Competition of the Coulomb and hopping based exchange interactions in granular magnets

0.47 MB·

by  O. G. Udalov

Competition of the Coulomb and hopping based exchange interactions in granular
magnets
O. G. Udalov1,2 and I. S. Beloborodov1
1Department of Physics and Astronomy, California State University Northridge, Northridge, CA 91330, USA
2Institute for Physics of Microstructures, Russian Academy of Science, Nizhny Novgorod, 603950, Russia
(Dated: February 2, 2017)
We study exchange coupling due to the interelectron Coulomb interaction between two ferro-
7
magnetic grains embedded into insulating matrix. This contribution to the exchange interaction
1
complements the contribution due to virtual electron hopping between the grains. We show that
0
theCoulomb and thehoppingbased exchangeinteractions are comparable. However, for most sys-
2
tem parameters these contributions have opposite signs and compete with each other. In contrast

b to the hopping based exchange interaction the Coulomb based exchange is inversely proportional
e to the dielectric constant of the insulating matrix ε. The total intergrain exchange interaction has
F a complicated dependence on the dielectric permittivity of the insulating matrix. Increasing ε one
1 can observe the ferromagnet-antiferromagnet (FM-AFM) and AFM-FM transitions. For certain
parameters no transition is possible, however even in this case the exchange interaction has large

] variations, changing its valueby threetimes with increasing the matrix dielectric constant.
l
l
a PACSnumbers: 75.50.Tt75.75.Lf75.30.Et75.75.-c
h
-
s
I. INTRODUCTION lating matrix barrier, but also on the dielectric proper-
e
m ties of the matrix. Such an effect appears due to charge
quantization and the Coulomb blockade effects in FM
. Granular metals posses complicated physics involving
t nanograins.
a size andchargequantizationeffects whichinterplaywith
m complicated morphology of these systems [1–10]. Many- The intergrain exchange coupling studied in the past
- body effects play crucial role in granular metals. Elec- was due to virtual electron hopping between the grains
d tronic andthermaltransportproperties ofgranularmet- andcanbe associatedwith the kinetic energyin the sys-
n alsarebroadlystudiedboththeoreticallyandexperimen- tem Hamiltonian. However, it is known that the many-
o tally. These properties are defined by conduction elec- body Coulombinteractionalsoleads to the magnetic ex-
c
[ trons in the systems [2]. The situation becomes more change interaction [29 and 30]. Recently, the Coulomb
complicated in granular magnets with magnetic metal- based exchange interaction was considered in MTJ [31].
1 licgrainsbeingembeddedintoinsulatingmatrix[11–14]. Itwasshownthatthiscontributiontothemagneticinter-
v The magnetic state of granular magnets is defined by action between magnetic leads separated by the insulat-
4
0 three main interactions: magnetic anisotropy of a sin- inglayeriscomparableandevenlargerthanthehopping
1 gle grain, magneto-dipole interaction between ferromag- based exchange coupling.
0 netic (FM) grains and the intergrain exchange interac- InthispaperweconsideracompetitionoftheCoulomb
0 tion. Magneticpropertiesofgranularmagnetswerestud- andthehoppingbasedexchangecouplinginthesystemof
2. ied in many papers. Numerous papers were devoted to two spherical magnetic grains embedded into insulating
0 magneticanisotropyandmagneto-dipoleinteraction[15– matrix. In contrast to the layered system the screening
7 22]. Much less is known about the exchange interaction of the Coulomb interaction in granular system is differ-
1 betweenmagneticgrains[23–27]. Theinfluenceofthein- entduetofinitegrainsizes. Thisleadstotheappearance
v: tergrain exchange coupling on the magnetic state of the of additional terms in the total exchange interaction be-
i whole granular magnet are currently understood, how- tween grains. Also, the hopping based exchange interac-
X
ever the microscopic picture of the intergrain exchange tion in granular and layered systems is different. Thus,
r interaction is still missing. Note that the intergrain ex- thecompetitionofhoppingandCoulombbasedexchange
a
change coupling is related to the conduction electrons. interaction in granular system results in essentially dif-
The theoryofsucha couplingextends the theoryofcon- ferent total coupling.
duction electrons in granular metals. InRef.[31]itwasshownthatthe Coulombbasedcou-
In most experimental studies the exchange coupling pling strongly depends on the insulator dielectric con-
betweenmagneticgrainswasexplainedusingSlonczewski stant. For granular system both the hopping and the
model [28], developed for magnetic tunnel junctions Coulomb based exchange depends on the matrix dielec-
(MTJ). Usually, the coupling between grains was esti- tric susceptibility.
matedusingthismodelbytakingintoaccountthegrains InthispaperwecalculatetheCoulombbasedexchange
surface area. Recently, it was shown that the intergrain interaction between FM nanograins and study the com-
coupling differs fromthe exchangecoupling inMTJ [27]. petition between two mechanisms of exchange interac-
In granular system the exchange coupling depends not tion.
only onthe distance betweenthe grainsandonthe insu- The paper is organized as follows. In Sec. II we in-
2
ergy Wˆ , the potential profiles of grains Uˆ and the ex-
k 1,2
(a) AFM ordering changeinteractionbetweenconductionelectronsandions
Hˆ [30] in each grain. Hˆ is the Coulomb interaction
1,2m C
between electrons.
We assume that the single particle potential energy
M1 a d M2 a is Uˆi = −UΠi, where Πi = 1 inside grain (i) and
Π = 0 outside grain (i). We consider only FM and
i
AFM collinear configurations of the grains magnetiza-
FM I FM tions M1,2. According to Vonsovskii s-d model the ions
influence the delocalized electrons through creation of
spin-dependent single particle potential of magnitude
(b) Hˆsp (r )= J σˆ M Π ; whereM takesonlytwo
1,2m i − sd z 1,2 1,2 1,2
^ ^ ^ ^ possible values 1.
U 1 +U2 +U1 m +U2m ±
Note that we choose the zero energy level at the top
0
EF <0 h B = -EF of the insulating barrier (see Fig. 1). This leads to the
e
al negative Fermi level, EF <0.
c d
s We introduce a single particle Hamiltonian for each
gy separategrain,Hˆg =Wˆ +Uˆ +Hˆ ,withtheeigen-
r 1,2 k 1,2 1,2m
ne functionsψs inthegrain(1)andφs inthegrain(2). The
E M I M i j
-U+Jsd subscript i stands for orbital state and the superscript s
denotes the spin state in a local spin coordinate system
spin up
-U
related to magnetization of corresponding grain. Due
-U-Jsd spin down to grains symmetry the wave functions are symmetric
x=0, y=0 2a 0 z ψarise(xǫ,sy,=z)ǫs==φǫsis(.x,y,−z). The energies of these states
1i 2i i
The creation and annihilation operators in grain (1)
are aˆs+ and aˆs, and in grain (2) are ˆbs+ and ˆbs. The
FIG. 1. (Color online) (a) Two FM metallic grains with ra- i i i i
totalnumberofelectronsisgivenbythe operatorsnˆ and
dius a and intergrain distance d embedded into insulating
matrixwithdielectricconstantε. M1,2 standsforgrainmag- mˆ ingrain(1)and(2),respectively. Thewholesystemis
neutral. The total charge of ions in each grain is en .
neticmoment. (b)Schematicpictureofpotentialenergypro- 0
−
files for electron with spin “up” (red line) and “down” (blue We introduce the zero-order many-particle wave func-
line)statesforAFMconfigurationofleadsmagneticmoments tions ΨAFM and ΨFM for AFM and FM configurations
0 0
M1,2. Red and blue lines are slightly shifted with respect to of leads magnetic moments M1,2. These wave functions
eachotherforbetterpresentation. Zeroenergycorrespondsto describe the non-interacting FM grains (d ). All
the top of energy barrier for electrons in the insulator. Sym- states ψs and φs with energies ǫs < E ar→e fi∞lled and
i j i F
bolsFMandIstandforFMmetalandinsulator,respectively. all states above E are empty (we consider the limit of
F
All other notations are defined in the text.
zerotemperature). Thewavefunctionsofcoupledgrains,
when d is finite, are denoted as ΨFM and ΨAFM for FM
troduce the model for granular system. In Secs. III and and AFM configurations, respectively.
IV we underline the main results for the hopping based We split the Coulomb interaction operator into two
exchange coupling in granular systems. In Sec. V we parts, HˆC = HˆdC + HˆiC. Here HˆdC describes direct
calculate the inter-electronCoulomb interaction and the Coulomb interaction of electrons in the grains. It has
intergrain exchange coupling. We discuss and compare the form [2 and 3]
the Coulomb and the hopping based exchange interac-
tion in Sec. VI. Finally, we discuss validity of our theory Hˆ =E (nˆ n )2+E (mˆ n )2+ e2 (nˆ n )(mˆ n ),
in Sec. VII. dC c − 0 c − 0 C − 0 − 0
m
(2)
where E = e2/(8πε ε a) is the grain charging energy
c 0 eff
II. THE MODEL in SI units with ε being the effective dielectric con-
eff
stantofthe surroundingmedia. In generalε candiffer
eff
WeconsidertwoidenticalFMgrainswithradiusa(see from the dielectric constant ε of the insulating matrix.
Fig.1). TheHamiltoniandescribingdelocalizedelectrons In granular magnets the effective dielectric constant de-
in the system can be written as follows pends on properties of insulating matrix and grains [2].
In inhomogeneous systems, such as layers of grains lo-
Hˆ =Hˆ +Hˆ , (1) catedontopofinsulatingsubstrate,thechargingenergy,
0 C
E ,isacomplicatedfunctiondependingonthegrainden-
c
wherethesingleparticleHamiltonianHˆ0 = i(Wˆk(ri)+ sity,dielectricpropertiesofthesubstrateandgeometrical
Uˆ (r )+Uˆ (r )+Hˆ (r )+Hˆ (r )) has the kinetic en- factors [32 and 33]. In Eq. (2), C is the mutual grains
1 i 2 i 1m i 2m i P m
3
capacitance. In the absence of spin-orbit interaction the spin and the
The second part of the Coulomb interaction describes spatial parts of wave functions are separated. The spin
the indirect spin-dependent Coulomb interaction - the partsare(10)T and(01)T forthespinupandspindown
exchange interaction [29] states, respectively. We introduce the following coordi-
nates: z is along the line connecting graincentres; z =0
Hˆ = Usaˆs+aˆsˆbs′+ˆbs′, (3) is the symmetry point between the grains; x and y are
iC − ij i i j j
i,j,s perpendicular to z, r⊥ = x2+y2. Grains surfaces are
X
closetoeachotheraroundpoint(x,y,z)=0. Ingeneral,
with p
thewavefunctionsarethesphericalwaveswithquantum
numbers (m,n,l). For d a and S = πa/κ πa2
Uisj = d3r1d3r2ψis∗(r1)φsj′(r1)UˆCψis(r2)φjs′∗(r2). (κ = 2m E /~2 is th≪e inverse chcaracterist0ic≪length
0 e F
ZZ (4) scale of e−lectron wave function decay inside the insula-
p
tor) we approximate the electron wave functions in the
Here s′ = s for FM and s′ = s for AFM configuration vicinity of grain surfaces with plane waves. We change
ofgrainmagneticmoments;an−dUˆ istheoperatorofthe quantumnumbers(m,n,l)with(k ,k ,k ). Inthevicin-
C x y z
Coulomb interaction between two electrons. In Eq. (3) ityofgrainscontactareaweusethefollowingexpressions
we keep only diagonal elements of the indirect Coulomb for wave functions
interactionwith repeating indexes. We do this assuming
that electron wave functions have random phases due to τs d r2
scattering on impurities. In this case only matrix ele- ψks(z,r⊥) k exp κks +z+ ⊥ eik⊥r⊥,
≈ √Ω − 2 2a
ments with repeating indices survive. Also we omit the (cid:18) (cid:18) (cid:19)(cid:19)
τs d r2
indirect Coulomb interaction between conduction elec- φsk(z,r⊥) k exp κks z+ ⊥ eik⊥r⊥.
trons in the same grain. On one hand this contribution ≈ √Ω − 2 − 2a
(cid:18) (cid:18) (cid:19)(cid:19)
does not produce any interaction between grains and on (8)
the other hand it leads to spin subband splitting which
is much smaller than the s-d interaction (and may be
This expression is valid in the insulator region out-
incRoerpceonrtaltyedthinetoexccohnasntgaentinJtsedr)a.ction between magnetic side the grains. Here τks = kz2+kizκks is the am-
grains was considered using the Hamiltonian Hˆ + plitude of the transmitted electron wave, k⊥ =
Hˆ [27]. However,later itwasshown[31] thatthe in0di- (kx,ky,0), r⊥ = (x,y,0), Ω = 4πa3/3 and κks =
redcCtCoulombinteractionmayalsoleadtomagneticcou- 2me(U −sJsd−~2kz2/(2me))/~2 is the inverse decay
pling between the FM contacts. In particular, this was length written in new notations. We neglect the surface
p
demonstrated for MTJ with infinite leads. The indirect interference effect and the penetration of electron wave
Coulombbasedinterlayerexchangeinteractionwasfound function beyond the grainin determining the normaliza-
to be comparable with hopping based exchange interac- tion factor.
tion. In the present paper we calculate the intergrain Below we will use the symbols i and j (instead of k)
exchange interaction based on the indirect Coulomb in- to describe a set of quantum numbers characterizingthe
teractionofelectrons,Hˆ . Wedenotethehoppingbased orbitalmotionofelectrons. Theoverlapofwavefunctions
iC
exchange interaction as Hex. It is given by the following ofelectronsi andj locatedindifferentgrainsexists only
h
expression between the grains in a small region in the vicinity of
r⊥ = 0. The in-plane area ((x,y)-plane) of the overlap
Hhex =hΨAFM|Hˆ0+HˆdC|ΨAFMi−hΨFM|Hˆ0+HˆdC|ΨFM(5i). irnetgrioodnuisceSdcija=boπve(λai⊥rje)a2,,wShe=reπλλi⊥2j ,=is th2ea/c(oκnita+ctκajr)e.aTfhoer
c ⊥
The contribution to the exchange coupling from the in- p
direct Coulomb interaction is given by electrons at the Fermi level (size λ⊥ = a/κ0).
Forelectronwavefunctionsinsidethegrainsweobtain
p
Hex = ΨAFM Hˆ ΨAFM ΨFM Hˆ ΨFM . (6)
iC h 0 | iC| 0 i−h 0 | iC| 0 i
For Coulomb based exchange interaction it is enough to eikz(cid:18)d2+z+r2⊥2a(cid:19)+ξse−ikz(cid:18)d2+z+r2⊥2a(cid:19)
average the operator over the ground state. The total ψks(z,r⊥) k eik⊥r⊥,
exchange interaction is defined as follows ≈ √Ω
Hex =Hhex+HieCx. (7) φsk(z,r⊥) eikz(cid:18)d2−z+r2⊥2a(cid:19)+ξkse−ikz(cid:18)d2−z+r2⊥2a(cid:19)eik⊥r⊥,
≈ √Ω
(9)
III. SINGLE GRAIN WAVE FUNCTIONS
Consider single spherical metallic grain with radius a. with ξks = kkzz+−iiκκkkss. Below we will use Eqs. (8) and (9) to
We will follow the approach and notations of Ref. [27]. calculate exchange interaction between the grains.
4
IV. HOPPING BASED EXCHANGE kmsax = 2me(U −sJ)/~2, (18)
INTERACTION
p
This mechanismwasconsideredfor grainsinRef. [27]. kFs = 2me(EF+U −sJ)/~2, (19)
We split the expression for the hopping based exchange
p
interaction into two parts
J˜ =2m J /~2. (20)
sd e sd
Hex =Hex Hex, (10)
h h0− hε We introduce the charging energy ǫ˜ = 2E e2/C ,
c c m
−
where whichcanbeestimatedas˜ǫ =e2/(8πaǫǫ )ford 1nm
c 0
≈
and a [1;10] nm.
πa kFs The∈matrixelementsTs ,Ps ,andVs inEqs.(11)and
Hex= dk((ks)2 k2)Vs 12 12 k
h0 (2π)2κ F − k− (12) are given by the following expressions
0 s Z0
X
− 8πa2κ0 s "Z0kF−dsk1Z0kdFsk2δ˜s(k1,k2)T1−2ss(P1−2ss)∗− Vks =−sJsd (|(τκisis|))2e−2κisd,
− Z0kFsdk1XZ0kFsdk2δs(k1,k2)T1s2s(P1s2)∗#. Tisjs′=−(sJsd+U)τ(i(sk∗iτ)js2′(+κ(isκ+jsκ′)js2′))e−κjs′d,
(11) Pisjs′ = τis(∗kτ2js′+(κ(κis+s)κ2)js′)e−κisd+ τis(∗kτ2js+′(κ(κis+s′)κ2js)′)e−κjs′d+
and j i i j
Hheεx= + 2τis∗τjs′e−(κis+(κκsjs′)d2sκins′h)((κis−κjs′)d2).
= a km−axdk ku−dpk ξ˜−(k1,k2)|T1−2−|2 + i − j (21)
−8π2κ0 (Z√2J˜sd 1Z0 2 ~2(k12−2km22e−2J˜sd) +ǫ˜c For semimetal with only one spin subband occupied
+ km+adxk1 ku+dpk2 ~2 (ξ˜k+2(k1,kk22+)|T21J+˜2+|)2+ǫ˜ − (tEheFo<ccuJpsdie−dUsp)iwnesusbubmanindE(sq=s.“(-1”1)). and (12) only over
Z0 Z0 2me 1 − 2 sd c
−Xs Z0kmsaZx0min(k1,kFds)k1dk2 ξs~(2k(1k2,m12−ke2k)22|)T+1s2−ǫ˜sc|2. V. COULOINMTBERBAACSETDIOENXCHANGE
(12)

Integral in Eq. (4) includes the operator of the
For simplicity we change all different squares Scij in the Coulomb interaction Uˆ . For homogeneous insulator it
integrals with characteristic contact area S0 = πa/κ0. has the form Uˆ = e2/C(4πε ε r r ), where ε is the
C 0 1 2
This change does not influence the resulting exchange | − |
mediumeffectivedielectricconstant. Inourcasethesys-
interaction a lot. We introduce the following functions
tem is inhomogeneous and the Coulomb interaction is
renormalized by screening effects due to metallic grains.
(k−s)2 k2, 2sJ˜ +k2 <k2,
δ˜s(k1,k2)= (kFs)2 −k2,1 2sJ˜ sd+k22>k2,1 (13) There are two regions contributing to Eq. (4): 1) The
(cid:26) F − 2 sd 2 1 regioninside the FM grains Ω1 (Ω2) where the Coulomb
interaction is effectively screened and is short-range [3
and 34]
(ks)2 k2, k <k ,
δs(k1,k2)=(cid:26)(kFFs)2−−k212, k12 >k21, (14) UˆL =Ω∆δ(r r )+2E +
C 2 1− 2 c
(22)
2E λ2 2E λ2
ξ˜s(k1,k2)= ((2(ks−J˜ssd)2+kk122−),kk22)>, kk1s<, kFs, (15) + caTFδ(|r1|−a)+ caTFδ(|r2|−a),
(cid:26) F − 2 1 F
where ∆ is the mean energy level spacing, Ω∆ =
6π2E /((k+)3 +(k−)3). In metals the Coulomb inter-
F F F
ξs(k1,k2)=(cid:26)((k(k12Fs−)2k−22)k,22k)1,k<1 >kFsk,Fs, (16) aTchtoiomnasis-Fsecrrmeeinleedngothn,tλhTeFle≈ngt(h sec2aklF3e/(o4fπtεh0e)EoFr)d−e1r o≈f
0.05 nm. The characteristic length scale of the electron
and notations density variationis κ−1 0.5 nmp. Thus, we can use the
0 ≈
localapproximationfordecayingelectronwavefunctions
kusp =min( k12+2sJ˜sd,kF−s). (17) since λTF ≪κ0−1.
q
5
The Coulomb based exchange coupling between infi- thechargesρ(1,2) areperiodicfunctionsinthe(x,y)plane
ij
nite magnetic leads was considered in Ref. [31], where it anddecayexponentiallyalongz direction. In the caseof
was shown that the Coulomb interaction inside the FM magneticgrainsthegeometryofthesystemismorecom-
leads also contributes to the total interlayer exchange plicated. We will use the following approximation: the
coupling. However,for infinite leads the lastthree terms region of interaction of electrons in states i and j is re-
in Eq. (22) disappear. In the present paper we take into stricted by the area Sij. The linear size of this area is
c
account these terms appearing due to finite grain sizes. muchlargerthantheFermilengthforlargeenoughgrains
2) The secondregioncontributing to Eq. (4) is the re- ( πa/κ >1/k ). Inthis casewecanmodeltheinterac-
F
gion between the grains where screening of the Coulomb tion regionas two leads with parallel surfaces neglecting
p
interaction is weak and the interaction is long-range. grainscurvature. Intheregionofinteractionwecalculate
However, due to metallic grains, the electric field of two the electric field created by charges ρ(1,2) as if we have
ij
interacting electrons is finite only inside this region. We the infinite parallel leads. The matrix element of the in-
denote the renormalized Coulomb interaction inside the teractionis givenby I =(ε ε/2)( d3rEijEij), where
insulating layer as UˆCI. Ω˜ is the volume restriijcted b0y the inΩ˜eIqualit1ies2z <d/2,
troInnsoinusridmeotdheelgerleacintrsodnos ninostidinetetrhaectinwsuitlhateoarchanodtheelre.c- r⊥I <aκij. In practice, we multiplyR the area-no|r|malized
matrixelementsinRef.[31]bythecontactareaSij. Fol-
The right hand side of Eq. (4) can be considered as c
lowingRef.[31]wederivethefollowingexpressionforthe
the Coulomb interaction between two effective charges,
Coulomb based exchange interaction
ρ(1) =eψs∗(r)φs′(r) and ρ(2) =eψs(r)φs′∗(r). Here s′ =
ij i j ij i j
s for FM and s′ = s for AFM ordering. One can see Iex =I˜˜ I˜+ I˜−, (25)
that ρ(1) =ρ(2)∗ =ρ−. ex− ex− ex
ij ij ij
We can write the matrix elements of the indirect where
Coulomb interaction as a sum of two terms
Uisj =Lsij +Iisj, I˜˜ex =−16eπ24aε0εZ0kF+Z0kF−dk1dk2|(κτ1+1+)+∗τκ2−2−|2e−d(κ1++κ2−)×
Ls = d3r d3r ρ (r )UˆLρ∗ (r ), k2max+k1max (k2max+k1max)/2
ij ZZΩ1+Ω2 1 2 ij 1 C ij 2 (23) ×Z0 qωI(q)dqZ0 kζ(k,q)dk.
Is = d3r d3r ρ (r )UˆIρ∗(r ), (26)
ij 1 2 ij 1 C ij 2
ZZΩI
wofhtehreeiΩn1s,u2la=tiΩngislatyheer.gTrahineivnodluexmseastnadndΩsIfiosrtshpeinvoilnudmexe I˜esx =−16πe42ε ε kFs k1dk1dk2|(κτs1s)+∗τκ2ss|2e−d(κ1s+κ2s)×
0 Z0 Z0 1 2
of electron wave function in grain (1). The spin state kmax+kmax (kmax+kmax)/2
2 1 2 1
of electron in grain (2) is the same (s) for FM and s qω (q)dq kζ(k,q)dk.
for AFM configuration. We can split the total Coulom−b ×Z0 I Z0
based exchange interaction into two contributions (27)
Hex =Lex+Iex. (24) The maximum value of perpendicular momenta are
C
kmax = (ks)2 k2 and kmax = (ks′)2 k2 , where
1 F − 1z 2 F − 2z
Below we consider these two contributions to the s′ = s in expression for kmax and kqmax in Eq. (26), and
Coulomb based exchange interaction separately. s =“+”,ps′ =“-” in Eq. (217). We al2so introduce the fol-
lowing functions
A. Contribution to the exchange interaction due to 0, (φ <φ ) or (φ <φ ),
the insulating region, Iex ζ(k,q)= φ φ2 , ot3herwise1, 3 (28)
1 3
(cid:26) −
To calculate the contribution to the exchange interac- where
tion due to the insulating region we will follow the ap-
0, k >kmax+q/2,
plartoeadchfoorfMRTefJ..[3I1n]twhhiseraepepxrcohaachngtehecoeulpelcitnrgicwfiaesldcaElciuj- φ1(k,q)= π+πsign(k21m1ax−q/2), k <|k1max−q/2|,
created by effective charges ρ(1,2) inside the insulati1n,g2  arccos k2+q2/4q−k(k1max)2 , otherwise.
ij
region was calculated by taking into account the screen- (cid:16) (cid:17) (29)

ing produced by the FM leads. The leads were treated
as ideal metal with zero screening length. The energy of
π, k <kmax q/2,
tthioins)fiIeiljd=(t(hεe0εp)a(rtΩcIodr3rreEspi1joEni2dji)nggivtoestthheemesuttiumaaltientoefrtahce- φ2(k,q)=(arccos k22+q2/−4q−k(k2max)2 , otherwise.
matrixelementofindirectCoulombinteraction. InMTJ (cid:16) (cid:17) (30)
R
6
φ3(k,q)=π−φ2(k,q). (31) the right grain. The charge ρij is non-zero only in the
small area Sij in the (x,y) plane and penetrates into
c
ThereducedmatrixelementωI(q)isgivenbytheexpres- the grain by the distance κ−1. Therefore the potential
sion 2Ecλ2TFδ(r a) interacts with the charge ρ only in
a | 2| − ij
the small area of the surface Sij 4πa2. Therefore
ωI(q)=ωIx(q)+ωIz(q), (32) c ≪
thispotentialgivesasmallcontributiontotheintergrain
where exchange interaction in comparison to the contribution
coming from the first term of Eq. (22), Ω∆δ(r r ).
ω = (α2+α2)sinh(dq) +α2sinh(d∆κ) +2α α d+ The directcalculations show that the small2param1 −eter2is
Iz 1 2 q 3 q 1 2 (aκ )−1(ak )−1(E /E ) 1. Forthisreasonweneglect
(cid:26) 0 F F c ≪
sinh((∆κ+q)d/2) sinh((∆κ q)d/2) the last two terms in Eq. (22).
+4α α +4α α − ,
1 3 ∆κ+q 2 3 ∆κ q
− (cid:27) The matrix element calculated using the second term
sinh(dq) sinh(d∆κ)
ω = (α˜2+α˜2) +α˜2 +2α˜ α˜ d+ in Eq. (22) is given by
Ix 1 2 q 3 q 1 2
(cid:26)
sinh((∆κ+q)d/2) sinh((∆κ q)d/2)
+4α˜ α˜ +4α˜ α˜ − ,
1 3 ∆κ+q 2 3 ∆κ q
− (3(cid:27)3) 2E d3r d3r ρ (r )ρ∗ (r )=2E τs 2 τs′ 2
c 1 2 ij 1 ij 2 c| i| | j | ×
washfeorlelo∆wκs =κ1s−κ2s′ andfunctionsαi andα˜i aredefined (κZisZ+Ωκ1+jsΩ′)22 e−κjs′dSjs′Sinc(qxλj⊥)Sinc(qyλj⊥) +
× Ω2 (cid:12) (ks)2+(κs′)2
−qd e(∆κ−q)d2 −qd e(∆κ−q)d2 (cid:12)(cid:12) i j
α1=e 2 σ2− q−∆κ , α˜1=−e 2 σ2− q−∆κ , e−κisdSisSinc(q(cid:12)(cid:12)(cid:12)xλi⊥)Sinc(qyλi⊥) 2.
α2=e−q2dσ1+ e−q(q++∆∆κκ)d/2, α˜2=e−q2dσ1− e−q(+q+∆∆κκ)d2 , (kjs′)2+(κis)2 (cid:12)(cid:12)(cid:12) (37)
(cid:12)
2∆κ 2q (cid:12)
α = , α˜ = − .
3 q2 ∆κ2 3 q2 ∆κ2
− −
(34)
Here Ss = πa/κs is the surface area and λi = Ss is
i i ⊥ i
The functions σ1,2 are defined as thelinearsize,andq=k1⊥ k2⊥isthemomentum. The
− p
contribution to the intergrain exchange coupling due to
σ0 eqd+σ0 this matrix element is
σ = 1(2) 2(1), (35)
1(2) eqd e−qd
−
with
e−qd/2 Lex = −e2 kFs kFsdk dk τs 2 τs 2δ(k ,k )
σ0 = e(q−∆κ)d/2 e−(q−∆κ)d/2 , Ec 64π3εε 1 2| 1| | 2| 1 2 ×
1 q ∆κ − 0 s Z0 Z0
σ20 = qe−+−q∆d/κ2 (cid:16)e−(q+∆κ)d/2−e(q+∆κ)d/2(cid:17). (36) ×(κ1s+κ2s)2(cid:26)X(k22+e−(κ2κ1s1s)d2)2κ1s + (k12+e−(κ2κ2s2s)d2)2κ2s+
(cid:16) (cid:17) e−(κ1s+κ2s)d
(k2+(κs)2)(k2+(κs)2)max(κs,κs) −
B. Contribution to the exchange interaction due to 2 1 1 2 1 2 (cid:27)
grains, Lex e2 kF+ kF−dk dk τ+ 2 τ− 2δ˜(k ,k )
− 32π3εε 1 2| 1 | | 2 | 1 2 ×
0Z0 Z0
is Idnefitnheisd riengiEonq.t(h2e2)o.peTrahteoropoefraCtoourlocmonbsisintsteorafcftoiounr (κ++κ−)2 e−2κ1+d + e−2κ2−d +
terms. Thelastthreetermscontributeonlyinthecaseof × 1 2 ((k22+(κ1+)2)2κ1+ k12+(κ2−)2)2κ2−
nanoscalegrains. Thesetermsvanishforinfinitemetallic e−(κ1++κ2−)d
leads. .
First, we consider the last two terms describing single (k22+(κ1+)2)(k12+(κ2−)2)max(κ1+,κ2−))
particle potential uniformly distributed over the grain (38)
surface. This potential is zero inside the grain. Con-
sider the interaction between an electron in some state
ψs located in the left grain and an electron in state
i
φs located in the right grain. Consider the interior of ThefirstterminEq.(22)givesthefollowingcontribution
j
7
to the intergrain exchange interaction Note that we do not consider the intergrain magneto-
dipole(MD)interaction[17–22,and36],whichcompetes
Lex = −3a(U +EF) γ(s,s′) withtheexchangeinteractionandleadstotheformation
loc 26π((k+)3+(k−)3)) × ofsuperspinglassstate. TheinfluenceofMDinteraction
F F s,s′
X onthemagneticstateofGFMwasdiscussedinRefs.[17–
ks ks
F Fdk dk ((ks′)2 k2)((ks)2 k2) 20, and 36].
× 1 2 F − 2 F − 1 ×
Z0 Z0
e−2dκ1s|τ1s|2 1+|r2s′|2 +Re (r2s′)∗ + B. Comparison with layered systems
×( κ1s 2κ1s κ1s+ik2!!
|τ2s′|2κe2s−′2dκ2s′ (cid:18)1+2κ|r2s1′s|2 +Re(cid:18)κ2s(′r+1s)∗ik1(cid:19)(cid:19)), cMoTuBpJoltiihnn,gttwhheeerpehaocspotp.nisnTidgheearreneddafrtoherealatCyloeeuraelsodtmtsbhtrrubeecatseuesdrseesenxtsciuahclahndgiafes-
(39)
ferences between granular and layered systems.
The first difference is related to the morphology of
we introduce the function
granular system. Due to spherical grain shape the effec-
′ 1, s=s′, tive area of interaction is small and it linearly depends
γ(s,s)= 1, s=s′. (40) on the grain size, a. Therefore the intergrain exchange
(cid:26)− 6 interaction in granular systems grows linearly with a in
contrast to the MTJ, where interaction grows as a2.
The second difference is the essential influence of the
C. Total exchange interaction
Coulomb blockade effect on the hopping based exchange
coupling. In MTJ the Coulomb blockade is absent while
The total intergrain exchange interaction is given by
inGFMtheCoulombinteractionsuppressestheFMcon-
the following expression
tribution to the hopping based magnetic intergrain cou-
pling.
Hex =Hex+Lex +Hex+Iex+Lex, (41)
h0 loc hε Ec The third difference appears due to finite grain sizes.
The Coulomb based exchange interaction has an addi-
where term Hex is given by Eq. (11), Lex by Eq. (39),
h0 loc tional contribution, Lex, appearing due to the second
HexbyEq.(46),IexbyEqs.(25-27)andLex byEq.(38). Ec
hε Ec term in Eq. (22). This contribution does not depend on
the grainsize a. Onone hand the interactionareagrows
linearly with a, and on the other hand this term is pro-
VI. DISCUSSION OF RESULTS
portional to the charging energy E 1/a.
c
∼
Thus,thetotalexchangeinteractionbetweenmagnetic
There are several contributions to the intergrain ex- grains can not be extracted from the known result of in-
changeinteractionin Eq.(41). These contributions have terlayerexchangecoupling in MTJ by simple multiplica-
different physical nature and different dependencies on tion of the later by the grain or effective contact area.
system parameters. In this section we will discuss these
contributions and compare the intergrain exchange cou-
pling with the interlayer exchange coupling in MTJ.
C. Comparison of different contributions to the
Coulomb based exchange coupling in granular
A. Granular magnets systems
First, we discuss the influence of intergrain exchange The Coulomb based intergrain exchange interaction
interactiononpropertiesofgranularmagnetswithmany has several contributions. The first contribution, Iex, is
grains forming an ensemble of interacting nanomagnets. due to the region between the grains. In this region the
Theexchangeinteractionbetweenthegrainsleadstothe Coulomb interaction can be considered as a long-range
formation of long-rangemagnetic order appearing below interaction. The electric field of a point charge pene-
a certain temperature [3, 23–25], which is called the or- tratesoverthewholevolumeoftheinsulatorbetweenthe
deringtemperatureT . ForIsingmodel[24and35]the grains. Thisfieldisreducedbythedielectricbetweenthe
ord
orderingtemperature in granularmagnets with FM cou- grains. Thus, the electron-electron interaction between
pling is related to the intergrain exchange interaction as the grains depends on the dielectric constant of the in-
T = z Hex, where z = 6 is the coordination number sulating matrix, ε. The secondcontributionappears due
ord n n
for three dimensional cubic lattice. Below we will plot to the Coulomb interaction between electrons inside the
the exchange interaction multiplied by the coordination grains, Lex. It consists of two terms: 1) the short-range
number,z =6,toshowthetemperaturewherecoupling term in Eq. (22), Lex, and 2) the size effect term, Lex.
overcomesntemperature fluctuations. Terms Iex and Lex lolicnearly grow with the grain sizeEac.
loc
8
d = 1 nm U = 5 eV a = 1 nm d = 1 nm U = 5 eV a = 1 nm
K) (K)
(10 (a) Js d = 4.5 eV (b) J s d = 3.8 eV 4 (a) h B = 0.5 eV 120 (b) h B = 0.2 eV
ex ex
1 zn Lloc zn Lloc 2 zn Iex 80 zn
40
0.1 ex
I
0.01 Iex Iex 0 LeExc 0 LeExc
-40
ex
1E-3 LeExc LeExc -2 Lloc -80 Lelxoc
1E-4
1.4 1.2 1.0 0.8 0.6 0.4 0.2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -4 -120
1 2 3 4 5 1 2 3 4 5
h B (eV) h B (eV) J s d (eV) Js d (eV)
FIG. 2. (Color online) The intergrain exchange interaction FIG. 3. (Color online) The intergrain exchange interaction
(multiplied by the coordination number) as a function of in- as a function of spin subband splitting, Jsd, for U = 5 eV,
sulating barrier height hB for U = 5 eV, ε = 4.5, d = 1 nm, ε = 4.5, d = 1 nm, a = 5 nm, and (a) hB = 0.5 eV, (b)
a=1 nm and (a) Jsd =4.5 eV, (b) Jsd =3.8 eV. Black lines hB =0.2 eV. Black lines show Leloxc (Eq. (39)), blue lines are
rsehdowlin|Leseloxca|re(Efoqr.(|I3e9x)|),(Eblqu.e(2li5n)e)s.aTrehefoyr-|aLxeEixsc|h(aEsql.og(3a8ri)t)hamnidc for LeExc (Eq.(38)) and red lines are for Iex (Eq. (25)).
scale. Dashed parts show the region where functions Leloxc,
Lex and Iex are negative.
Ec to the Coulomb based exchange interaction Lex, Lex,
and Iex on the spin subband splitting of electlorcons Einc-
side the grains, J , for a = 5 nm grains. In this case
TThheerceofonrteribthuetioinnflLueEexnccdeooesf nthoitsdteepremndinocnretahseesgrwaiitnhsidzee-. tphaeracmonetterirbsu.tTiohneLcsodeEnxctrisibnuetigolnigidbulee tino gthraeinwshLoleex riasnngeegao-f
creasingthegrainsizea. However,ourcalculationsshow loc
tive(AFM)forsmallsplittingandpositive(FM)forlarge
thatevenforverysmallgrainswitha 1nmthe contri-
butionLeExc ismuchsmallerthantwoot≈hercontributions. strpilbituttiniogn(cwohmeninognflryomonethsepiinnssuulbabtianngdriesgfiiollne,dI).exTchheancognes-
Figure 2 shows the behavior of these contributions to itssigntwice. ForsmallJ thecouplingispositive(FM),
sd
the Coulombbasedexchangeinteractionasa functionof for intermediate J the contribution is negative (AFM)
barrier height, hB = −2meEF/~2 (which is the differ- and for large splittsidng Iex >0 (FM).
encebetweentheenergiesoftheinsulatorbarrierandthe
p For large spin subband splitting (when only one sub-
Fermilevel). Thecurvesareshownforverysmallgrains,
band is filled) and for large barrier h the contribu-
with grains diameter 2a = 2 nm. Even in this case the B
tionIex exceedsthe contributioncomingfromthe grains
contribution Lex exceeds two other contributions only
when Lex or IeExcchange its sign. However,in this region (Fig. 3(a)). For small barrier the situation is the oppo-
loc site. For small splitting and for the case when both spin
the intergrain coupling due to the Coulomb interaction subbands are filled (J < E +U) the contribution due
is very small 10−2K. Thus, with a good accuracy we sd F
can neglect th∼e contribution Lex in most cases. to grains exceeds the contribution due to the insulating
Ec region(Iex < Lex ). InthisregionLex isofAFMtype
Contributions Lex and Iex are comparable. Figure 2 | | | loc| loc
loc and thus the whole Coulomb based coupling is of AFM
showshowthesetwocontributionschangetheirsignwith
type.
changingthe barrierheight, h . For largebarrierthe in-
B Note that for small barrier height the Coulomb based
teractionisweakandpositive(FMtype),whileforsmall
coupling Lex canbe ratherlargereaching100K.Thus,
barrier the interaction is negative (AFM type). One can | loc|
the intergrain Coulomb based exchange coupling can be
see that for large barrier the contribution due to the in-
observed in experiment.
tergrainregion,Iex,exceedscontributionfromthegrains,
Lex. For small barrier the situation is the opposite,
loc
Lex >Iex.
loc
Notethatthecontributionduetointergrainregionde- D. Coulomb vs hopping based exchange
pends on the dielectric constant of the insulator, Iex interactions
ε−1, while Lex does not depend on ε. Thus, changin∼g
loc
the matrix dielectric constant, ε one can change the ra- Figure 4 compares the hopping Hex and the Coulomb
h
tio of Lex and Iex. Figure 2 shows the case for ε = 4.5, Hex basedexchangeinteractionsasafunctionofthebar-
loc C
corresponding to Si insulator. rier height h for the following parameters: U = 5 eV,
B
Figure 3 shows the dependence of three contributions d = 1 nm, a = 5 nm, ε = 4.5 and (a) J = 5.0 eV,
sd
9
d = 1 nm U = 5 eV a = 5 nm d = 1 nm U = 5 eV a = 1 nm
K) (K)
1(00 (a) Js d = 5.0 eV (b) J s d = 3.8 eV 6 (a) h B = 0.5 eV 120 (b) h B = 0.2 eV
zn HCex zn HCex 4 zn Iex 80 zn
10
ex 40 ex
ex 2 Hh Hh
Hh
1 0
0
-40
0.1 Hhex -2 HCex HCex
-4 -80
0.01
1.4 1.2 1.0 0.8 0.6 0.4 0.2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -6 -120
1 2 3 4 5 1 2 3 4 5
h B (eV) h B (eV) J s d (eV) Js d (eV)
FIG. 4. (Color online) The intergrain exchange interaction FIG. 5. (Color online) The intergrain exchange interaction
(multiplied by the coordination number) as a function of in- as a function of spin subband splitting, Jsd, for U = 5 eV,
sulating barrier height hB for U = 5 eV, ε = 4.5, d = 1 nm, ε = 4.5, d = 1 nm, a = 5 nm, and (a) hB = 0.5 eV, (b)
a = 5 nm and (a) Jsd = 5 eV, (b) Jsd = 3.8 eV. Black lines hB =0.2eV.BlacklinesshowthehoppingbasedcouplingHhex
show the hopping based coupling |Hhex| (Eq. (10)) and red (Eq. (10)) and red lines are for the Coulomb based coupling
lines are for the Coulomb based coupling |HCex| (Eq. (24)). HCex (Eq. (24)).
The y-axis has logarithmic scale. Dashed parts show the re-
gion where functions Hhex and HCex are negative.
Hex on h and J are quite similar but have the oppo-
C B sd
sitesign. Thereasonforsuchasimilarityisrelatedtothe
(b) Jsd = 3.8 eV. One can see that the Coulomb and fact that both Hhex and HCex are defined by the density
the hopping based exchange couplings are comparable. of states in the vicinity of the Fermi surface. Consider
For large spin subband splitting, Fig. 4(a), the Coulomb the first term in Eqs. (11) and (39). The first integral
based coupling exceeds the hopping based coupling. For describes one of the hopping based contributions. The
weak splitting (Jsd <EF+U) both contributions change second integraldescribes one of the Coulomb based con-
their sign. This happens almost for the same barrier tributions. These two integrals are the most simple to
height. Contributions Hex and Hex have the opposite analyse. Due to the presence of the exponential factor,
h C
sign for almost all parameters. For large spin subband e−2κd only electrons in the vicinity of the Fermi surface
splitting Hhex is negative, Hhex < 0 (AFM) for any hB contribute to the integrals. We assume that the matrix
while the Coulomb based coupling is positive (FM). For elements do not depend on the electron energy (besides
small splitting (Jsd < EF+U) the Coulomb based inter- the exponential factor). In this case we can estimate
action Hex is positive for large barrier, and negative for
C
smFailglubraerri5er,shwohwilse Hthheex shhoopwpsintgheboapspeodsitHeebxehaanvdior.the Hhex0∼VF−N−−VF+N+−... (42)
h
CoulombbasedHex contributions to the totalintergrain and
C
ebxacnhdanspgelititnintegraJcstdiofnoratshae ffuonllcotwioinngopfairnatmerentaelrss:piUn s=ub5- Lex γ(s,s′) kFs kFsd′k dk ((ks′)2 k2)((ks)2 k2)
eV, d = 1 nm, a = 5 nm, ε = 4.5 and (a) hB = 0.5 loc ∼s,s′ Z0 Z0 1 2 F − 2 F − 1 ×
eV, (b) h = 0.2 eV. One can see that both contribu- X
B ′
tions are comparable and have the opposite sign. For ×(Lse−2dκ1s+Ls′e−2dκ2s )=(N−0−N+0)(L+N+−L−N−),
smallsplittingthehoppingbasedcontributionispositive ks
F
(FM),whiletheCoulombbasedcontributionisnegative, N0 = dk((ks)2 k2),
HCex <0. Forlargesplittingthesituationistheopposite. s Z0 F −
ks
N = Fdk((ks)2 k2)e−2dκs,
s F −
Z0
(43)
1. A toy model
±
whereV andL aretheparametersindependentofinte-
F s
ThemainfeatureofthehopingbasedandtheCoulomb gration variables. The key element of both the formulas
based contributions is the sign change as a function of is the integral of the form ((k )2 k2)e−2κddk. This
F
−
the barrierheighth andthe spinsubbandsplitting J . integral defines the number of electrons participating in
B sd
Moreover, one can see that the dependencies Hex and the exchange interaction. ERquation (42) has only single
h
10
integralsbecausethistermisthefirstorderperturbation
theory correction to the system energy and it is propor- ex d = 1 nm U = 5 eV a = 1 nm
znH (K)
tional to the number of electrons in the system. Equa-
100
tion(43)hasdoubleintegralssinceitdescribesthemany (a) (b)
body interaction and it is proportionalto the number of 40 h B = 0.2 eV
10
electrons squared. The different spin subbands give con-
tributions to the exchange interaction of opposite sign. 20
1 J s d = 5.0 eV
Forsemi-metals(onlyonespinsubbandisfilled, J >
sd h B = 0.5 eV
(U +EF)) only the integrals overmajority spin subband 0.1 0
are survived. Therefore, the majority spin subband de-
h B = 0.3 eV
fines the signofthe exchangeinteraction. For smallspin 0.01
subband splitting, J E (and κ k ) we have J s d = 4.5 eV -20
sd F 0 F
≪ ≪
1E-3
κ3
Z ((kFs)2−k2)e−2κsddk ∼ dk0Fse−2κ0d. (44) 1E-4 1.4 1.2 1.0J s 0d . 8 = 03..68 e0V.4 0.2 -40 1 2 3 4 5
Thisresultmeansthatthespinsubbandwithhigherden- hB (eV) J s d (eV)
sity of states at the Fermi surface (higher k ) gives the
F
smaller contribution to the exchange interaction mean- FIG. 6. (Color online) Total intergrain exchange interaction
ing that at small J the minority spin subband defines Hex (Eq. (7)) as a function of (a) the barrier height hB, and
the sign of the excshdange interaction. This causes the (b) spin subband splitting, Jsd, for U =5 eV, ε=4.5, d=1
nm, a = 5 nm. In plot (a) the y-axis has logarithmic scale.
sign change of the exchange coupling at a certain Jsd. DashedpartsshowtheregionwherefunctionHexisnegative.
Toestimate the transitionpointweestimate the integral
((ks)2 k2)e−2κsddk at small Fermi momentum k+
κ . TFhe−estimate in Eq. (44) does not work in this lFim≪it
0 Note that both the Coulomb and the hopping based
R(aknFd→((0k)F−. )W2−ehka2v)ee−R2(κ(dkdF+k)2∼−(kκ203))e/−(2dκkdF−d)ke−∼2κ(k0dF+.)3Teh−e2κex0d- cthoentirnibsuultaiotinnsgdmepaetnridx.onThteheCdoiuelloemctbriccopnetrrmibiutttiivonitycaonf
changeinteractionchangesitssingwhentheintegralsfor be written as
R
both spin subbands are equal. This point is defined by
the condition κ3 dk−(k+)3. Usually, κ2 E and Iex
therefore, the tra0n≈sitionFapFpears close to th0e p≪ointFk+ = HCex =Leloxc+ 1ε , (45)
F
0, i.e. close to the case of semimetal (J (U +E )).
sd ≈ F whereIex istheCoulombbasedexchangecouplinginside
This is in agreement with our calculations. The condi- 1
the insulator with ε = 1. Note that Iex can be either
tion also shows that the sign change appears with vary- 1
positiveornegativedependingonthesystemparameters.
ing the barrierheighh ,whichis alsoin agreementwith
B The dependence of the hopping contribution Hex on the
our calculations. This toy model explains the behavior h
dielectric constant is more complicated (see Ref. [27]).
of the exchange interaction and the reason for similarity
Approximately it can be written as
between the Coulomb and the hopping based exchange
contributions.
d√2m˜ǫ γ~√h
Hex =Hex+Hex 1 carctan B ,
h h0 h1 −s γ~√hB sd√2mǫ˜c
E. Total exchange interaction
  (46)
where γ 3.43 and Hex > 0. The dielectric permittiv-
In granular systems the Coulomb and the hopping ≈ h1
ity in this equation enters through the effective charging
based exchange interactions compete with each other.
energy,ǫ˜ 1/ε,forsimplicityweomitthedifferencebe-
These two contributions have the opposite sign for al- c ∼
tween ε and ε. The second term in Eq. (46) increases
most all parameters. eff
with increasing ε. This is in contrast to the Coulomb
Figure 6 shows the total intergrain exchange interac-
based coupling. Also, we note that ǫ˜ depends on the
tion, Hex as a function of (a) the barrier height h , and c
B grain size, a. Decreasing the grain size leads to the en-
(b)thespinsubbandsplitting,J ,forU =5eV,ε=4.5,
sd forcement of the Coulomb blockade effect making Hex
d = 1 nm, a = 5 nm. The sign and the magnitude of h
more sensitive to variation of ε.
the total exchange interaction depends on the value of
spin subband splitting, J and the barrier height, h . Using Eqs. (45) and (46) we can write
sd B
For small splitting J the coupling is AFM while for
sd Hex =Hex+
large splitting it is FM. Depending on J the coupling 0
sd
changes its sign one or three times. Due to the competi- Iex d√2mǫ˜ γ~√h
tion between the Coulomb and the hopping mechanisms + 1 +Hex 1 carctan B .
themagnitudeofthetotalexchangeinteractionissmaller ε h1 −s γ~√hB sd√2mǫ˜c
than the magnitude of the Coulomb based contribution.   (47)