Chemical trends of substitutional transition metal dopants in diamond: an ab initio study

1.5 MB·English

by  Thomas Chanier

Chemical trends of substitutional transition metal dopants in diamond: an ab initio
study
T. Chanier, C. Pryor, and M. E. Flatt´e
Optical Science and Technology Center and Department of Physics and Astronomy,
University of Iowa, Iowa City, Iowa 52242, USA
Theelectronicandmagneticpropertiesofneutralsubstitutionaltransition-metaldopantsindia-
mondare calculatedwithindensityfunctionaltheory usingthegeneralized gradientapproximation
to the exchange-correlation potential. Ti and Fe are nonmagnetic, whereas the ground state of
V, Cr and Mn are magnetic with a spin entirely localized on the magnetic ion. For Co, Ni, and
Cu, the ground state is magnetic with the spin distributed over the transition-metal ion and the
nearest-neighborcarbonatoms; furthermoreaboundstateisfound inthegapthatoriginates from
2
the hybridization of the 3d-derived level of the dopant and the 2p-derived dangling bonds of the
1
nearest-neighborcarbons. Ap–dhybridizationmodelisdevelopedinordertodescribetheoriginof
0
themagneticinteraction. Thismodelpredictshigh-spintolow-spintransitionsforNiandCuunder
2
compressive strain.

n
a PACSnumbers: 71.10.Hf,71.27.+a,71.30.+h
J

7
I. INTRODUCTION diamond due to the use of the TM as a solvent cata-

] lyst during the high-pressure-high-temperature (HPHT)
i
c growth process17–20. Among the Ni-related centers, the
s Diamondhasattractedgreatinterestrecentlyforquan-
NE8 center, formed by a substitutional Ni with four NN
- tum information science because of its wide band gap
l nitrogen atoms, is responsible for a ZPL at 793 nm with
r and the existence of more than 500 optically-addressible
t a narrow phonon sideband, but its low reproducibility
m centers1, many with long spin coherence times. An
impedes its use for applications. Among Cr-related cen-
extensively-studied color center is the NV center, con-
. ters, a ZPL at 749 nm15 has been attributed to an in-
at sistingofacarbonvacancywithanearest-neighbor(NN) terstitial chromium and two other ZPL at 744 and 764
m substitutional nitrogen atom. In its negatively-charged
nm have been detected but the source has not yet been
state, theNV− centercarriesaspinS =1, whichissplit
- identified16. TM dopants can also be introduced by ion
d by the C symmetry environment of the spin center2;
3v implantation14,15 or during chemical vapor deposition
n its spin state has been proposed as a qubit for quan-
o (CVD)13,16,whichofferthepossibilityofintroducingthe
tum computation. Spin initialization and readout has
c dopants substitutionally and without additional atoms
been demonstrated by optical pumping and absorption
[ or as part of a defect complex. Co- and Ni-related cen-
measurement3 at the zero phonon line (ZPL) at 1.945
ters, with different nitrogen, boron and vacancy com-
1 eV. Spin manipulation has been achieved by microwave
v manipulation4–6, and also through the interaction of the plexes, have been studied in various charge states by
2 GGA calculations21,22, finding neutral substitutional Co
NV− centerspinwithanitrogennuclearspin4 oranother
4 to be stable in pure diamond for a Fermi energy E be-
nearby substitutional nitrogen electron spin7. However, F
5 tween3.0and3.6eV,andneutralsubstitutionalNitobe
1 the NV− center has a large phonon sideband at room
stable for E between 2.6 and 3.0 eV.
. temperaturethatmaylimititsuseforquantuminforma- F
1 Transition-metal spin centers in diamond offer sev-
tion processing8.
0 eral potential advantages over other spin centers, includ-
2 Other applications, like quantum optics and quan- ing (1) the availability of both strongly-hybridizing (t -
1 tum cryptography, require stable, bright, and narrow symmetry) and weakly-hybridizing (e-symmetry) state2s
:
v linewidth single photon sources (SPS). Potential optical associated with the d levels23,24, (2) the high symmetry
i centers for SPSs in diamond are the NV− center, the of the substitutional dopant (tetrahedral group, which
X
silicon-vacancy (SiV) center9,10, Ni-related centers9,11–14 meansthecrystalfielddoesnotsplitangular-momentum
ar and Cr-related centers15,16. The large phonon sideband 1 triplets25), and (3) the availability of larger spin-orbit
of the NV− center spreads the emission over 100 nm re- interactions through the d electrons, which may permit
sultinginaDebye-Wallerfactoroflessthan5%,whichis faster manipulation of a spin through an electric field
notidealforquantumkeydistribution8. TheSiVcenter, (e.g. Ref. 26). Challenges for these spin centers include
formedbyasubstitutionalSinexttoavacancy,hasbeen the large strain introduced when they are placed substi-
proposed as a stable SPS with a ZPL at 738 nm, but its tutionally (although formation energies have previously
fast nonradiative decay degrades its fluorescence10 and beenfoundtobemoderate),andthepotentiallowerspin
ultimately its reliability. coherence time due to the larger spin-orbit interaction.
Transition-metal(TM)dopantshavealsobeenstudied To clarify these phenomena we present a first-
indiamond,howeverinitialstudiesfocusedonTMimpu- principlesstudyofsubstitutionaltransition-metal(TM )
s
rities in natural diamond samples, or in some synthetic dopants in diamond. Whereas TM dopants may exist
s
2
in several different charge states in diamond, we concen- V0 and Mn0, which we shall see below become magnetic
s s
trateourstudyontheneutralsubstitutionalTMdopant in a similar fashion to Cr0. A different origin is found
s
(TM0) and the chemical trends within the 3d row of the for magnetism in Co0 ,Ni0 and Cu0, which have much
s s s s
periodic table. For each TM0 dopant we determine the smaller magnetic energies ∼−100 meV. Sc0 and Zn0 are
s s s
ground-stateconfiguration,includingthechargeandspin includedtocompletethe3drow,howeverthemagnetism
state. We find for Ti through Cu a single p–d hybridiza- of these ions differs from that of all the other 3d ions.
tion model explains the electronic and magnetic proper-
ties of the defect. Sc and Zn, which occur at the ends of TM R[%] EM[meV] MT[µB] MTM[µB] MC[µB] NTM NC
the row, are not described by this model. In Section II, Sc 23.2 -49 1.0 0.2 0.2 19.3 6.4
after a description of the method,we present the results Ti 19.0 0 0.0 0.0 0.0 20.1 6.5
for3dtransition-metaldopants,inSectionIIIwedescribe V 16.7 -249 1.0 1.0 0.0 21.2 6.5
Cr 14.8 -1006 2.0 2.0 0.0 22.3 6.4
changes under strain of the ground-state spin for Ni and
Mn 13.3 -246 1.0 1.0 0.0 23.5 6.4
Cu and contrast with Cr (which is largely unchanged by
Fe 12.2 0 0.0 0.0 0.0 24.6 6.4
strain). In Section IV we comment on the implications
Co 12.9 -42 1.0 0.6 0.1 25.9 6.3
of these results and the strain-induced spin changes for
Ni 14.2 -71 2.0 0.9 0.2 26.9 6.3
quantum information processing.
Cu 15.7 -127 3.0 0.8 0.4 28.1 6.2
Zn 16.6 -100 2.0 0.2 0.3 29.0 6.3
II. DOPANT CONFIGURATION TABLE I. TMC SGGA Results : R is the NN relaxation,
63
CALCULATIONS AND p−d HYBRIDIZATION E = E −E the magnetic energy, M the total
M SGGA GGA T
MODELS magnetization, M (M ) the on-site magnetization on the
TM C
TM ion (NN C atom), N (N ) the calculated number of
TM C
A. Method electrons on the TM ion (NN C atom).
The calculations were performed with the scalar
relativistic version of the full potential local orbital
FPLO9.00-33 code27 using the (spin-polarized) gen-
eralized gradient approximation ((S)GGA) with the 4
parametrization of Perdew, Burke and Ernzerhof28. The
convergence of the results with respect to k-space in-
tegrals was carefully checked. We found that a 8x8x8 3
k-point grid were sufficient to obtain convergence of the
]
total energy difference between spin-polarized (SGGA) B
and spin unpolarized (GGA) calculations. mM [ 2
We used a 64-atom TMC (TM = Sc, Ti, V, Cr, Mn,
63
Fe,Co,Ni,Cu,Zn)supercellcorrespondingtoa2×2×2
1
multiple of the diamond conventional cell with a TM
substituted for one carbon. We fixed the supercell to
correspond to the experimental lattice constant29 of dia-
0
monda =3.5668˚A,andalltheatomicpositionswithin Sc Ti V Cr Mn Fe Co Ni Cu Zn
0
the supercell were allowed to relax with a precision of 1
FIG. 1. TMC SGGA total magnetic moment.
meV/˚A. We considered here only T -symmetric atomic 63
d
relaxation. As a check, we did a fully-relaxed-atoms cal-
culation without any symmetry constraints (C )for the
1
NiC supercell. The C relaxed calculation is about 2 Figure 1 shows the calculated total magnetization of
63 1
meV higher in energy than the T relaxed calculation, TMC supercells. Ti and Fe are non-magnetic. We
d 63
which rules out such a symmetry-breaking distortion. obtain for Cr and Ni a total magnetization M = 2 µ ,
T B
corresponding to a total spin S = 1, although we shall
see later that these two spin centers differ greatly due
B. Transition-metal dopant ground-state spin to the importance of different d orbitals in creating the
configurations ground-statespin. Zn,alsowithS =1,differsfromboth
as it has a closed d shell. For Sc, V, Mn and Co, we
The SGGA results are summarized on Table I. The obtain a total magnetization MT =1 µB, corresponding
most stable magnetic solution is obtained for Cr0, with to a total spin S = 1. The highest total magnetization
s 2
a magnetic energy EM = −1006 meV, corresponding to is obtained for Cu with MT = 3 µB corresponding to a
anS =1groundstateE lowerinenergythanthenon- total spin S = 3.
M 2
magnetic GGA solution. Less stable magnetic solutions For V0, Cr0 and Mn0, the total magnetization of
s s s
(E ∼ −250 meV) are obtained for the 3d neighbors M = 1.0, 2.0, and 1.0 µ is entirely localized on the
M T B
3
TM ion, whereas for Co0, Ni0 and Cu0, the total mag- GGAfunctional30. AsubstitutionalTMionisatthecen-
s s s
netization of M = 1.0, 2.0, and 3.0 µ is distributed ter of a tetrahedron formed by the 4 NN carbon atoms,
T B
between the TM ion and the NN carbon atoms with a implying a crystal field of T symmetry. This T crystal
d d
TM ion magnetization of M = 0.6, 0.9 and 0.8 µ field will split the single particle 3d manifold into a dou-
TM B
and a NN carbon atom magnetization of M = 0.1, 0.2 bly degenerate e level and a triply degenerate t level.
C 2
and 0.4 µ , corresponding to a total magnetic moment For tetrahedral bonding the t levels are expected to be
B 2
0.4,0.8and1.6µ distributedovertheNNC.Thenum- higher in energy than the e levels. The symmetry of the
B
ber of electrons located on the TM ion compared to its bondsfromthefournearestneighborsiscompatiblewith
atomic number, and the number of electrons on the NN t symmetry23, but not with e symmetry. Thus the 3d-
2
carbonatomsallowustoidentifytheconfigurationofthe derivedt levelhybridizeswiththeNNcarbon2p-derived
2
TM ion and the induced charge on the NN C’s. t states to form bonding t and antibonding t hy-
2 B AB
FromTitoFe,theTMoccupationnumberisconsistent brid states. Due to the strong hybridization between the
withaTM2+ configuration4s03dn (n=2,3,4,5,6)andthe TMionandtheNNcarbonatoms,the3d-derivedhybrid
TM2+ ion induces two extra electrons distributed over states will form bound states in the gap24,31,32.
the 4 NN carbon dangling bonds with a local charge on For the non-magnetic Ti0 impurity, there are 3 bound
s
each NN carbon atoms QC ∼ 0.5. For Co, Ni, Cu, the states in the gap : a totally occupied (6-fold degenerate)
TMoccupationnumberisconsistentwithaTM+ config- t level situated at E +0.6 eV, a totally empty (4-fold
B v
uration 4s03dn+1 (n = 7, 8, 9) and the TM+ ion induces degenerate) e level at E +3.3 eV and a totally empty
v
oneextraelectrondistributedoverthe4NNcarbondan- (6-fold degenerate) t level at E + 4.8 eV. For V0,
AB v s
glingbondswithalocalchargeoneachNNcarbonatoms Cr0,Mn0, thespinupandspindown(3-folddegenerate)
s s
QC ∼0.2. Sc and Zn differ greatly from all the other 3d tB levels are totally occupied and nearly spin degener-
dopants, and hence will be discussed only at the end of ate. For V, the t forms a bound state at E +0.7 eV
B v
Section II. and for Cr0 to Cu0, the t state forms a crystal field
s s B
resonance state inside the valence band which (with in-
creasing atomic number) decreases in energy. For V, Cr
andMn,theoccupationdifferenceofthespinupandspin
down e bound states is responsible for the total magne-
tization localized on the TM ion; the Cr e level config-
(0) uration is (e↑)2(e↓)0, corresponding to a total magnetic
moment M = 2.0 µ (S = 1). The absence of mag-
(0) T B
(0) (0) (0) (0) (0) (0) (0) netismforFeisduetoatotallyoccupiedspin-degenerate
(1) (1) e level (4-fold degenerate). For the next three dopants
(1) (2) in the series (Fig. 1), Co, Ni and Cu, the e levels remain
(3)
(6) (6) (2) (2) totally occupied and enter the valence band. For these
(4) (2) dopantsthespinupantibonding(3-folddegenerate)t↑
(2) AB
(3) state starts to become occupied, with 1, 2 or 3 electrons
(3) (6) (3) respectively, while the spin down antibonding t↓ level
AB
(3) remains totally empty. For Co0, Ni0 and Cu0, the oc-
s s s
cupation difference of the spin up and spin down t
AB
bound states is responsible for the total magnetization
that is distributed over the TM ion and the NN carbon
atoms through the hybrid t level; for example, the
FIG. 2. Single-particle TM 3d-derived SGGA Kohn-Sham AB
Ni t level configuration is (t↑ )2(t↓ )0, correspond-
(KS) energy levels of TMC63 calculation. TM 3d levels are AB AB AB
split by the tetrahedral crystal field into eσ and tσ2 levels. ing to a total magnetic moment MT = 2.0 µB (S = 1)
The bonding and antibonding levels (tσ and tσ ) are the distributed over the Ni ion and the NN carbon atoms.
B AB
resultsofthehybridizationbetweentheTMt2andNNcarbon Ap–dhybridizationmodel23,24 helpsexplaintheorigin
2p-derived t2 levels. Ev (energy reference) and Ec are the ofthemagneticinteractionanddifferentiatethebehavior
SGGAvalencebandmaximumand(direct)conductionband
of thosedopants to theleft ofFe fromthose to theright.
minimum of pure diamond. In parenthesis are indicated the
In pure diamond, each carbon shares 4 electrons to form
occupation number of the KS energy levels.
covalent bond. If one carbon atom is removed to form a
vacancy, 4electronswilloccupythedanglingbondswith
the neutral configuration 2s22p2. When a TM ion is in-
The TM 3d-derived Kohn-Sham states within the se- sertedatthevacancysitetoformasubstitutionaldopant,
ries are presented in Figure 2. The GGA direct and it will induce an extra charge on the dangling bond dis-
indirect band gap for pure diamond are respectively tributedoverthe4NNcarbons. ForTM=V,CrandMn,
ED = E −E = 5.6 and EInd = 4.3 eV, both of the magnetization is driven by the e bound states with
GGA c v GGA
whicharesmallerthantheexperimentalgapEInd =5.47 the magnetization entirely localized on the TM ion in
G
eV due to the well known gap underestimation of the the TM2+ configuration 4s03dn and two extra electrons
4
shared by the non-magnetic NN carbon atoms. These
two extra electrons will occupy the dangling bonds and
form a defect level of configuration 2s22p4. The TM 3d- 30
derived t2 levels hybridize with the 2p-derived t2 states spin up Total
occupied by 4 electrons. For TM=Co,Ni and Cu, the 20 Ti
C
magnetization is driven by the antibonding t bound NN
AB
states. The TM is in the TM+ configuration 4s03dn+1 V] 10
e
which induces one extra electron distributed over the es/
NN carbon atoms. This extra electron will occupy the Stat 0
dangling bonds and form a defect level of configuration OS [
2s22p3 which induces a non-zero magnetic moment on D-10
the NN carbon atoms. The TM 3d-derived t levels will
hybridize with these 2p-derived t states occ2upied by 3 -20 spin down
2
electrons.
-30
-8 -6 -4 -2 0 2 4
In the next subsections, we will present the electronic
Energy [eV]
structure of the different TM0 dopants sorted by their
s
spin values. Dopants with the same spin values can have
dramatically-differentoriginsforthemagnetism,depend-
ing on whether the magnetism is driven by the e states
or the t states. This difference will also lead to a very
AB
different response of the ground-state spin to strain.
Ti2+ C DB
NN
C. The S =0 centers : Ti0 and Fe0
s s
Fig. 3 shows the total, Ti and NN carbon partial den-
sity of states (DOS) of the TiC supercell. The Ti 3d
63
levels are split by the T crystal field into e and t lev-
d 2
els. Ti t levels hybridize with the NN carbon 2p levels
2
to form bonding t and antibonding t hybrid levels. FIG. 3. (color online) Top : Total (black solid line), Ti (red
B AB
The spin-degenerate bonding t levels are totally occu- dashed line) and NN C (blue, dot-dashed line) partial DOS.
B
pied and form a bound state at 0.6 eV above the valence Fortheseandsubsequentsuchplotstheupperportionofthe
plot is for the majority spin and the lower portion is for the
bandmaximum(VBM).Thespin-degenerateelevelsare
minorityspin;herethepartialDOS’sarethesameastheion
totally empty and form a bound state at 3.3 eV above
isnonmagnetic. Thebondingstatet isdelocalizedacrossthe
the VBM. The spin-degenerate antibonding t levels B
AB Ti3dandNNC2pstates. Bottom: Schematicrepresentation
aretotallyemptyandformaboundstatesituatedat4.7 of the p–d hybridization model for Ti0.
s
eV above the VBM. The number of electrons located on
the Ti ion is 20.1 which corresponds to a Ti2+ configu-
ration 4s03d2 and spin S = 0. Ti2+ induces two extra
1
electrons distributed over NN carbon dangling bonds of
configuration 2s22p4 and spin S =0. The Ti (t )2 level
2 2
hybridizes with the NN carbon 2p4 t defect level. The
2
spin-degenerate bonding t states are totally occupied
B
andtheelevelsaretotallyempty,whichexplainstheab-
sence of magnetism. Fig. 3 explains the observed DOS stateat0.9eVabovetheVBM.Thespin-degeneratean-
by a p–d hybridization model. The two Ti 3d electrons tibondingt levelsaretotallyemptyandformabound
AB
occupy the spin-degenerate Ti t level (violating Hund’s state situated at 3.8 eV above the VBM. The number of
2
rule)toformalowerenergyclosed-shellbondingt level electrons located on Fe is 24.6, which corresponds to a
B
with the four t electrons contributed by the nearest- Fe2+ configuration of 4s03d6 and spin S = 0. Fe2+ in-
2 1
neighbor carbons. ducestwoextraelectronsdistributedovertheNNcarbon
Fig. 4showsthetotal,FeandNNcarbonpartialDOS dangling bonds, which have a 2s22p4 configuration and
of the FeC supercell. The Fe 3d levels are split by spin S = 0. The Ti (t )2 level hybridizes with the NN
63 2 2
the T crystal field into e and t levels. Fe t levels hy- carbon 2p4 t defect level. The spin-degenerate bonding
d 2 2 2
bridizewiththeNNcarbon2plevelstoformbondingt t and e levels are totally occupied explaining the ab-
B B
and antibonding t hybrid levels. The spin-degenerate sence of magnetism. Fig. 4 explains the observed DOS
AB
bondingt levelsaretotallyoccupiedandformacrystal by a p–d hybridization model, showing it is very similar
B
fieldresonancestateat-1.7eVbelowtheVBM.Thespin- to Ti, but with a full e level instead of the empty e level
degenerateelevelsaretotallyoccupiedandformabound of Ti.
5
40 30
spin up Total spin up Total
Fe 20 V
C C
20 NN NN
V] V] 10
e e
es/ es/
Stat 0 Stat 0
S [ S [
O O
D D-10
-20
spin down -20 spin down
-40 -30
-8 -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4
Energy [eV] Energy [eV]
Fe2+ C DB
NN
V2+
C DB
NN
FIG.4. (coloronline)Top: Total,FeandNNCpartialDOS.
Bottom : Schematic representation of the p–d hybridization
modelforFe0. Theestatesareindicatedinredtoemphasize
s
their localized character; t states are indicated in black and
2
are delocalized between the Fe and NN C. Symbols are the FIG.5. (coloronline)Top: Total,VandNNCpartialDOS.
same as in Fig. 3. As before the upper portion of the plot is for the majority
spinandthelowerportionisfortheminorityspin. Bottom:
Schematic representation of the p–d hybridization model for
V0. Symbols are the same as in Fig. 3.
D. The S = 1 centers : V0, Mn0 and Co0. s
2 s s s
Fig. 5 shows the total, V and NN carbon partial DOS V(t )2 levelhybridizeswiththeNNcarbon2p4 t defect
2 2
of the VC supercell. The V 3d levels are split by the level. Thebondingt statesarefullyoccupied(asforTi
63 B
T crystal field into e and t levels. V t levels hybridize andFe),butinthecaseofVtheelevelsarepartiallyoc-
d 2 2
with the NN carbon 2p levels to form bonding t and cupied in the configuration (e↑)1(e↓)0 corresponding to
B
antibonding t hybrid levels. The bonding t levels a total magnetic moment M = 1.0 µ entirely local-
AB B T B
are totally occupied and form a nearly spin-degenerate ized on V. Fig. 5 explains the observed DOS with a p–d
bound state at 0.7 eV above the VBM. The spin up and hybridization model.
spin down e levels are localized on the V site, giving rise Fig. 6showsthetotalMnandNNcarbonpartialDOS
two bound states situated at 2.5 eV and 3.5 eV above of the MnC supercell. The Mn 3d levels are split by
63
the VBM and separated by a Hund exchange splitting the T crystal field into into e and t levels. Mn t levels
d 2 2
JH =1 eV. The V e level contribution to the total mag- hybridize with the NN carbon 2p levels to form bonding
netization is about 60 % and the bonding t states are t and antibonding t hybrid levels. The bonding t
B B AB B
weakly spin-polarized. The antibonding t states are levels are nearly spin degenerate and form a crystal field
AB
totally empty and form a nearly spin-degenerate bound resonance state located around -2.2 eV below the VBM.
stateat5.1eVabovetheVBM.Thenumberofelectrons The spin up and spin down e levels are localized on the
located on the V ion is 21.2, which corresponds to the Mn site, giving rise to two bound states situated at 0.6
V2+ configuration 4s03d3 and spin S = 1. V2+ induces eV and 1.8 eV above the VBM separated by a Hund ex-
1 2
two extra electrons distributed over the NN carbon dan- changesplittingJH =1eV.TheMnelevelcontribution
glingbondsofconfiguration2s22p4 andspinS =0. The tothetotalmagnetizationisabout70%andthebonding
2
6
20 40
spin up Total
spin up
Co
C
10 NN
20
V] V]
e e
es/ es/
Stat 0 Stat
S [ S [ 0
O O
D D
-10
Total
Mn -20 spin down
C
spin down NN
-20
-8 -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4
Energy [eV] Energy [eV]
Mn2+ Co2+
C DB C DB
NN NN
FIG.6. (coloronline)Top: Total,MnandNNCpartialDOS. FIG.7. (coloronline)Top: Total,CoandNNCpartialDOS.
Bottom : Schematic representation of the p–d hybridization The antibonding state t has approximately a 50 % Co 3d
AB
model for Mn0s. Symbols are the same as in Fig. 3. character (50 % NN C 2p character). Bottom : Schematic
representation of the p–d hybridization model for Co0. Sym-
s
bols are the same as in Fig. 3.
t statesareweaklyspin-polarized. Theantibondingt
B AB
levels are weakly spin-polarized and form a bound state
situated at 5.1 eV above the VBM. The number of elec- The spin up and spin down e levels are localized on the
tronslocatedontheMnionis23.5,whichcorrespondsto Cosite,givingrisetotwopeaksatthetopofthevalence
a 4s03d5 configuration and spin S = 1. Mn2+ induces band. The number of electrons located on the Co ion
two extra electrons distributed ove1r the2NN carbon dan- is 25.9 which corresponds to a Co+ configuration 4s03d8
gling bonds with 2s22p4 configuration and spin S2 = 0. and a spin S1 = 0. Co+ induces one extra electron dis-
The Mn (t )2 level hybridizes with the NN carbon 2p4 tributedoverNNcarbondanglingbondsofconfiguration
2
t2 defect level. The bonding tB states are fully occupied 2s22p3 and spin S2 = 12. The Co (t2)4 level hybridizes
and, as in the case of Mn, the e levels are partially oc- with the NN carbon 2p3 t2 defect level. The 2p3 config-
cupied in the configuration (e↑)2(e↓)1, corresponding to uration of the NN carbon dangling bond is responsible
a total magnetic moment MT =1.0 µB entirely localized for the S = 21 spin (MT = 1.0 µB) which is distributed
on the Mn. Fig. 6 explains the observed DOS by a p–d overtheCo(MCo =0.6µB)andthe4NNcarbonatoms
hybridization model. (MC = 0.1 µB). Fig. 7 explains the observed DOS by a
Fig. 7showsthetotal,CoandNNcarbonpartialDOS p–d hybridization model.
of the CoC supercell. The Co 3d levels are split by the
63
T crystal field into e and t levels. The Co t levels
d 2 2
hybridize with the NN carbon 2p levels to form bonding E. The S =1 centers : Cr0s and Ni0s.
t and antibonding t hybrid levels. The bonding t
B AB B
levels are weakly spin-polarized and form a crystal field Fig. 8 shows the total, Cr and NN carbon partial
resonance state located around -2.7 eV below the VBM. SGGADOSoftheCrC supercell. TheCr3dlevelsare
63
7
total magnetic moment M = 2.0 µ entirely localized
T B
on the Cr ion. Fig. 8 explains the observed DOS by a
20 p–d hybridization model.
spin up Total
Cr
C
NN
10
V]
e
es/ 20 spin up Total
S [Stat 0 NCNiN
O 10
D V]
-10 es/e
spin down Stat 0
S [
-20-8 -6 -4 -2 0 2 4 DO-10
Energy [eV]
-20 spin down
-8 -6 -4 -2 0 2 4
Energy [eV]
Cr2+
C DB
NN
Ni+
C DB
NN
FIG.8. (coloronline)Top: Total,CrandNNCpartialDOS.
Bottom : Schematic representation of the p–d hybridization
model for Cr0. Symbols are the same as in Fig. 3.
s
FIG.9. (coloronline)Top: Total,NiandNNCpartialDOS.
splitbytheT crystalfieldintoeandt levels. Crt lev- Bottom : Schematic representation of the p–d hybridization
d 2 2
elshybridizewiththeNNcarbon2plevelstoformbond- model for Ni0. Symbols are the same as in Fig. 3.
s
ing t and antibonding t hybrid levels. The bond-
B AB
ing t levels are nearly spin-degenerate and form crystal Fig. 9 shows the total, Ni and NN carbon partial
B
field resonance states around -1.5 eV below the VBM. SGGA density of states (DOS) of the NiC supercell.
63
The spin up and spin down e levels are localized on the The Ni 3d levels are split by the T crystal field into e
d
Cr site, giving rise to two bound states approximately and t levels. Ni t levels hybridize with the NN car-
2 2
0.8 eV and 2.8 eV above the VBM and separated by a bon 2p levels to form bonding t and antibonding t
B AB
Hund exchange splitting JH =2 eV. The Cr e level con- hybridlevels.Thespinupandspindownbondingt lev-
B
tribution to the total magnetization is roughly 80 % of els forms a crystal field resonance state situated around
the total magnetization and the bonding t states are -2.1 eV below the VBM. The spin up and spin down
B
weakly spin-polarized. The antibonding t levels are e levels are localized on the Ni site, giving rise to two
AB
totallyemptyandformstwoboundstatesaround4.3eV peaks approximately 1 eV below the valence band maxi-
above the VBM. The number of electrons located on the mum(VBM)andseparatedbyaHundexchangesplitting
Cr ion is 22.3, which correspond to a Cr2+ configuration JH = 0.4 eV. The spin up and spin down antibonding
4s03d4andspinS =1. Cr2+inducestwoextraelectrons t levels are located at 2.5 and 3.0 eV above the VBM.
1 AB
distributedoverNNcarbondanglingbondsofconfigura- Theantibondingt levelcontributiontothetotalmag-
AB
tion2s22p4andspinS =0. TheCr(t )2levelhybridizes netizationismorethan70%,with50%fromNi3dlevel
2 2
with the NN carbon 2p4 t defect level. The bonding t and 20 % from the NN carbon 2p level. The number of
2 B
statesarefullyoccupiedandtheelevelsarepartiallyoc- electronslocatedontheNiionis26.9,whichcorresponds
cupiedintheconfiguration(e↑)2(e↓)0 correspondingtoa to a Ni+ configuration 4s03d9 with a spin S = 1. Ni+
1 2
8
induces one extra electron distributed over the four NN els are situated at 1.9 and 2.7 eV above the VBM. The
carbon dangling bonds of configuration 2s22p3 and spin number of electrons located on the Cu ion is 28.1, which
S = 1. The Ni t levels hybridize with the NN carbon corresponds to a Cu+ configuration 4s03d10 and a spin
2 2 2
2p3 t2 defect level. This p–d hybridization picture iden- S1 =0. Cu+ induces one extra electron distributed over
tifiestheoriginofthecalculatedSGGADOSasduetoa the NN carbon dangling bonds of configuration 2s22p3
FM interaction between S1 and S2 as shown on Fig. 9. and spin S2 = 32. The Cu t2 level hybridizes with the
NN carbon 2p3 t defect level. The 2p3 configuration of
2
theNNcarbondanglingbondisresponsibleforthetotal
F. The unusual case of Cu0s : a S = 23 center. spin S = 32 (MT = 3.0 µB), which is distributed over
the Cu (M = 0.8 µ ) and the 4 NN carbon atoms
Cu B
(M =0.4 µ ). Fig. 10 explains the observed DOS by a
C B
p–d hybridization model. Cu+ 3d levels are nearly spin
degeneratecorrespondingtoanon-magnetic3d10 closed-
shell configuration.
30
spin up Total
20 Cu
C G. Outliers: Sc0 (S = 1) and Zn0 (S =1).
NN s 2 s
V] 10
e Fig. 11 shows the total, Sc and NN carbon partial
es/
S [stat 0 S3dGGlevAeldsehnysibtryidoifzsetasltiegsht(lDyOwSi)thoftthheeNSNcCc6a3rsbuopnertc2ellel.veSlcs
O
D-10 and give rise to 2 spin-up and 2 spin-down bound states
situated at 0.7 and 1.2 eV (1.0 and 1.6 eV). The spin-
-20 spin down down t2 level at 1.6 eV is partially empty (1 hole) which
is responsible for the total spin-polarization M = 1 µ
T B
-30-8 -6 -4 -2 0 2 4 whichisdistributedontheScandNNcarbonatoms. The
Energy [eV] two spin-degenerated peaks at 4.6 eV arises from the Sc
e levels. There are no antibonding levels and the density
of states cannot be explained by the p−d hybridization
model used above for Ti through Cu.
30
Total
Cu+ 20 SCc
C DB NN
NN
V]10
e
es/
Stat 0
S [
O
D-10
-20
FIG. 10. (color online) Top : Total, Cu and NN C par- -30
-8 -6 -4 -2 0 2 4
tial DOS. Bottom : Schematic representation of the p–d hy- Energy [eV]
bridizationmodelforCu0. SymbolsarethesameasinFig.3.
s
Fig. 10 shows the total, Cu and NN carbon partial FIG. 11. (color online) Total, Sc and NN C partial DOS.
DOS of the CuC supercell. The Cu 3d levels are split
63
by the T crystal field into e and t levels. Cu t levels Fig. 12 shows the total, Zn and NN carbon partial
d 2 2
hybridize with the NN carbon 2p levels to form bonding SGGAdensityofstates(DOS)oftheZnC supercell. Zn
63
t and antibonding t hybrid levels. The bonding t gives rise to two bound states of t symmetry at 1.6 and
B AB B 2
levels are weakly spin-polarized and are located around 2.1 eV above the VBM which are almost entirely of NN
2.6eVbelowtheVBM.Thespinupandspindownelev- carbon2pcharacter. Thepartiallyempty(2holes)bound
els are localized on the Cu site, giving rise to two nearly stateat2.1eVabovetheVBMisresponsibleforthetotal
spin-degenerate peaks situated around 2.4 eV below the magnetization M = 2 µ which is distributed over the
T B
VBM. The spin up and spin down antibonding t lev- ZnandNNcarbonatoms. ThenumberofelectrononZn
AB
9
is 29.0 corresponding to a configuration 3d104s1. Thus fixed-spincalculations. TheHS(LS)stateofCr0 andNi0
s s
theZn 4s level isalmost depletedandthe Zn4s electron corresponds to a total magnetic moment M = 2 µ
HS B
occupiesaboundstateofmostlyNNcarbon2pcharacter (M = 0 µ ) whereas the HS (LS) state of Cu0 corre-
LS B s
toloweritsenergy. TheZn+3dlevelsaredelocalizedand sponds to M = 3 µ (M = 1 µ ). For Cr0, we do
HS B LS B s
are located at about -6 eV below the top of the valence not obtain a transition due to the large magnetic energy
band and are nearly spin degenerate, corresponding to a oftheS =1groundstate,about1eVenergeticallylower
non-magnetic 3d10 closed-shell configuration. thantheS =0non-magneticsolution. Theenergeticsta-
bilityofthemagneticgroundstateisassociatedwiththe
e character of the highly spin-polarized states, whereas
both Ni and Cu have magnetism driven by the t states.
2
20 The results for Ni0 and Cu0 are presented in Fig. 13.
s s
Total We obtain a transition from high-spin S =1 (S = 3) to
2
Zn low-spin S = 0 (S = 1) under compressive hydrostatic
10 CNN strain with a transition2at e =−7 % ( e =−6 %) for
H H
V] Ni0 (Cu0).
es/e s s
Stat 0
S [
O
D
20
-10
Total
Ni
C
10 NN
-20
-8 -6 -4 -2 0 2 4
Energy [eV] V]
e
es/
Stat 0
FIG. 12. (color online) Total, Zn and NN C partial DOS. OS [
D
-10
III. HIGH-SPIN TO LOW-SPIN TRANSITION
-20
WITH STRAIN -8 -6 -4 -2 0 2 4
Energy [eV]
100
NiC
63
CuC
63
50
Ni+ C DB
V] NN
e
m
[LS 0
E
-HS
E
-50
-100 FIG. 14. Low-spin state of Ni0, S = 0 : Top : Total and Ni
-10 -8 -6 -4 -2 0 s
Hydrostatic Strain [%] and NN C partial DOS. Bottom : Schematic representation
of the p−d hybridization model for Ni0.
s
FIG. 13. Total energy difference between low-spin and high- Fig. 14and15presentstheDOSofthelow-spinSGGA
spin fixed spin moment SGGA calculations as a function of ground state of Ni0 and Cu0 under compressive hydro-
s s
hydrostatic strain. static pressure e = −10 %. The corresponding total
H
magnetic moment for the (spin unfixed) SGGA solution
Wenowstudytheeffectofhydrostaticpressureonthe is M =0.1 µ for Ni0 and M =1.4 µ for Cu0. The
LS B s LS B s
magnetic properties of high-spin TM0 dopants (TM = TM+ configuration 4s03dn+1 remains unchanged, which
s
Cr, Ni and Cu). For that purpose, we take the total en- induces an extra electron on the NN carbon atoms, cre-
ergydifferencebetweenhigh-spin(HS)andlow-spin(LS) ating a 2s22p3 defect level. Fig. 14 and 15 explain the
10
observed DOS with a p − d hybridization model. For and low-spin state is also observed in transition-metal-
Ni0, the spin-degenerate Ni 3d levels hybridize with the ion compounds, where it was shown theoretically to be
s
spin-degenerate 2p3 dangling bonds giving rise to a non- due to a competition between localization (induced by
magnetic interaction, corresponding to an antiferromag- theHund’sexchangecouplingwhichfavorsthehigh-spin
netic interaction between the Ni S = 1 and a spin state) and a tendency towards delocalization induced by
1 2
S =−1 distributedovertheNNcarbondanglingbonds. the crystal field (which favors the low-spin state)34.
2 2
ForCu0, thespin-degenerateCu3dlevelshybridizewith
s
the 2p3 dangling bond. The 2p3 configuration is respon-
sible for the total spin S = 1 distributed over the NN IV. CONCLUSION
2
carbon dangling bonds.
We have studied by first-principle calculations the
chemicaltrendsofneutralsubstitutionaltransition-metal
dopants in diamond. The origin of the magnetism for
20 these dopants has been explained by a p–d hybridization
model. Ti and Fe impurities are non-magnetic corre-
Total
Cu sponding to closed bonding hybridized t and an empty
C 2
10 NN (Ti) or full (Fe) e level respectively. For V, Cr and Mn,
V] themagnetizationisentirelylocalizedontheTMionand
e
es/ driven by the e levels. For Co, Ni and Cu, the magneti-
Stat 0 zationisdistributedovertheTMionandtheNNcarbon
OS [ atoms with a magnetization due to the antibonding t2
D
levels. Electron paramagnetic study of these magnetic
-10
centerswouldbeofgreatinterestinordertoconfirmthe
calculated chemical trend.
The calculated total magnetic moments for the
-20
-8 -6 -4 -2 0 2 4 TM ion series are in agreement with previous GGA
Energy [eV]
calculations35. Ref. 35 finds the same relaxation with
point goup T symmetry for Ti, Cr and Fe. They found
d
aNNrelaxationofD forV,MnandCoandaC sym-
2d 1
metry relaxation for Ni. To verify the validity of our
results, which were all-atom relaxation calculations per-
formedwithT symmetry,wedidnearest-neighboratom
d
relaxationcalculationswithinGGA(1meV/˚Aprecision)
with no symmetry constraints, fixing the other atomic
positions to their ideal value. For Ni, the tetrahedra cal-
Cu+
culated this way is slightly distorted to C symmetry,
C DB 1
NN howeverthedistortionislessthana1%differenceindis-
tance and less than a 0.1% difference in angle, with a
magnetic energy E = 48 meV and a total spin S = 1.
M
Hence our T -symmetry results represent well the state
d
of these transition-metal ions.
The nearly-degenerate S = 1 and S = 0 states of
thesubstitutionalnickelimpurityindiamondcanbeun-
derstood as an exchange-coupled system of two electron
FIG. 15. Low-spin state of Cu0s, S = 12 : Top : Total and Cu spins: onelocalizedonthenickelionandonedelocalized
and NN C partial DOS. Bottom : Schematic representation on the four nearest-neighbor (NN) carbon atoms. Such
of the p−d hybridization model for Cu0.
s nearly-degenerate S = 1 and S = 0 states can be used
to construct an effective two-state decoherence free sub-
The nearly-degenerate S = 1 and S = 0 states of space from the S = 0 triplet (T ) and the singlet (S)
z 0
thesubstitutionalnickelimpurityindiamondcanbeun- states. This approach would be similar to that imple-
derstood as an exchange-coupled system of two electron mentedfordoublequantumdots36 withelectrostaticgat-
spins: one localized on the nickel ion and one delocal- ing. For the nickel ion, strain modulation could be used
ized on the four nearest-neighbor (NN) carbon atoms33. tomanipulatetheenergysplittingbetweentheS andthe
Strain changes the exchange coupling and as a result T states,insteadoftheelectrostaticgatingusedfordou-
0
modifies the ground state from parallel to antiparallel ble quantum dots36. This is one way that the electronic
alignment. AsimilarphenomenonoccursforCu,butthe configurations of transition-metal dopants could be used
transition is between S = 3/2 and S = 1/2. The exis- to encode quantum information in a form amenable for
tenceofapressureinducedtransitionbetweenahigh-spin solid state quantum information processing.