Band model for the understanding of ferromagnetism in semiconductors and insulators

0.68 MB·English

by  Mirek Hänsel

Band model for the understanding of ferromagnetism in semiconductors and
insulators
M. H¨ansela,∗, W. Noltingb, M. Potthoffa
aI. Institut fu¨r Theoretische Physik, Universit¨at Hamburg, Jungiusstr. 9, 20355 Hamburg, Germany
bInstitut fu¨r Physik, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
Abstract
5To investigate ferromagnetic semiconductors and insulators, such as the famous EuO, EuS, or CrBr3, we propose a hy-
1bridizedKondo–latticemodel, where, inadditiontotheconductionelectrons, localizedmoments(e.g., the4f–electrons)
0aremodeledasastronglycorrelatedbandsystem. Thequasi–emptyconductionbandisweaklyfilledduetothehybridiza-
2
tion term. This activates the intraatomic exchange coupling between conduction and localized electrons. Temperature–

ndependent phase diagrams and quasiparticle densities of states are presented for various coupling and hybridization
astrengths. Moreover, the influence of the one–particle energy of the localized electrons E is discussed. A comparison
J f
with mean field calculations is given at the end of this work.
5
Keywords: ferromagnetism, ferromagnetic semiconductors, ferromagnetic insulators

]
l
e
-
r1. Introduction For our purposes, the well–known Kondo–lattice model
st [13], with local spins replaced by correlated f–orbitals,
. Collective magnetism requires the existence of perma- is a formidable starting point as the interaction between
t
anent magnetic moments, which arise from either local- subsystems (in our case: localized and itinerant electrons)
m
ized or itinerant electrons. The distinction of the mag- may lead to magnetic ordering. Note, however, that the
-netic moment’s type is of enormous conceptual impor- interaction is of indirect nature (e.g. Ruderman-Kittel-
d
ntance: While itinerant electrons cause band magnetism, Kasuya-Yosida interaction [14, 15, 16], hereafter referred
oparticularly known from the classical ferromagnets Fe, Co to as RKKY), thus, in search of spin–ordering effects, re-
cand Ni, and usually examined within the Hubbard model quiringatleastaminimumofelectronsineachsubsystem.
[[1, 2, 3], local moment magnetism can be found in insu- The problem one is now confronted with is that semicon-
1 lators and semiconductors (such as EuO, EuS and CrBr3 ductors,andinsulatorslikewise,havequasi–emptyconduc-
v[4, 5, 6]), in the huge class of the heavy fermion systems, tion bands, making indirect electronic exchange impossi-
8in diluted magnetic semiconductors (which are promising ble. In our work, we therefore extend the Kondo–lattice
4candidates for spintronic applications [7, 8, 9, 10]), and model by a hybridization term, which gives localized elec-
9
0also in some metallic systems like Gd. Such local moment trons the opportunity to virtually transform into conduc-
0systemsareusuallydescribedbyspinmodels(e.g. Heisen- tionbandelectronsandviceversa. Thissolvestheproblem
.bergorIsingmodel[11,12])withcouplingconstantsdueto of beforehand absent conduction electrons, hence allowing
1
0direct exchange or any type of superexchange. It remains avirtualRKKYinteraction,butalsodestabilizesthemag-
5an important aspect of modern magnetism fundamental neticmomentsinthedifferentsubsystems. Aswecanread
1research to find a unified theory which depicts the variety offfromourresults,thelastfactisofgreatimportancefor
v:of magnetic phenomena. A small step along this path will theunderstandingofthemagnetismfoundwithourmodel
ibe presented in this paper. proposed.
X
Since collective magnetism is exclusively realized in The setup of this paper is as follows: In Sec. 2 we
arsolidstatebodies,whoseelectronicstructureisrepresented present the Hamiltonian, which characterizes the many–
by energy bands and gaps that consequently determine bodyproblemthatwesolvebyGreen’sfunctionsmethods
the magnetic properties, it seems to be the more natural in Sec. 3. Within this solution, a set of self–energies is re-
choice to use band models instead of pure spin models. In quired, but so far undetermined. Approximations to com-
thepresentwork, weproposeabandmodelthatdescribes pute the self–energies are presented in Sec. 4, finalizing
semiconductorsandinsulators,i.e.,localmomentsystems. the theoretical framework. The results and a correspond-
ingdiscussionarepresentedinSec. 5. Asummaryisgiven
in Sec. 6.
∗Correspondingauthor
Email address: [email protected]
(M.H¨ansel)
Preprint submitted to Elsevier January 6, 2015
2. Model The last part of the Hamiltonian in Eq. (1),
For the above described systems we introduce the fol- H =V (cid:88)(cid:16)c† f +f† c (cid:17) , (7)
lowing Hamiltonian H: V iσ iσ iσ iσ
i,σ
H =H +H +H +H +H . (1)
s f f(U) sf V allowsfor(virtual)electronictransitionsfromthe4f–level
into the conduction band and vice versa, where V is the
Thekineticenergyoftheconductionbandelectronsiscon-
hybridization strength. While the s–f–exchange, Eq. (6),
cerned by H ,
s originates from the non–classical part of the Coulomb in-
H = (cid:88)(T −µδ )c† c , (2) teraction between the conduction electrons and the local-
s ij ij iσ jσ izedf–electrons,thehybridizationterm,Eq. (7),isaone–
i,j,σ
particle scattering term which mimics, in the most sim-
wherec† (c )arecreation(annihilation)operatorsofcon- ple way, the hybridization between the respective bands.
iσ iσ
Whileconcretevaluesmightbeobtainedbystandardself–
duction band (s-)electrons at lattice site R with spin σ,
i
consistent band–structure or constraint RPA calculations,
respectively. The chemical potential is denoted by µ, and
there is a priori no reason why those couplings should not
T are the usual hopping matrix elements, which are re-
ij
co–exist.
lated to the Bloch energies (cid:15)(k) via
As a consequence of the hybridization, the respective
1 (cid:88) average occupation numbers
T = (cid:15)(k)e−ik(Ri−Rj) . (3)
ij N
k n =(cid:68)c† c (cid:69) , n =(cid:68)f† f (cid:69) ,
sσ iσ iσ fσ iσ iσ
Forlowbandoccupations,aCoulombinteractionbetween (cid:88) (cid:88) (8)
s–electrons can be neglected safely. ns = nsσ , nf = nfσ ,
Assuminganon-degeneratef–levelE ,theone–partic- σ σ
f
le energy of the localized f–electrons with creation (anni-
are not necessarily constants when varying a model pa-
hilation) operators fi†σ (fiσ) is represented by rameter. Contrary, the total occupation number
(cid:88)
H = (E −µ)f† f . (4) n=n +n (9)
f f iσ iσ s f
i,σ
is kept fixed at a constant value by proper adjustment of
In order to disfavor double occupancies, a large intra-
the chemical potential µ. Since needed for later purposes,
atomic Coulomb interaction is taken into account by
we also introduce the dimensionless magnetizations m ,
s,f
Hf(U) = U2f (cid:88)nfiσnfi−σ , (5) ms =(cid:88)zσnsσ ,
i,σ σ
(10)
(cid:88)
m = z n ,
wheren =f† f isthenumberoperator. Sincethe4f– f σ fσ
fiσ iσ iσ σ
wave functions’ overlap is negligible, a direct f–electron
exchange is fairly small and hence not part of the Hamil- using z =δ −δ .
σ σ↑ σ↓
tonian(1). Note,moreover,thatEqs. (4)and(5)areonly Tofixsomeofthemodelparameters,wesettheon–site
validforsystemswithatotalspinof1/2,thusbeinganap- energies of the conduction electrons to T =0 eV. Hence,
ii
proximationinmodelingthe4f–levels. Thissimplification the conduction band center of gravity defines the energy
is done in order to keep mathematics on a tractable level, zero. EnergyunitsareessentiallyfixedbychoosingW =1
whereas the underlying physical effects are not expected eVforthewidthoftheconductionband. Furthermore,we
to be affected significantly. assume a strong Hubbard interaction U (cid:29)W.
f
To allow for magnetic ordering, interactions between The one–particle energy of the 4f–levels, E , is a pa-
f
conductionbandandf–electronsmustbeconsidered. The rameterthatdecisivelyaffectsthephysicsofthemodel. A
s–f–exchange is commonly described by an intraatomic stable local magnetic moment on the f-levels is formed in
spin–spin–coupling, the limit E (cid:28) 0 and E +U (cid:29) 0 where each f-level is
f f f
exactly occupied by one electron. Here, however, we also
(cid:88)
Hsf =−J σi·Si , (6) consider a parameter regime where Ef comes close to the
i lower edge of the conduction band. This implies the pres-
ence of charge fluctuations and thereby the hybridization
where σ and S represent the conduction and f–electron
i i
term,Eq. (7)isactivatedwhicheventuallygeneratesanef-
spinoperatorsatlatticesiteR ,respectively. Thecoupling
i
fectivef–f magneticexchange. Notethatthisimpliesthat
strength is given by J, where positive (negative) values of
the Schrieffer–Wolff transformation [17] does not apply to
J stand for a preferred (anti)parallel alignment of s- and
this parameter regime. Consequently, H +H +H
f–electron magnetic moments. f f(U) V
2
cannot be replaced by a local antiferromagnetic exchange where Σ(s),sf(E) is the Fourier–transformed self–energy,
kσ
which would trivially compete (or cooperate) with H . G0 is the correlation–free s-Green’s function,
sf k
Itisworthmentioningthatthemodel(1)reducestothe
(cid:126)
conventionalperiodicAndersonmodelifthes–f–exchange, G0(E)= , (16)
Eq. (6), is neglected (J = 0). On the other hand, for k E+µ−(cid:15)(k)
V =0,themodelessentiallyreducestoafermionizedvari-
andwhereGV=0 standsforthehybridization–free(V =0)
ant of the famous Kondo-lattice model. kσ
conduction band electrons’ Green’s function:
(cid:126)
3. Theory GV=0(E)= . (17)
kσ E+µ−(cid:15)(k)−Σ(s),sf(E)
kσ
In order to determine n and m self–consistently,
s,f s,f
we attack the many–body problem posed by the Hamilto- Since needed for later purposes, we also introduce the free
nian(1)usingGreen’sfunctionmethods. Withthedefini- propagator G0(E):
tion of the pure one–particle s- and f–Green’s functions,
1 (cid:88)
G (E)= G0(E) . (18)
1 (cid:88) (cid:68)(cid:68) (cid:69)(cid:69) 0 N k
Gkσ(E)= N eik(Ri−Rj) ciσ;c†jσ E , k
i,j (11) Inanalogytothesystemofconductionbandelectrons,
F (E)= 1 (cid:88)eik(Ri−Rj)(cid:68)(cid:68)f ;f† (cid:69)(cid:69) , the equation of motion of the localized electrons can be
kσ N iσ jσ E derived:
i,j
1 (cid:16)
as well as with two mixed Green’s functions, F (E)= (cid:126)+(E −µ)F (E)+VP(1)(E)+
kσ E f kσ kσ
Pk(1σ)(E)= N1 (cid:88)eik(Ri−Rj)(cid:68)(cid:68)ciσ;fj†σ(cid:69)(cid:69)E , + N1 (cid:88)eik(Ri−Rj)(cid:68)(cid:68)(cid:2)fiσ,Hf(U)(cid:3)−;fj†σ(cid:69)(cid:69)E+
i,j (12) i,j
Pk(2σ)(E)= N1 (cid:88)i,j eik(Ri−Rj)(cid:68)(cid:68)fiσ;c†jσ(cid:69)(cid:69)E , + N1 (cid:88)eik(Ri−Rj)(cid:68)(cid:68)[fiσ,Hsf]−;fj†σ(cid:69)(cid:69)E .
i,j
all given in k–space with N being the number of lattice
(19)
sites, one easily obtains the following equation of motion:
As in Eq. (14), we use the following substitutions for the
1 (cid:16)
G (E)= (cid:126)+((cid:15)(k)−µ)G (E)+VP(2)(E)+ f–electrons:
kσ E kσ kσ
+ N1 (cid:88)eik(Ri−Rj) (cid:68)(cid:68)[ciσ,Hsf]−;c†jσ(cid:69)(cid:69)E(cid:17) . (cid:2)fiσ,Hf(U)(cid:3)− −→(cid:88)Σi(mf)σ,U(E)fmσ ,
i,j m
(20)
(13) [f ,H ] −→(cid:88)Σ(f),sf(E)f ,
iσ sf − imσ mσ
In Eq. (13), the angular brackets represent a higher m
Green’s function. To find a solution of the many–body
whichareonlyvalidwithintheGreen’sfunctions. Defining
problem, we use the following ansatz according to the rig-
a hybridization–free f–Green’s function FV=0,
orous Dyson equation [18]: kσ
(cid:126)
[ciσ,Hsf]− −→(cid:88)Σ(ims)σ,sf(E)cmσ , (14) FkVσ=0(E)= E+µ−Ef −Σk(fσ),U(E)−Σk(fσ),sf(E) , (21)
m
which is only valid within the expression for the Green’s with Σ(f),U(E) and Σ(f),sf(E) being the Fourier–transfor-
kσ kσ
function (cid:68)(cid:68)[c ,H ] ;c† (cid:69)(cid:69) . The equation above de- med expressions of the self–energies given through Eq.
iσ sf − jσ E (20), the full f–Green’s function can be written as
fines the electronic self–energy Σ(s),sf(E). Note, that Eq.
imσ (cid:32) (cid:33)
(14)doesnotrepresentanoperatoridentity. Insertingthe VP(1)(E)
F (E)=FV=0(E)· 1+ kσ . (22)
self–energy ansatz (14) into the equation of motion (13), kσ kσ (cid:126)
one obtains:
1 ThemixedGreen’sfunctionsP(1) andP(2) areyetun-
G (E)=G0(E)+ G0(E)Σ(s),sf(E)G (E)+ kσ kσ
kσ k (cid:126) k kσ kσ known. Usingthe substitutions (14)and(20) intheequa-
+ V G0(E)P(2)(E) tions of motion for Pk(1σ) and Pk(2σ), and using the hybri-
(cid:126) k kσ (15) dization–free Green’s functions (17) and (21), one gets:
(cid:32) (cid:33)
VP(2)(E)
=GVkσ=0(E)· 1+ k(cid:126)σ , P(1)(E)= VGVkσ=0(E)F (E) , (23)
kσ (cid:126) kσ
3
and ansatz (14), the following self–energy was obtained:
VFV=0(E)
P(2)(E)= kσ G (E) . (24)
kσ (cid:126) kσ Σ(s),sf(E)=− Jz m +
kσ 4 σ f
NohigherGreen’sfunctionsappearintheseresults,which
enablesustosolvethesystemofs-andf–Green’sfunction + J2 · Sf(Sf +1)− zσ2mf (cid:0)zσ2mf +1(cid:1) .
equations by inserting Eqs. (23) and (24) into Eqs. (15) 4 (cid:0)G (cid:0)E− Jm z (cid:1)(cid:1)−1− J
0 4 f σ 2
and (22):
(28)
(cid:18) 1 V2 (cid:19)−1 Here, G is given by Eq. (18) and S is the f–spin quan-
G (E)= − FV=0(E) , (25) 0 f
kσ GV=0(E) (cid:126)2 kσ tumnumber. AsmentionedinSec. 2,wetreatthef–band
kσ
as a spin–1/2–system, i.e. S =1/2. Note, however, that
f
(cid:18) 1 V2 (cid:19)−1 Eq. (28) is not restricted to this case.
F (E)= − GV=0(E) . (26)
kσ FV=0(E) (cid:126)2 kσ Thenextstepisfindingtheself–energiesΣ(f),U(E)and
kσ kσ
Σ(f),sf(E) of the hybridization–free local moment system.
Of course, this represents a reformulation of the full kσ
Eq. (21) shows, that knowing the f–electron’s Green’s
many–body problem only, since expressions for the self–
function is sufficient for obtaining the sum of the wanted
energies or rather the hybridization-free Green’s functions
GV=0 and FV=0 are still missing. We will work them out self–energies. The latter are approximated by the self–
kσ kσ
energies of the fermionized Kondo–lattice model in the
in the next section.
zero–bandwidth limit. The resulting self–energy is k–
independent. This approximation is exact for the case of
4. Self-energies an empty conduction band and should remain reliable for
low conduction band filling.
The last sections’ preparatory work shifts the focus
Starting once again with a mean field (MF) treatment
from the original problem towards the set of (so far un-
ofthelocalmomentsystem,oneobtainsa(k–inde–pendent)
determined)self–energies, whichhavetobecalculatedap-
f–electron’sGreen’sfunctionwhichisequivalenttotheso-
proximately.
lution of the zero–bandwidth Hubbard model [18]:
To find an analytical solution, we make an effective
medium approach, meaning that the self–energies are (cid:32)
1−n
worked out within the two respective subsystems, assum- F(MF,V=0)(E)=(cid:126) f−σ +
σ E+µ−E + Jz m
ing V = 0. The main idea is that hybridization effects f 4 σ s
(29)
are already taken into account for the Green’s functions (cid:33)
n
tosomeextent(seeEqs. (25)and(26)). Physicallyspeak- + f−σ .
E+µ−E + Jz m −U
ing,conductionbandelectronsthenencounteraneffective f 4 σ s f
f–electron potential and vice versa.
As mentioned before, this result will be primarily impor-
For the hybridization–free conduction band system,
tant for later comparison with the effective medium ap-
which is described by the Hamiltonian parts (2) and (6),
proach. The theory of the latter, of course, also requires a
the associated Green’s function is given by Eq. (17). Us-
local moment Green’s function, on which we will focus in
ingthestaticmeanfield(MF)approximation,oneobtains
the following paragraphs.
the following expression for the s–electron’s self–energy:
Thesystemturnsouttobeexactlysolvableinthezero–
bandwidth limit [20]. This yields the following Green’s
Σ(ksσ),sf(E)≡Σ(σs),MF =−J2zσ(cid:10)Sfz(cid:11)=−J4zσmf , (27) function:
where Sfz stands for the z–projection of the f–system’s FV=0(E)≡FV=0(E)=(cid:88)4 αjσ . (30)
spin operator S . In this paper, the mean field result is kσ σ E−E
f j
j=1
introduced for comparison only.
A more reliable expression for Σ(s),sf(E) was given by Coulomb- and s–f–interaction are responsible for a split-
kσ
Nolting et. al [19], who found an approximate solution to ting into four sublevels, which are located at energy posi-
the self–energy by interpolating between several exactly tions E ,
j=1,...,4
known cases, such as the zero–bandwidth limit, the fer-
J
romagnetically saturated semiconductor and the second–
E =T −µ− ·S ,
orderperturbationtheory. Thederivedself–energyistrust- 1 0 2 c
worthy for arbitrary temperatures and coupling strengths J
E =T −µ+ ·(S +1) ,
J, as well as for low band occupations n . Likewise using 2 0 2 c
s (31)
J
E =T −µ+U − ·(S +1) ,
3 0 2 c
J
E =T −µ+U + ·S ,
4 0 2 c
4
each of them connected to a spectral weight α , where 5. Results and Discussion
jσ
j =1,...,4:
Referring to systems like EuX (X=O,S,Se,Te), a face-
1 (cid:16) z m (cid:17)
α = S +1+ σ s +∆ −(S +1)n , centered cubic (f.c.c.) lattice structure is considered. Cal-
1σ 2S +1 c 2 −σ c f−σ
c culationsaredoneforanintraatomicCoulombinteraction
1 (cid:16) z m (cid:17)
α2σ =2Sc+1 Sc− σ2 s −∆−σ−Scnf−σ , aexmpoenrgimtehnetalolcdaalitzaed(seeelecet.gro.n[s22o]f).URfe=ca1ll0tehVatatchceorcdoinndgutco-
1 tion band center of gravity is fixed at T =0 eV and that
α = (S n −∆ ) , 0
3σ 2Sc+1 c f−σ −σ the width of the conduction band is assumed as W = 1
1 eV.Keepalsoinmind, thatallcalculationsareperformed
α = (∆ +(S +1)n ) .
4σ 2S +1 −σ c f−σ for a total spin of 1/2 for both conduction band (S ) and
c c
(32) local moment system (S ).
f
We divide this Section into several parts, beginning
In the equations above, S = 1/2 stands for the total
c with metallic systems without hybridization influences
spin of the conduction band system. The average occu-
(part 5.1). Part 5.2 addresses the case of a semiconductor
pationnumbersn andconductionbandmagnetization
f−σ or insulator but still V = 0. Finally, the results of the
m werealreadydefinedinEqs. (8)and(10), whereasthe
s fulltheorywithfinitehybridizationarediscussedandalso
parameter ∆ is a higher correlation function:
σ compared with mean field calculations in part 5.3.
(cid:68) (cid:69)
∆ = Sσf† f +z (cid:104)Szn (cid:105) . (33)
σ c −σ σ σ c σ
5.1. Magnetic response of a metallic conduction band for
Here, Scσ = Scx +izσScy is the ladder operator for a con- V=0
duction electron’s spin S , and Sz its z–component. In
c c The first special case to be discussed is the metallic
the conventional Kondo–lattice model, this two–particle
Kondo–lattice limit [13, 23], where the hybridization is
correlation can, surprisingly, be expressed via the single
switched off (V = 0). The conduction band is taken as
f–electron’s Green’s function [21]:
half–filled.
1 (cid:88) (cid:90)∞ Some characteristic features of the half–filled Kondo–
∆ = f (E)(E−E ) Im F (E−µ)dE , lattice model can be understood by looking at the quasi-
σ JπN − f kσ
k −∞ particle density of states shown in Fig. 1. For simplicity,
(34) we assume to have a fixed f–electron’s magnetization m ,
f
where f (E) is the Fermi function. By combining Eq. which - instead of being calculated self–consistently - is
−
(30) with Eq. (34), and after tedious, but straightforward treated here as an external parameter. Its temperature
work,oneobtains∆ ,asafunctionalofn ,n (andthus dependence is given via a Brillouin function with a Curie
σ fσ sσ
m ) and ∆ itself. Instead of calculating n within the temperature of 300 K.
s σ s,fσ
hybridization–free subsystems, we chose to evaluate them The spin resolved quasiparticle densities of states
in the full system in order to allow for a better inclusion showninFig. 1splitintotwosubbandswhichareroughly
of hybridization effects. We therefore use the respective located at energies −JS /2 and J(S +1)/2. This is due
f f
Green’s functions (25) and (26), as well as the spectral to the interpolating self–energy ansatz Eq. (28) and is
theorem [18]: reminiscent of the splitting found in the zero–bandwidth
limit. A known shortcoming of the self–energy Eq. (28) is
∞
1 (cid:90) 1 (cid:88) that even for weak coupling strengths J the gap between
n =− f (E) Im G (E−µ)dE ,
sσ (cid:126) − Nπ kσ both subbands persists (see Fig. 1(a)). Nevertheless, this
−∞ k (35) doesnotseriouslyaffectourweak–J calculationssincethe
∞
1 (cid:90) 1 (cid:88) gap is negligibly narrow.
nfσ =− (cid:126) f−(E)Nπ Im Fσ(E−µ)dE . By comparing the results for different mf, one finds
−∞ k theexpectedproportionalitytotheconductionband’spo-
larization m . Note, that decreasing m goes along with
The k–summation in Eq. (35) can be replaced by an inte- s f
an enhancement of temperature. Moreover, the spectral
grationoverx,weightedbythefreeBlochdensityofstates
weight of the upper ↑–band decreases as the temperature
(cid:37) :
0
1 (cid:88) is reduced, vanishing in the case of T = 0 K. This effect
(cid:37) (x)= δ(x−(cid:15)(k)) . (36)
0 N issimilartothespecialcaseoftheferromagneticallysatu-
k ratedsemiconductor(forfurtherdetailsseeRefs. [24,19]).
In self–consistent, numerical calculations, the occupa-
Insuchsaturatedsystems(m =1),the↑–electroncannot
f
tionnumbersn (andhencethemagnetizations,seeEq.
s,fσ flip its spin, and thus the lower ↑–quasiparticle density of
(10)) can now be evaluated. A non–vanishing total mag-
states has the shape of the free (f.c.c.)–Bloch density of
netization,
states. Of course, this is only valid for the saturated elec-
m =m +m , (37)
tot s f tron sort.
indicates a ferromagnetic phase.
5
6 however, implies that n = 1 and that the conduction
f
4 band is empty. A ferromagnetic state cannot emerge in
2 σ=↑ this case.
S
DO 0 Ferromagnetism is possible for V = 0 and n = 1, in
Q -2 σ=↑ principle, if there is a non-vanishing energy overlap of
the lower f-electron subband with the (split) conduction
-4
bands. Otherwise, conduction band electrons are missing
-6
-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 tomediateaneffectivemagneticexchangebetweenthelo-
energy [eV] cal moments in the f–subsystem. As the total filling is
fixed at n=1, the total magnetization m +m ≤1. On
(a) J =0.1eV s f
the other hand, an overlap between s- and f–type bands
6 notonlyimpliesns >0butatthesametimenf <1. This
4 meansthattherearechargefluctuationswhichdestabilize
2 σ=↑ the formation of f–magnetic moments and therefore dis-
S
DO 0 favor ferromagnetic order. Concluding, we expect that
Q -2 σ=↑ ferromagnetic order is unlikely for V = 0 (and n = 1),
irrespective of the f–level position E .
-4 f
To address this question numerically, we have studied
-6
- 1 .00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 the model in an Ef–range where ferromagnetism could in
energy [eV] principle be possible according to the discussion above.
We have scanned the entire regime of coupling strength
(b) J =1eV
J. If at all, ferromagnetism was expected for strong J.
However,onlyparamagnetismhasbeenfound,irrespective
Figure 1: Spin–resolved quasiparticle densities of states (QDOS) of
ofthevalueofthecouplingstrengthJ. Fromthenumerical
theconductionelectronsintheKondo–latticemodelforahalf-filled
pointofview,thisresultcruciallydependsonthefactthat,
conduction band and for different coupling strengths J with fixed
f–magnetizations m =0.5 (solid black lines) and m =1 (dotted unlike in similar approaches (e.g. [25]), our local moment
f f
redlines). Thechemicalpotentialµismarkedbythedashedvertical system is treated fully self–consistently.
line. µ is only very slightly temperature–dependent, i.e. dependent
Weconcludethattheself–consistentapproachwellde-
onm .
f
scribes the essential spin–flip processes preventing a fer-
romagnetic order of local moment and conduction band
The quasiparticle density of states changes qualitati- electrons.
vely in the weak coupling regime, since both subbands
(almost) merge (see Fig. 1(a)). When increasing the cou- 5.3. General case (V >0)
plingstrengthJ,acorrelationgapoccurs,whichseparates
As the discussion above has shown, a hybridization–
theupperbandfromtheFermiedge(seeFig. 1(b)). Then,
free system cannot explain the ferromagnetic nature of
exclusivelythelowerbanddeterminestheelectronicprop-
magnetic semiconductors and insulators, since there are
erties, which results in a rather constant magnetization
no charge carriers within the conduction band (due to
m . Later we will see, that this saturation effect plays an
s the finite gap between the latter and the local moment
important role in our model.
energy positions). By switching on the hybridization–
Theseobservationshavebeenmadeforagivenf–mag–
term, hybridization–induced virtual transitions of local-
netization and illustrate the reaction to an effective exter-
ized f–electrons into the conduction band lead to a non–
nal magnetic field. Of course, also the back reaction of
vanishing amount of charge carriers that participate in
the s–magnetization on the local moment system must be
s–f–interaction processes. These are responsible for the
considered in addition to get a self–consistent picture.
alignment of the electron spins.
Thenumericalstudiespresentedbelowarebasedonan
5.2. Self–consistent treatment of the Kondo–lattice limit
implementation of the full theory, as detailed in Secs. 3
(V=0)
and4. Weagainfixthetotaloccupationnumberatn=1,
Let us now come back to the full theory for V = 0. but systematically vary the energy position of the f–level
Here, mf is no longer treated as an external parameter. Ef aswellasthecouplingstrengthJ andthehybridization
Both mf and ms are rather obtained from the subsystem V. For the Hubbard–U, we choose U =10 eV as above.
occupationnumberswhicharecalculatedself–consistently. A typical example for the resulting quasiparticle den-
To address the case of a semiconductor or insulator, the sities of states is shown in Fig. 2. For two different values
totaloccupationnumberisfixedatn=1byadjustingthe of J, referring to the weak- and the intermediate coupling
chemical potential accordingly. Let us discuss two param- regime, a ferromagnetic solution is easily stabilized. This
eter regimes: is achieved with a small but finite hybridization V = 0.3
If Ef (cid:28) µ (cid:28) Ef + U, there are well-defined local eV. Note, that with Ef = −1 eV there would not be any
magneticmomentsformedinthef-electronsystem. This,
6
f-electrons 500 1.0
4
s-electrons
400 0.8
V] 2 σ=↑ K]
OS [1/e 0 ature [ 300 PM 0.6
D er
Q -2 σ=↑ mp 200 0.4
e
T
-4 100 FM 0.2
-1.5 -1.0 -0.5 0.0 0.5 0.0
energy [eV] 0.0 0.2 V [eV] 0.4 0.6
(a) J =0.2eV
(a) mtot forfixeds–f–interactionstrengthJ =1eV
f-electrons
4 1.0
s-electrons
250
V] 2 σ=↑ 0.8
OS [1/e 0 ure [K]200 PM 0.6
D at150
Q -2 σ=↑ mper 0.4
e100
T
-4
0.2
50 FM
-1 .5 -1.0 -0.5 0.0 0.5
energy [eV] 0.0
0.0 0.5 1.0 1.5 2.0
(b) J =1eV J [eV]
(b) mtot forfixedhybridizationV =0.3eV
Figure2: Quasiparticledensitiesofstates(QDOS)inthefullsystem
for different coupling strengths J with fixed hybridization V = 0.3
eV.ThelocalmomentenergyliesatEf =−1eV.Coulombrepulsion: Figure3: Totalmagnetizationmtot=ms+mf,representedbythe
U =10eV.Notethatf–levelslocatedat∼E +U arenotshown. color scale, for various s–f–interaction strengths J and hybridiza-
f
The chemical potential µ is located at µ = −1.074 eV for J = 0.2 tions V. Ef is fixed at -1 eV and the total occupation number is
eV(upperfigure)andµ=−1.276eVforJ =1eV(lowerfigure). fixedatn=1. FMistheferro-,PMtheparamagneticphase,both
separatedbytheblackline,whichmarkstheCurietemperature.
energy overlap of s- and f-subsystem bands for V = 0.
resembles a hopping process. Moreover, the spin–resolved
Thefact,thatthes-andf–quasiparticledensitiesofstates
quasiparticle densities of states are repulsively shifted (as
have the same support is a mere consequence of the finite
V acts like a perturbation), see Fig. 2. The latter effect
hybridization.
increasesthespinasymmetryinthequasiparticledensities
Variation of the s–f–coupling strength J and the hy-
of states, and favors ferromagnetism. This explains point
bridization V has significant influences on the total mag-
(ii) of the above list.
netization, and thus on the Curie temperature, as can be
Regarding the J–dependence, let us emphasize once
seen in our results in Fig. 3. Summarizing, our main re-
more that conventional coupling mechanisms such as
sults are:
RKKY in the weak–J regime or double exchange in the
(i) Ferromagnetism occurs for finite V only. strong–J regime do not apply here. The point is that the
(ii) The Curie temperature increases with hybridization total occupation number n is fixed to 1. Therefore, any
strength V. increaseofthef–electronnumberimpliesadecreaseofthe
(iii) Around J = 0 eV the Curie temperature takes its conduction electron number. This is found to destabilize
highest values and the system is highly polarized in the magnetic order.
the low–temperature regime. For J =0 eV, ferromagnetism is found to be very sta-
ble with a high Curie temperature (see Fig. 3(b)). Fer-
(iv) For large J, the Curie temperature saturates.
romagnetic order originates in this case from the strongly
(v) Positive values of J show ferromagnetism, negative
correlated f–band where the Hubbard–U is much larger
ones commonly paramagnetism.
than the effective bandwidth resulting from the virtual
The limiting case V = 0 eV is a clearly paramagnetic hybridization processes introduced by the finite V.
situation,asdiscussedinSec. 5.2,seeFig. 3(a). ForV >0 A finite J brings up two contrary effects, a splitting
eV, virtual s- and f–electron transitions occur, causing a of ↑- and ↓–quasiparticle densities of states due to the
broadeningofthequasiparticledensitiesofstates,whichis linear–in–J (Ising-)term on the one hand and a reduc-
clear since the hybridization–term in the Hamiltonian (7) tionoforderedspinstructuresowingtospin–flipprocesses
7
given by terms with quadratic J on the other hand. From 250 J = -0.2 eV
our results we must conclude, that the latter dominates. J = 0.2 eV
200 J = 1 eV
As spin–flip processes close the gap between ↑- and ↓–
quasiparticle densities of states, the system’s total polar- K] 150
[
ization decreases. Nevertheless, the Curie temperature C
T
saturates for J →∞ and a finite but small magnetic mo- 100
ment is retained (see point (iv)).
50
Last but not least, the sign of J is quite important
when looking for ferromagnetic phases. Starting off with 0
the hybridization–induced shifts of the ↑- and ↓–quasi– -2.0 -1.5 -1.0 -0.5 0.0 0.5
Ef[eV]
particle densities of states, negative coupling strengths
produce an Ising term that splits the densities of states (a) FixedparametersV =0.3eVandn=1.
into the opposite direction, hence reducing the spin asym-
metry. As the sign of J is of importance only in the Ising 250 n = 0.5
term, the paramagnetic phase for J < 0 and the ferro- n = 0.7
magneticphaseforJ >0resultfromthisfirst–order–in–J 200 n = 1
term only, just like static mean field calculations would [K] 150
predict. C
T
Anotherwayofunderstandingourresultsisasfollows: 100
For a system with non–vanishing hybridization and a ma-
50
jority of f –electrons, a majority of occupied s –states
↑ ↑
evolve due to hybridization. Such a parallel alignment of 0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
most electrons is either energetically favorable (J >0 eV)
Ef[eV]
orunfavorable(J <0eV).Inthelattercase,bothelectron
sorts tend to order antiparallelly, which makes the occu- (b) FixedcouplingJ =0.2eVandhybridizationV =0.3eV.
pation of states with opposite spin configuration more at-
tractive. The less effective level–repulsion effect described Figure4: Curietemperatureasafunctionofthelocalmomentenergy
level position E for different parameters such as the s–f–coupling
above leads to a decrease of the spin asymmetry. f
J (Fig. 4(a))andtheoccupationnumbern,Fig. 4(b).
FerromagnetisminthissystemstronglydependsonE ,
f
i.e. on the energy distance between the local moment and
conduction band system. Closing (opening) the gap be- of the f–electron moments. Hence, TC increases, see Fig.
tween both subsystem’s densities of states increases (de- 4(b). Furtherdecreaseofnreducesthenumberofoccupied
creases)hybridizationinfluences. Hence, shiftingEf leads f↑–states, and will finally lower the Curie temperature.
toaneffectivehybridization,whicheventuallychangesthe When pushing the f–level into the conduction band
magnetization and the Curie temperature (see Fig. 4(a)). (secondcase), theferromagneticphasebreaksdownfaster
With increasing E the s- and f–subbands start to over- forsmalloccupationnumbersthanforlargerones. Thisis
f
lap. On the one hand, this produces charge carriers in due to the filling order, starting now with the conduction
the conduction band which promote ferromagnetic order band (which in this situation gives the lowest states to
as they mediate an effective magnetic coupling between be occupied). As soon as all s–electron states are filled,
the f–electrons’ spins. On the other hand, increasing E the remaining electrons will occupy the f–states given at
f
simply introduces charge fluctuations in the f–levels and energy Ef within the conduction band. The smaller n is
destroysthelocalf–moments. Thisleadstoarapidbreak- chosen,thefewerf–stateswillbeoccupied,hencereducing
down of the ferromagnetic phase as soon as the f–level is the corresponding magnetization mf.
pushed into the conduction band. A similar E depen- As a last result, we want to point out the differences
f
dence, though with different values of the Curie temper- betweenmeanfieldtheory(seealsoRefs. [26,27])andour
ature, has been found in previous works using the mean effective medium approach. As seen in Sec. 4, our ansatz
field ansatz (see particularly Refs. [26, 27]). goes beyond mean field as we allow for higher–order con-
As our model is not necessarily restricted to semicon- tributions of the s–f–electronic correlation. Fig. 5 shows
ductorsandinsulators(suchastheEuropiumchalcogenides), that both theories give the same results in case of J = 0
we additionally investigate the physics for different occu- eV. As the self–energy vanishes in both cases, this is a
pation numbers n. As is seen in Fig. 4(b), a spectacular trivial result (see Eqs. (27) and (28)). Strong deviations
influence on the Curie temperature is found. Again, one are found for J (cid:54)= 0 eV. All differences are due to the
needs to distinguish between the open gap situation (E quadratic–in–J term contributions. The saturation effect
f
belowtheconductionband)andtheclosedgapregime(E for large J, for instance, is not found in the mean field
f
inside the conduction band). In the first case, decreasing results. The Curie temperature rather increases with in-
nmeanstointroducemoreandmorechargecarriersinthe creasing J without any limit.
conductionbandsystemthatmediateaneffectivecoupling Even with the mean field treatment of the spin–spin
8
[28], which results from an interpolation between a large
1 2 0 0
amount of exactly solvable special cases, such as second
1 0 0 0 order perturbation theory, the zero–bandwidth limit and
themagneticpolaron. Asforthelocalmomentsystem,the
8 0 0
] stronglocalizationoff–electron’swavefunctionsallowsus
[K 6 0 0
C to use an analytically exact zero–bandwidth solution. By
T
4 0 0 connectingthesubsystemswithfinitehybridizationinour
full theory, we are able to calculate occupation numbers,
2 0 0
polarizations, correlations, etc., in a self–consistent way.
0 Note in this context, that in particular magnetic ordering
-2 -1 0 1 2 was investigated in a self–consistent manner.
J [e V ]
Our numerical calculations show that ferromagnetism
canbefoundwiththetheorypresentedinSecs. 3and4of
Figure 5: Curie temperature as a function of the coupling strength
J for mean field (dashed lines) and effective medium (solid lines) this paper. In addition, magnetic ordering (and thus the
approach. RedgraphsarecalculatedforconstanthybridizationV = Curie temperature) of systems with energy gap between
0.3eV,blacklinesforV =0.5eV.E isfixedat-1eV.
f f–level and conduction band is dominated by the chosen
hybridization strength. Surprisingly, the s–f–interaction
strength rather acts as a destabilizing parameter. This is
coupling,Eq. (6),andevenatJ =0,thereisanon-trivial
most likely due to our self–consistent treatment of the f–
correlationeffectthatisinducedbyU . Theferromagnetic
f
electron’s magnetic moments, which therefore differs from
orderinthecaseJ =0solelyoriginatesfromthisHubbard
conventional RKKY behavior. Contrary, fully closing the
interaction. In fact, due to the charge fluctuations intro-
energy gap between f– and s–band increases electronic
duced by V, the f–level develops a finite but small width
fluctuations and, hence, destroys the ferromagnetic order-
suchthatonecaneffectivelythinkofastronglycorrelated
ing.
Hubbardmodelwithasmallbandwidthwhichapparently
As our band model shows highly interesting physical
sustains a stable ferromagnetic phase.
behavior, the model could be used for further research on
the system’s transport properties, thermodynamic quan-
6. Summary tities, or antiferromagnetic ordering properties. Another
possible extension could bring different self–energies into
We have presented a band model for describing fer-
focus, for example a k–dependent one (see [24]), or a self–
romagnetic semiconductors and insulators. Referring to
energy which can be obtained via moment conserving de-
local moment systems like EuO, EuS, Gd, etc., the model
coupling approach treatment. Also, inclusion of realistic
consists of two subsystems, namely a band of conduction
DFT–results would improve the description of real sub-
(s-)electronsonthe one, andstronglylocalized f–electron
stances, as previous works have shown (see for example
levelsontheotherhand. Withrespecttotheabovenamed
[29, 30]).
classofsubstances,localmomentferromagnetismrequires
singly occupied f–levels, which can be achieved by use of
sufficiently large on–site Coulomb interaction. Although Acknowledgement
the conduction band is rather sparsely occupied, or even
The work is supported by the Deutsche Forschungsge-
empty in the extreme case of temperatures close to ab-
meinschaft (project A14 of the SFB 668).
solute zero, we expect it to be necessary for ferromag-
netism in local moment systems. We therefore include
a hybridization term which allows for (virtual) electron
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