Multiband tight-binding model of local magnetism in GaMnAs

0.24 MB·English

by  Jian-Ming Tang

Multiband tight-binding model of local magnetism in Ga Mn As
1−x x
Jian-Ming Tang and Michael E. Flatt´e
Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242-1479
Wepresentthespinandorbitallyresolvedlocaldensityofstates(LDOS)forasingleMnimpurity
and for two nearby Mn impurities in GaAs. The GaAs host is described by a sp3 tight-binding
4
Hamiltonian, and the Mn impurity is described by a local p-d hybridization and on-site potential.
0
Local spin-polarized resonances within the valence bands significantly enhance the LDOS near the
0
band edge. For two nearby parallel Mn moments the acceptor states hybridize and split in energy.
2
Thusscanningtunnelingspectroscopy candirectly measuretheMn-Mninteraction asafunction of

n distance.
a
J PACSnumbers: 75.50.Pp,75.30.Hx,75.30.Gw,71.55.-i

7
2
The successful growth of ferromagnetic diluted mag- magnetic interaction. The LDOS near the valence band

] netic semiconductors (DMSs) based on III-V com- edge is significantly enhanced by these resonances. Each
i
c pounds[1,2,3,4,5]hasattractedmuchattentiondueto MndopantcanenhancetheLDOSbyasmuchasafactor
s
the nearby metal-insulator transition and also for their of2upto the second-nearestneighbors(See Fig.1), cor-
-
l potential application to nano-scale nonvolatile storage responding to a ∼ 7% increase of the average LDOS for
r
t devices and to quantum computation [6, 7]. In order 1%Mnconcentration. ThestrongLDOSenhancementis
m
to turn a nonmagnetic semiconductor into a magnet, a consistentwithrecentangle-resolvedphotoemissionmea-
.
t sizable amount (∼ 1%) of magnetic dopants are intro- surements[16],andqualitativelyexplainstheincreasesof
a
m duced. Experimental studies [2, 8, 9, 10, 11, 12] have the absorption coefficients in the intraband [12] as well
shown strong correlations between the ferromagnetism as the interband absorption spectroscopies [9, 10]. In
-
d and the hole carriers that are also contributed by the particular, the experimental result that the band-edge
n Mn doping in the III-V DMSs. The emerging picture of absorption coefficient increases in finite magnetic fields
o this ferromagnetism is that the Mn magnetic moments more for one polarization of light is consistent with our
c
are localized, and the ferromagnetic coupling is medi- result that the resonances are both orbitally- and spin-
[
ated through the delocalized hole carriers [6, 13]. A polarized.

3
deeper understanding of this ferromagnetic coupling has Whereas the spin-polarized band-edge resonances will
v
been hampered by insufficient knowledge of the valence alter the Mn-Mninteraction,the acceptorstates provide
8
1 band structure. The assumption that the holes reside adirectmeansofmeasuringthatinteraction. Theaccep-
1 in the unperturbed valence bands of the host semicon- tor level of Mn splits when two Mn magnetic moments
5 ductors [2, 8, 14] has recently been questioned by both arein a parallelconfiguration[19]. The range ofinterac-
0
theoretical and experimental studies [10, 12, 15, 16]. Ef- tion between the Mn dopants can be probed by the size
3
fective mass theories provide good spectral resolution at of the splitting as a function of the separation. We find
0
/ the band edge, but cannot describe distortions on dis- thatthe splittings areaslargeastens ofmeVevenwhen
t
a tances of one or two lattice constants for they are only twoMndopantsareseparatedbyafewlatticeconstants.
m accurateformomentaclosetotheΓpoint. Densityfunc- This magnetic interaction is anisotropic with respect to
- tionalcalculations [17, 18], however,describe localprop- theaxisconnectingthetwoMndopantsduetospin-orbit
d ertieswell,butstateoftheartsupercellcalculationshave interaction,inqualitativeagreementwiththecontinuum
n
not had sufficient spectral resolution to resolve the shal- model [20]. Both the local enhancement of the valence
o
c low bound states in the gap and the sharp resonances in bandedgeandthesplittingoftheacceptorlevelcouldbe
: the valence bands. probed by scanning tunneling spectroscopy (STS) [21].
v
i In this Letter we study the local density of states As pointed out by Vogl and Baranowski [22], the
X
(LDOS)ofverydiluteconcentrationsofMninGaAsasa fivefold degenerate Mn 3d-orbitals split in a cubic lat-
ar prototype III-V DMS, and show that the valence bands tice to an E-symmetric (dx2−y2-like) doublet, which is
are strongly altered by the Mn dopants, and in return only weakly coupled to the tetrahedral host, and a T2-
influence the interaction between the Mn magnetic mo- symmetric (d -like)triplet, which caneffectively couple
xy
ments. In order to obtainboth sufficient spectral resolu- to the neighboring dangling sp3-hybrids. The antibond-
tion and proper dispersion relations throughout the full ingstatesoftheT2-symmetricstatesandthesp3-hybrids
Brillouinzone,amultibandtight-bindingapproachincor- form the acceptor states within the gap. These anti-
porating spin-orbit interaction is employed. Our results bondingstatesaredelocalizedinspacesincetheyoverlap
show that the hybridization between the Mn 3d-orbitals stronglywiththehostvalencebands. Inourcalculations
andtheGaAsvalencebandsleadstospin-polarizedreso- thehybridizationofthemajorityMnd-orbitalsistreated
nances within the valence bands and to delocalizedferro- asaneffectiveextendedspin-dependentpotentialU1 act-
2
ingonthefournearest-neighborsitesofMn,becauseonly 4 2
Hsaamndiltponoirabni.taTlshaeroepeexrpatliocritfloyrmuseodf4oinuroMurntipgohtte-nbtiinadliinsg -1(eV) 23 (a) Mn ↑↓ (b) As (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) 11.5
V = U0Xc†ℓs(R0)cℓs(R0)+U1XXc†ℓ↑(Rj)cℓ↑(Rj), ates 1 0.5
ℓ,s j=1 ℓ st
f
(1) y o
(swchℓesel(erRcet)rU)o0nisisitnhtehtehcreoenaℓ-tsiooitrnebi(otaarnblniatiahtlilsaeintteieornRg)y.odpTieffrheaerteoMnrcnoef,daco†ℓsspp(aRinn)t- ocal densit00..46 (c) Ga (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (d) Ga (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) 00..46
is located at R0, and the four nearest-neighbor sites are L0.2 0.2
labeled by R1-R4. We neglect the difference of the po-
tentialmatrixelementsbetweenthe s andporbitals,be- 0 0
-1.5 -1 -0.5 0 -1 -0.5 0 0.5
causethes-orbitalsmakeanegligiblecontributiontothe
Energy (eV)
LDOSnearthevalencebandedge. Thequantizationaxis
for the spin is aligned with the Mn core spin. The mi-
FIG.1: LDOSspectraforthecharged(ionized)Mndopantin
nority Mn d-orbitals are assumed to be much higher in
Ga1−xMnxAs(correspondingtobulkn-doped)at(a)theMn
energy, and the hybridization energy of them with the site,at(b)oneofthenearest-neighborAssites,andat(cand
spin-down states is neglected. We find that the effective d) one of the second-nearest-neighbor Ga sites. The shaded
potential U1 is of critical importance in inducing the ac- balls in each panel show the corresponding atomic sites, and
ceptor level at the experimentally observed energy. The theblackballshowstheMnsite. TheMncorespinisaligned
with the z-axis of the lattice, and the z-axis is aligned ver-
significanceof the hybridizationto the acceptorlevelen-
tically. In this particular case, the spectra at four nearest-
ergy has also been noted recently by Dietl et al. [23]. As
neighborsitesareidentical, andhavetwodifferentfunctional
for the delocalized nature of the acceptor state, our re-
forms at the second-nearest-neighbor sites. The total LDOS
sults show that only about 10%of the spectralweightof isshownbythesolidlines. Thedashedlinesshowtheprojec-
the acceptor state is concentrated at the Mn site, 20% tions onto the two spin polarizations relative to the Mn core
is distributed over the four nearest-neighbor sites, and spin. The dotted lines show the LDOS of the bulk GaAs.
the remaining 70% is extended to farther sites. Further- The zero of energy is the valence maximum of bulk GaAs.
The valence band LDOS of bulk GaAs is more concentrated
more, the large spin-orbit interaction of GaAs splits the
at the As sites than at the Ga sites, because the LDOS is
gap states into three different energy levels, and intro-
thedensity of states modulated by theprobability density of
duces anisotropy to the spatial structure.
electrons. FortheneutralMn dopant (corresponding tobulk
WeusetheKoster-Slatertechnique[24]tocalculatethe p-doped Ga1−xMnxAs) only the highest energy peak of the
Green’s function whose imaginary part gives the LDOS. three localized peaks apparent above in each panel would be
Thismethodhasbeenproventogivethecorrectchemical visible.
trend of impurity levels in semiconductors [25], and we
have previously applied it to successfully predict STM
where Gˇ(ω) is the full matrix representation using all
spectra near impurities in superconductors [26]. Start-
ing with the tight-binding Hamiltonian Hˆ0(k) of homo- asittoemRic iosrbgiitvaelns bayt all lattice sites. The LDOS at each
geneous GaAs, one first calculates the retarded Green’s i
function, Gˆ0(k,ω) = [ω − Hˆ0(k) + iδ]−1. To obtain 1
A(R ,ω) = − Im tr Gˆ(R ,R ,ω) , (3)
a good description of the valence band structure, we i π h α i i i
use the sp3 model including spin-orbit interaction [27].
Then, by Fourier transforming Gˆ0(k,ω), we construct where the trα is taken with respect to the orbitals of the
the homogeneous Green’s function in coordinate space, α atom, depending on which type of atom is actually
Gˆ0(Ri,Rj,ω),whereRi andRj labelthezincblendelat- located at the site Ri.
tice sites. This step consumes the majority of computa- TodeterminethevaluesofU0 andU1,wefirstconsider
tion time. It takes about one day for one link (R −R ) the possibility that the potential is nonzero only at the
i j
with a spectral range of 2 eV on a personal computer. Mn site (U1 =0), and the size of the on-site potential is
The number of links used in this study is about 500. A assumed to be the same for all orbitals. When the Mn
constantlinewidth of δ =10 meV produces a good spec- atomreplacesaGaatom,the strengthofthison-sitepo-
tral resolution within a reasonable computation time. tentialis estimated to be U0 ∼1 eV basedonthe energy
The final Green’s function is obtained by solving the difference of the ionizationenergies of Ga (4s24p)and of
Dyson’s equation, Mn (3d54s4p), which are about 6 and 5 eV respectively.
Notonlyisthison-sitepotentialtooweaktobindanyac-
Gˇ(ω) = Gˇ0(ω)+ 1ˇ−Gˇ0(ω)Vˇ −1Gˇ0(ω), (2) ceptorlevelinthegap,itisnotpossible(evenwithanun-
(cid:2) (cid:3)
3
p + i p p p − i p
x y z x y
7
(a) (b) (c)
)
1 6
-V
e
(
s 5
e
at
st4
f
o i
sity 3
n ii
e
d2
cal iii
o1
L
iv
0
0 0.1 0.2 0 0.1 0.2 0 0.1 0.2
Energy (eV)
FIG.3: Theorbitally resolved LDOSfor(a)px+ipy,(b)pz
FIG.2: (color) Thespatial structureof(a) theMn acceptor and (c) px−ipy orbitals. The Mn core spin is aligned with
levelat 113 meVand of (b)thevalencebandLDOSat −500 the z-axis. Each panel contains four LDOS curves. From
meV.TheLDOSpeaksarecenteredattheMnatom,andare top to bottom, the curvesshow theLDOS at (i) theMn, (ii)
extended mainly on the (001) plane. In (b), far away from the nearest-neighbor (As) sites, and (iii and iv) the second-
theMn site, thesites with higher LDOS are occupied by the nearest-neighbor (Ga) sites. Cases (iii) and (iv) correspond
Asatom,andwithlowerLDOSareoccupiedbytheGaatom. to the configurations (c) and (d) in Fig. 1 respectively. The
We have assumed that the squared modulus of the Wannier LDOS spectra are shifted by multiples of one unit for easier
functions to be a Gaussian with a width of half the distance visualization.
between neighboring Ga and As atoms.
spin-polarized (parallel to the Mn core spin) and split
realisticallylargeon-sitepotential,U0 ≫10eV)tobinda into three energylevelsdue to the spin-orbitinteraction.
holemorethan60meVabovethevalencebandedge. The Forbulkn-dopedGa1−xMnxAstheFermilevelliesabove
Mndopantintroducesanacceptorlevelat113meVabove
the upper acceptor state, so the impurity is ionized and
the valence band edge, therefore, the effective potential
all three levels would be visible in a tunneling experi-
mustbeextendedatleasttothefournearest-neighborAs
ment. If the Fermi level lies below the upper acceptor
sites. If we fix U0 = 1 eV, and tune U1 to give the cor- state,suchasinbulkp-dopedGa1−xMnxAs,thendueto
rect acceptor level energy, the required nearest-neighbor
the electron-electron interaction only one localized state
potential is about 3.59 eV. This nearest-neighborpoten-
will be visible above the Fermi level and none below.
tial is too large to be accounted for by the tail of the
Due to band-bending at the surface it is often possible
screened Coulomb potential. The screened Coulomb po-
to see both the ionized and neutral dopants in the same
tential at the nearest-neighbor sites of a singly charged
sample (as in Ref. 21). As the Mn core spin rotates,
Mn center is only about 0.5 eV, or even smaller if the
the orbital character of the acceptor states changes, but
system is doped with carriers, as in our case. Therefore, the three energy levels remain the same within the sp3
the large effective potential at the nearest-neighbor sites
model. There is some experimental evidence for the ex-
must resultfrom the hybridizationof the Mn 3d-orbitals
istence of the two additional acceptor energy levels [29].
and the host sp3-hybrids. The exchange interaction ef-
Thesignificantenhancementofthespin-upLDOScontin-
fectively pushes the valence-band states with the same
uesintothevalencebandto1.5eVbelowthe bandedge.
spin polarization as the Mn core spin into the gap. The
The spin-polarizedresonances emerge at an energy close
states with the opposite spin polarization essentially re- to the split-off band top (∼ 350 meV below the band
main unperturbed.
edge). Onthe other hand, the spin-downband is weakly
Figure1showstheLDOSspectraattheMnsiteandat perturbed and non-resonant. Figure 1 (c) and (d) also
neighboring sites. In this particular case we have chosen show that the large enhancement of the valence band
the Mn core spin to align with one of the (100) crystal LDOS extends to the second-nearest neighbors. From
axes (which is the bulk easy axis [28]). As a result, the the spectral weight of the acceptor state, ∼ 5% at each
spectra at the four nearest-neighbor As sites are identi- As site,we estimate the p-dexchangeinteractionβN0 to
cal due to the residual symmetry operations of C2 and be 52U1 ×20% ≈ −0.3 eV, within a factor of 3 of that
S4 about the crystal axes. The twelve second-nearest- obtainedfromtransportmeasurements(seeChapter1in
neighbor Ga sites are divided into two classes with four Ref. 6). The discrepancy may come from the enhanced
andeightsiteseach. Theacceptorstatesarealmostfully LDOS near the Mn spin, which is not included in the
4
1) 4 orbital character more extended in the plane perpendic-
-V 3 (a) 18 17 14 18 ular to the Mn core spin, the splitting of the acceptor
e
s ( 2 level is the largest when the Mn core spins are oriented
e
stat 1 71 62 63 54 perpendicular to the axis that joins them.
f In summary we have presented calculations of the
sity o 43 (b) [001] 100 LduDcOesSspniena-rpoMlanriziendGreasoAnsa.ncTeshtehapt-denhhyabnrcieditzhaetioLnDOinS-
n
de 2 near the valence band edge. The spin-orbit interaction
al 1 Mn 140 splitstheacceptorlevel,andintroducesanisotropytothe
c
Lo 0 -0.1 0 0.1 0.2 [−110] MThne-MranngineteorfatchteioMn nin-Madndiintitoenratcotitohneicsrsyhsotawlnantoisoetxrtoepnyd.
Energy (eV) through several lattice constants. For quantitative com-
parisons with STS measurements, significant surface ef-
FIG.4: Splittingoftheacceptor levels. The left panelshows fectsareexpectedforaMnatomatthetoplayerbecause
the spectra at the Mn site with two Mn dopants separated theMnatomhasonelessAsneighbor,andthehybridiza-
by (a) (0,0,3) and (b) (−3/2,3/2,0). The dashed line shows
tionchanges. However,ithas beenreportedthat dopant
the spectrum when there is only one Mn dopant. In (b) the
atoms as deep as in the fifth layer are seen by STS [21],
twosplitpeaksofthe113meVlevelareindicatedbythetwo
and the spectra for these cases should closely resemble
heavy-line segments. The right panel shows the amount of
the splitting of the 113 meV level. One of the Mn atoms is the bulk.
locatedatthe(0,0,0)site(bottom-leftcorner),andtheother
This work was supported under the ARO MURI
is at one of the circles. The number in the circle shows the
DAAD19-01-1-0541.
splittingenergyinmeV.BothMncorespinsarealignedwith
thez-axis. The black dots show the Assites.
transport modeling.
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x y [13] T. Dietl et al., Science 287, 1019 (2000).
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[17] S.Sanvito,P.Ordej´onandN.A.Hill, Phys.Rev.B 63,
further split into “bonding” and “antibonding” states.
165206 (2001).
When the two Mn core spins are antiparallel, one of the
[18] M. van Schilfgaarde and O. N. Mryasov, Phys. Rev. B
U1 terms acts on spin-down As-orbitals, and the poten- 63, 233205 (2001).
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133 (1985).
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