeneracy of elec-
m
votedtofindingotherHMmaterialsofHeuslerfamily[4], tronic bands, thus resulting in robust half-metallicity.
. transitionmetal(TM) oxides[5], andbinary compounds This makes the HH C1 structure, space group F¯43m
t b
a such as zinc-blende TM pnictides or chalcogenides [6–8], (No. 216), particularly interesting from both theoretical
m
and non-TM-basedsp-electron HM ferromagnets [9, 10]. and technological point of view [28–31]. More explicitly,
- More recently, spin gapless semiconductor (SGS) is in- the crystal structure of ternary HH compounds, such as
d
troduced as another class of HM materials character- NiMnSb, consists of fcc Bravais lattice with three atom
n
ized by an open semiconductor band gap for one spin basissituatedonthecubediagonal: main-groupspatom
o
direction, and nearly closed gap for the other [11]. The at (0,0,0), high-valent TM1 atom at (1/4,1/4,1/4), and
c
[ SGS materials predicted or fabricated so far are mainly low-valent TM2 atom at (1/2,1/2,1/2) in Wyckoff coor-
the TM-based full-Heusler and inverse full-Heusler com- dinates. In the d0-d HH alloys, introduced in this let-
3
pounds with nearly zero but indirect band gaps [12–15]. ter, the low-valent TM2 atom is replaced by a d0 atom
v
7 Tothebestofourknowledge,therehasbeennoreportof of alkali or alkaline-earth metals defined by the valence
9 SGSinstoichiometrichalf-Heuslercompounds(see,also, electronic configurationns1,2,(n−1)d0 such that the d0
3 Ref. [15]). A particularly interesting kind of SGS that atom and the TM atom are first neighbors, whereas the
8 hasbeenshowntoexistinamodel2Dferrimagneticsys- d0 atom and the sp atom are second neighbors, sepa-
0 tem[16],istheDirachalf-semimetal(DHS),alsoreferred ratedby the TM atom. This arrangementis found to be
.
1 toasDiracSGS,withitscharacteristicDiracnodelinear the energetically most favorable. Despite the intuitive
0 dispersion for one spin channel–as in graphene [17]–and chemical bond view that atoms devoid of d electrons are
7 an open semiconductor band gap for the other. The ex- unabletoformd-dcovalentbonds[32],itisshownbyrig-
1
perimental realization of DHS materials is envisioned to orousdensity functional electronicstructure calculations
:
v pave the way for development of ultra-fast and power- that d-d bond formation is indeed possible between the
i efficient spintronics of the future [18]. Although there d0 atoms,as defined above,and the TM atoms. Because
X
have been a number of theoretical proposals for the re- the empty(n−1)dorbitalsofd0 atomareonlyaboutan
ar alization of quasi-2D DHS materials [19–27], the com- electron-volt higher in energy than the occupied ns1,2,
plexity of the structures and/or low magnetic transition thepromotionofelectronstothe empty(n−1)dorbitals
temperatures, have rendered their fabrication and pos- costs a small amount of energy, which is more than re-
sible device application a formidable challenge [18]. It gainedorovercompensatedbythecovalentbondingwith
is therefore important to find stoichiometric DHS com- neighboring tetrahedrally coordinated TM atoms. This
pounds characterized by thermal and magnetic stability provides the covalent d-d hybrids that stabilize the HH
at room temperature. To this end, we present, among structure, and the double exchange that is mainly re-
others, the d0-d Dirac half-semimetal HH-CoKSb. sponsible for their magnetism and half-metallicity, as in
the prototyped0-dhalf-metallic systemHH-MnSrP (see,
also, Fig. 1 and Fig. 2).
∗CorrespondingAuthor: [email protected] In this letter, the band structure results, the chemical
stability tests, and magnetic transition temperatures T
c
2
arereportedfortheprototyped0-dHHalloys. Itisamat- such as NiMnSb with more than 18 valence electrons in
ter of considerable interest to find that HH-CoKSb ex- total,follow the Slater-PaulingrelationM =Z −18
tot tot
hibitsDirachalf-semimetallicity,apropertythatisbeing [35], where Z is the total number of valence electrons
tot
reportedforthefirsttimeinreal3Dmaterials. Thisfind- per formula unit and M is the total magnetic moment
tot
ing paves the way for ultra-fast and power-efficient spin- in units of Bohr magneton. The Slater-Pauling relations
tronics of the future. Furthermore, by considering HH- are generally obtained by considering the covalent hy-
CrKSb,itisshownthattheintroductionofd0 atomssta- bridization of orbitals on neighboring sites [32, 35]. For
bilizes the otherwise instable chemical structure of half- d0-d HH-MnSrP, with Z < 18, similar considerations
tot
metallic zinc-blende TM pnictides–in this case CrSb– give a Slater-Pauling relation M = 18 − Z . The
tot tot
thus makingtheir applicationinspintronic devices feasi- total magnetic moment per formula unit is found to be
ble. The electronic structure calculations are performed M =4.00 for MnSrP that is consistentwith the above
tot
on the basis of spin-polarized density functional theory Slater-Pauling relation. The atomic magnetic moments
within the framework of self-consistent field plane wave are M = 3.98, M = 0.06, and M = −0.04. The
Mn Sr P
pseudo-potential method as implemented by the PWscf bulk of magnetic moment is carried by the TM atom
code in Quantum Espresso [33]. The generalized gra- Mn. Among the different magnetic structures, the fer-
dient approximationwith ultra-soft pseudo-potentials in romagnetic coupling within fcc sublattice of Mn atoms
theschemeofPBEwereused[34]. Adense18×18×18k- is found to be the magnetic ground state. The dif-
mesh was employed for Brillouin zone integration. The ference between non-magnetic and ferromagnetic total
high kinetic energy cut-offs of 60 Ry and 600 Ry were energy is ∆ENM−FM = 1.7eV/f.u. On inserting this
applied to the plane wave expansion of wave functions into the classical Heisenberg spin-Hamiltonian [36], the
andtheFourierexpansionofchargedensity,respectively. nearest-neighbor exchange coupling is obtained to be
Self-consistency is considered to be achieved when the J = ∆ENM−FM/(12MM2n) = 8.85meV. By Monte Carlo
total energy converges to better than 10−6 Ry/f.u. (Ry- simulationofthenearest-neighborIsingmodelonthefcc
dberg per formula unit). sublattice of Mn atoms, the Curie temperature of HH-
MnSrP is obtained to be T = 820K that is well above
c
the room temperature [37].
(a) Majo rity-spin (b) PDOS ( states/eV) (c) Mino rity-spin
3
MnSrP Mn-d MnSrP
Sr-d
2 P-p

-300.46
eV) 01 -300.47 Energy (Ry)--229974 HHOTeaerttlxrhfaa-oggHroohennouaamslllberic
E-E (F-1 y)-300.48 -3000.4 0.6 0.8 c1/.0a 1.2 1.4 1.6
R
-2 gy (-300.49
-3 Ener-300.50
-4
-300.51
-5
W L XWK-2 -1 0 1 2 W L XWK
-300.52
FIG.1: Spin-resolvedbandstructureandpartialdensity 350 400 450 500 550 600
3
of states of HH-MnSrP. Volume (a.u.)
FIG. 2: The total energy vs. volume curves for MnSrP
in four different competing structures. Data points are
Figure 1 shows the band structure of HH-MnSrP for fitted by the Murnaghan equation of state. The inset
both majority- and minority-spin direction. Unlike the shows the total energy vs. tetragonalization parameter
usual half-metallic HH compounds such as NiMnSb, c/a for HH-MnSrP (the line is guide to the eye).
in HH-MnSrP the majority-spin electrons are semicon-
ducting while the minority-spin electrons are metallic.
The minimum energy for spin excitation or the half- The rival structures of cubic half-Heusler in which the
metallic gap is E = 0.18eV, resulting in 100% spin- ternaryintermetallicalloyscanberealizedarethehexag-
HM
polarization of conduction electrons, i.e. half-metallicity. onal Ni In, orthorhombic TiNiSi, and tetragonal Fe As
2 2
The spin-resolved site and orbital projected density of structure [38]. The stability of HH-MnSrP is checked
states (PDOS) shown in Fig. 1 (b), indicate that the against the rival structures by calculating the total en-
HM property arises mainly from the exchange splitting ergyofMnSrPinalltheabovestructures. Asitisshown
of the d bands of TM atom Mn and d0 atom Sr near in Fig. 2, half-Heusler is the ground state structure of
the Fermi level. The usual half-metallic HH compounds MnSrP. Also in the inset of Fig. 2, the stability of HH-
3
MnSrP is tested with respect to constant-volume tetrag- will be both massive and massless holes in the majority-
onalization. There is a global minimum at c/a = 1, in- spin channel at finite temperature. The Fermi velocity
dicatingthat the cubic HH-MnSrPis stablewith respect of Dirac fermions is found to be v = 8.4 × 105m/s
F
to tetragonalization. Furthermore, the formation energy that is 84% of that in graphene [17]. CoKSb shows the
∆H = E −(E +E +E ) of the HH-MnSrP highest Fermi velocity compared to the quasi-2D DHS
f MnSrP Mn Sr P
is calculated to check the stability of material against materials that have been theoretically predicted so far
phase separation. The negative value ∆H = −0.92eV [19, 21, 23, 24, 27].
f
indicates that the formation of HH-MnSrP is favored by
the constituent elements. All of the above indicate that
the HH structure is the thermodynamic ground state of

MprnotSortPy.pTehhealsf-ammeetasltlaicbsilyitsytecmonHcHlu-sMionnSsroPb,taalsinoeadrefoforutnhde 01..50 Minority-Spin CCCooo---pet2g g
to apply to the other prototype systems discussed next. 0.0
-0.5
V) Majority-Spin
e -1.0
3 (a) MCaojKo rSitby-spin (b) DOS ( states/eV) (c) MCionKo rSitby-spin S (states/000...001050 Minority-Spin KKK---pet2gg
O-0.05
2 D Majority-Spin
P-0.10
1 1.0 Minority-Spin Sb-p
0.5
)
V 0 0.0
e
( -0.5
E F-1 -1.0 Majority-Spin
E- -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
-2 E-EF (eV)
FIG. 4: Spin-resolved site and orbital projected density
-3
of states of HH-CoKSb.
-4
-5W L XWK-2 -1 0 1 2W L XWK The total magnetic moment of CoKSb is Mtot = 3.00
thatisconsistentwiththeSlater-PaulingrelationM =
tot
FIG. 3: Spin-resolved band structure and total density 18−Ztot. The distribution of magnetic moments among
of states of HH-CoKSb. the atoms on three interpenetrating fcc sublattices is
M = 2.99, M = 0.02, and M = −0.01. So it
Co K Sb
appears that HH-CoKSb has the right combination of
A model system that exhibits Dirac half- three-sublatticeferrimagneticorder,andvalenceelectron
semimetallicity consistsofitinerantconductionelectrons count Z = 15 that sets the Fermi level right on the
tot
magnetically coupled by a Kondo-type Hamiltonian Dirac node [16]. Both Z = 15 and the order of bands
tot
to a three-sublattice up-up-down ferrimagnet defined play significant roles in the DHS property of CoKSb as
on triangular lattice [16]. For a given Fermi level or explainedbelow. ThecrystalfieldsymmetryinHHstruc-
concentration of conduction electrons, the model system ture splits the atomic d states into doubly degenerate e
g
exhibits Dirac nodes in the band structure with 100% and triply degenerate t subspace at the Γ point. Be-
2g
spin polarization, thus suggesting the possibility of causetheelectronegativityofCoandSbarecomparable,
the realization of Dirac half-semimetals in realistic ∆X = 0.17, there is a strong tendency for covalent in-
transition metal and rare-earth compounds [16]. Here, teraction between the two. As indicated by the PDOS
we report for the first time the DHS property in the showninFig.4,thecovalentbondingmainlyinvolvesthe
real 3D stoichiometric material HH-CoKSb. Figure 3 p-statesofCoandSbneartheFermilevel. Themajority-
shows the band structure of HH-CoKSb. There is a spin bands are of t -, e -, and p-type in the ascending
2g g
sizable gap in the minority-spin band structure and the order of energy, which is inverted with respect to the
Fermi level falls within the gap. In the majority-spin normal HH half-metals such as NiMnSb [32], and a con-
channel, however, the conduction and the valence bands duction p-bandtouches the valence p-bands at the Dirac
touch directly at the Γ point and the Fermi level. The node. Onthe other hand,the minority-spinbandsarein
linear dispersion of the conduction band and one of the normal order p-, e -, and t -type, with the t -type
g 2g 2g
the valence bands in majority-spin channel, indicates bands exceeding the Fermi level. Also taking to account
that both carriers(electrons and holes) have vanishingly the singles-statedeepinenergy,inthe majority-spindi-
small band-mass, high mobilities, and both are 100% rectionallthe nine statess throughp are completely oc-
spin-polarized. However,becausethere arethree valence cupied. The remaining six valence electrons of Z =15
tot
bands touching the conduction band at the Fermi level, enter the minority-spin channel that is pushed up with
two of which have ordinary parabolic dispersion, there respect to the majority-spin by the spin splitting, thus
4
filling the states s through p completely, while the top Itiswellknownthatzinc-blende(ZB)transitionmetal
t states remain completely empty. A necessary condi- pnictides and chalcogenides are chemically instable (see,
2g
tion for DHS behavior in CoKSbis therefore that allthe also, Fig. 6) [8]. The introduction of d0 atoms remedies
valence p-bands of both spins are completely occupied this defect. Figure 5 showsthe spin-resolvedbandstruc-
while all the higher bands remain completely empty so ture of HH-CrKSb. The system exhibits a wide half-
that the Fermi level comes right on the Dirac node in metallic gap E = 0.71eV, which indicates that the
HM
the majority-spin, and an open semiconductor gap ap- HM property in CrKSb is robust with respect to col-
pears in the minority-spin separating the valence p-type lapse of spin-polarization with the temperature [43, 44].
bands from t -type conduction bands. Because all the As can be noted from the PDOS in Fig. 5 (b), the HM
2g
bandsthroughvalenceparecompletelyoccupiedinboth gap arises mainly due to the spin splitting of TM atom
spin channels,the considerationsthat lead to the Slater- d and sp atom p bands near the Fermi level. The total
Pauling relation M = 18−Z [32], remain valid. A magnetic moment per formula unit is M = 4.00 with
tot tot tot
significantfeatureofthemajority-spinbandsinCoKSbis the atomic distribution M = 4.85, M = 0.02, and
Cr K
thed-pband-inversion,whiletheminority-spinbandsare M = −0.87. As in the case of ZB transition metal
Sb
in the normal p-d order, which is a clear indication of a pnictides and chalcogenides, the corresponding Slater-
topologicallynon-trivialelectronicstructurethatleadsto Pauling relation is of the form M = Z −8, and the
tot tot
aquantumanomalousHallstate[39]. Asalsopointedout HM gap results mainly from the covalent p-d hybridiza-
byWang[18],theDHSisatthecriticalpointoftransition tion directed by the local tetrahedral symmetry [32, 42].
toamagnetictopologicalorCherninsulatorthroughthe Therefore,HH-CrKSbretainsallthedesirableproperties
applicationofspin-orbitcoupling[40],thusresultingina of ZB-CrSb with an added bonus, i.e. chemical stability.
quantum anomalous Hall effect [41], which is character- Figure 6 shows the total energy as a function of tetrago-
izedby100%spin-polarized,massless,anddissipationless nalizationparameterc/aforthe HH-CrKSbandthe ZB-
surface or edge states. Therefore, DHS materials such CrSb. Whereas the ZB-CrSb shows a local maximum
as CoKSb are of vital interest for ultra-fast and power- instability at c/a = 1, the HH-CrKSb has a global min-
efficientspintronicsofthefuture. TheCurietemperature imum at c/a = 1. In fact all the stability criteria that
of DHS CoKSb is obtained to be T = 730K, which to- were checked for the other prototype systems, also are
c
getherwiththermodynamicstabilityandsimplestoichio- found to apply to HH-CrKSb, with a formation energy
metric 3D structure, makes it the most viable candidate ∆H =−1.71eV.
f
for room temperature applications so far. It must also
be pointedoutthatthe bandstructureofanotherproto-
type systemHH-MnKSb(not shown),displaysparabolic
-242.8
half-semimetallicity with an inverted band order simi- CrKSb CrSb
-260
lar to HH-CoKSb, but with parabolic p-bands (massive
carriers) touching directly at the Γ point and the Fermi -243.3
level. HH-MnKSb is found to have a T = 870K. Both -262
c
Dcoinrasicdheraeldf-steombime uetnaiqlsuaenmdaptaerriaablsolfiocrhtahlef-speumrpimoseetaolfsdairse- y (Ry)-264 -243.8 Energ
g y
sipationless spintronics [18]. ner -244.3 (R
E y
)
-266
-244.8
(a) Majo rity-spin (b) PDOS (states/eV) (c) Mino rity-spin -268
3 -245.3
CrKSb Cr-d CrKSb
K-d
2 Sb-p 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
c/a
1
FIG. 6: Totalenergy is plotted againsttetragonalization
)
V 0 parameter c/a for HH-CrKSb and ZB-CrSb. The lines
e
E (F-1 are guide to the eye.
-
E
-2
To sum up, therefore, the band structure calculations
-3 reveal that chemically and magnetically stable d0-d HH
alloys can produce all kinds of HM behavior, including
-4
Dirac half-semimetallicity that is reported for the first
time in stoichiometric 3D materials. This finding opens
-5
W L XWK-2 -1 0 1 2W L XWK
new frontiers for massless and dissipationless spintronic
applications of the future. Furthermore, the d0 atoms
FIG.5: Spin-resolvedbandstructureandpartialdensity
are shown to stabilize the otherwise instable chemical
of states of HH-CrKSb.
structure of zinc-blende transition metal pnictides and
5
chalcogenides without altering the p-d exchange that is conductors because the bulk-like environment at the in-
mainlyresponsiblefortheirhalf-metallicity. Becausethe terface tends to preservethe interfacialHM property. In
HH structure is the thermodynamic groundstate ofd0-d view of the above, a large pool of chemically and mag-
ternary alloys, equilibrium preparation techniques such netically stable HH d0-d spintronic materials is awaiting
as arc-melting of stoichiometric quantities of constituent experimental realization.
elementsinaninertgasenvironmentmaybeemployedto
producebulkmaterialsandfreestandingthinfilms. The The authors wish to thank Dr. Hiroaki Ishizuka for
d0-d HH alloys containing group II d0 atoms, may also a number of useful communications. Support from the
be better suited as electrode contacts with II-VI semi- Research Council of Shiraz University is acknowledged.
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