Spatial trends of non-collinear exchange coupling mediated by itinerant carriers with different Fermi surfaces

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by  Hsin-Hua Lai

Spatial trends of non-collinear exchange coupling mediated by itinerant carriers with
different Fermi surfaces
Hsin-Hua Lai
Physics Department, California Institute of Technology, Pasadena, CA 91125, U.S.A.
Wen-Min Huang
Department of Physics, National Tsing Hua University, 300 Hsinchu, Taiwan
Hsiu-Hau Lin
Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan and
9 Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
0
(Dated: January 28, 2009)
0
2 Westudytheexchangecouplingmediatedbyitinerantcarrierswithspin-orbitinteractionbyboth
analytic and numeric approaches. The mediated exchange coupling is non-collinear and its spatial
n
trendsdependontheFermisurfacetopologyoftheitinerantcarriers. TakingRashbainteractionas
a
J an example, the exchange coupling is similar to the conventional Ruderman-Kittel-Kasuya-Yosida
type in weak coupling. On the other hand, in the strong coupling, the spiral interaction domi-
8
nates. Inaddition,inclusionoffinitespinrelaxationalwaysmakesthenon-collinearspiralexchange
2
interaction dominant. Potential applications of our findings are explained and discussed.

]
l PACSnumbers: 75.50.Pp,75.30.Et,72.25.Rb,75.70.Cn
l
a
h
- I. INTRODUCTION anism leads to exponential decay14. But, in strong
s
coupling (or equivalently, the low-density regime), the
e
m The central theme of spintronics is to manipulate DP relaxation mechanism changes into a peculiar power
. the extra spin degrees of freedom in condensed matter law15. Furthermore, in strong coupling, the cancella-
at systems1,2,3,4,5,6,7. Oneoftheproposalsistomanipulate tion of the spin-Hall conductivity for spin-conserving
m the spin polarization of electrons by the spin-orbit (SO) momentum scattering16,17 is not related to the vanish-
- interaction2,8 such as the Rashba interaction from the ing of the vertex function, as happened in the weak
d structural inversion asymmetry at a surface or interface regime. Instead, the cancellation is between the on-
n in semiconductors9. To make these proposals feasible in Fermi surface and off-Fermi surface contributions18. An-
o spintronics,itisimportanttoachieveaflexibletuningof otherexampleisthedisorder-inducedlocalizationdueto
[c the interaction strength. So far, many surprising works the Rashba coupling19. For a two-dimensional electron
have demonstrated how to enhance the strength of the gas with strong Rashba coupling, it behaves effectively
1 Rashba interaction either in a semiconductor or in some as an one-dimensional system20, which results in infi-
v metal spin-splitting surface states. For instance, by dop- nite bound states even for a short-range potential19,21.
7
ingBiintoGaAs,itwasshownthatthespin-splittinggap The topological change of the FS also plays an inter-
7
5 is significantly enhanced in the alloy GaAs1−xBix10,11, esting role in superconductivity22. It was shown that
4 which opens up an alternative approach to enhance the the critical temperature of the superconductor can be
. Rashbacouplinginsemiconductors. Furthermore, itwas tunedbytheRashbacoupling,suggestingmaterialswith
1
0 shown recently that the surface states of Bi/Ag show stronger Rashba coupling are good candidates for en-
9 a giant Rashba-type spin splitting due to the surface- hanced superconductivity22. Finally, in our previous
0 potentialbarrier12. Inaddition,itwasalsodemonstrated work23, we demonstrated that the exchange coupling
: that, by doping Pb into Bi/Ag, the Fermi energy of the mediated by itinerant carriers with Rashba coupling is
v
i surface states is tunable13. These exciting discoveries strongly tied up with the FS topology. We showed nu-
X open up the possibilities to manipulate the Rashba cou- merically that the non-collinear exchange coupling re-
r pling over a wide range of interaction strengths. There- vealstwospatialtrendsdependingontheFStopologies–
a fore, it is important and exciting to explore the related one is the RKKY-like oscillatory behavior and the other
physicspropertiesinbothweakandstrongregimeswhere is the spiral.
the Fermi surface (FS) topologies are different. Here we extend our previous work23 by a complemen-
NotethattheRashbainteractioninatwo-dimensional tary analytic approach within the path-integral formal-
electron gas gives rise to two different FS topologies ism. This method is particularly powerful in describing
that may play a significant role in dictating the re- the asymptotic behavior of the mediated exchange cou-
sponse functions. The classical Dyakonov-Perel (DP) plingandprovidesacomprehensiveunderstandingofthe
mechanism of spin relaxation14 due to Rashba cou- generaltrends. IntheweakRashbaregime,theexchange
pling shows different signatures when the FS topology coupling consists of the RKKY oscillations at shorter
changes. In weak coupling, the DP relaxation mech- length scale and a spiral background25. On the other
2
tweentheferromagneticlayersisdescribedbyaneffective
Heisenberg Hamiltonian, H = (cid:80) J SiSj. Within
eff ij ij L R
the linear response theory31,32, the mediated exchange
couplingJ isproportionaltothestaticspinsusceptibil-
ij
ity tensor,23,24,29,30
 
g +g cos2θ g sin2θ g cosθ
0 2 2 1
χij(r,θ)= g2sin2θ g0−g2cos2θ g1sinθ . (2)
−g cosθ −g sinθ h
1 1 0
FIG.1: Noncollinearexchangecouplingmediatedbyitinerant
carriers with Rashba interaction and finite spin relaxation. It is rather remarkable that the symmetry arguments
The hard magnet on the left is pined while the direction of
maketheangulardependenceexplicitandreducethenu-
the soft magnet on the right is determined by the effective
merical task down to evaluation of four real scalar func-
exchange coupling.
tions,g (r),g (r),g (r)andh (r). Afterintegratingout
0 1 2 0
the angular parts, the first three functions can be casted
into the following integral forms,
hinagndth,ethsteroRnKgKcoYupinlitnegrarcetgiiomnei.sIstuipsprraetshseerdnwicheetnhaetnttehre- g (r)=(cid:90) d2k1(cid:90) d2k2(cid:88) Iλ1λ2(cid:20)f((cid:15)k1λ1)−f((cid:15)k2λ2)(cid:21), (3)
a 2π 2π a (cid:15) −(cid:15) −iη
analytic results agree with previous numerics. We also λ1λ2 k2λ2 k1λ1
use a phenomenological approach to investigate the ef-
where the dispersion for the particle with momentum k
fect of the finite spin relaxation in the itinerant carriers.
and chirality λ = ±1 is (cid:15) = (k−λk )2/(2m∗)−E .
Since the finite relaxation rate breaks the time-reversal kλ R R
Here we introduce the Rashba momentum k = m∗γ
symmetry,itleadstothesuppressionoftheRKKYinter- R R
and the Rashba energy E = k2/2m∗. The oscillatory
action. Thus, the non-collinear spiral interaction stands R R
kernels (after integrating out the angular parts) in the
out and becomes dominant.
integral are
Therestofthepaperisorganizedasfollows. InSec. II,
we revisit the ferromagnetic/normal/ferromagnetic tri-
Iλ1λ2 =J (k r)J (k r), (4)
layer junction mediated by a two-dimensional electron 0 0 1 0 2
gas with Rashba interaction. The band structure and I1λ1λ2 =−λ1J1(k1r)J0(k2r)−λ2J0(k1r)J1(k2r), (5)
the Fermi surface topologies at different carrier concen- Iλ1λ2 =−λ λ J (k r)J (k r), (6)
2 1 2 1 1 1 2
trations are explained in detail. In Sec. III, we derive
the mediated exchange coupling within linear response where J (x) is denoted the Bessel function of the first
n
theory. After integratingout theangularpart, theresul- kind. For the Rashba Hamiltonian, its simpleness gives
tant formula can be implemented in numerics to calcu- rise to an extra relation h (r) = g (r)+g (r) beyond
0 0 2
latetheeffectiveexchangecoupling. InSec. IV,weusea the symmetry argument. As a result, we only need to
complimentary analytic approach to derive the effective evaluate three independent scalar functions.
exchange coupling. In Sec. V, the effects of finite spin Supposing the ferromagnet on the left of the trilayer
relaxationarediscussedinbothweakandstrongRashba magnetic junction is aligned along the z-axis, we are in-
regimes, followed by a brief conclusion. terested in the mediated exchange coupling proportional
to χ (r,θ = 0), where r is the width of intermediate
iz
layer. The induced moment is captured by the spiral an-
II. LINEAR RESPONSE THEORY gle (shown in Fig. 1), which can be expressed in terms
of χ and χ ,
xz zz
Consider a ferromagnetic/normal/ferromagnetic
χ (r,0)
(F/N/F) trilayer magnetic junction (TMJ) as shown cos[φ (r)] = xz , (7)
x (cid:112)
χ2 (r,0)+χ2 (r,0)
in Fig. 1, where the intermediate layer contains a two xz zz
dimensional electron gas (2DEG) with the Rashba cou- χ (r,0)
sin[φ (r)] = zz . (8)
pling. The 2DEG in the intermediate layer is described x (cid:112)χ2 (r,0)+χ2 (r,0)
by the Hamiltonian9, xz zz
Therefore,byevaluatingg (r)numerically,wecanstudy
(cid:90) (cid:20)−∇2 (cid:21) the spatial trends of the spairal angle φ (r).
H = d2r Ψ† 1−iγ (∇ σ −∇ σ ) Ψ, x
R 2m∗ R y x x y Inournumericalcomputations, wewillchangethepa-
√
(1) rameter k /k , where k = m∗γ and k = 2m∗(cid:15)
R F R R F F
where γ is the strength of the Rashba interaction and to calculate the mediated exchange coupling in different
R
m∗istheeffectivemassoftheitinerantcarriers. Thetwo- regimes. Forballisticcarriers,η →0isassumedinallcal-
component spinors Ψ†, Ψ are the creation/annihilation culations. For realistic materials or in the weak Rashba
operatorsfortheitinerantcarriers. Afterintegratingout regime, we choose the spin splitting ∆ = 2k γ = 5
R F R
theitinerantcarriers23,24,29,30,theexchangecouplingbe- meVandtheFermienergy(cid:15) =60meV.Orequivalently,
F
3
" "
"/2 "/2
! !
x 0 x0
!"/2 !"/2
!" R/L !" R/L+
0 1 2 3 4 F 0 1 2 3 4 5 s
FIG.2: Thenon-collinearangleφ intheweakReshbaregime FIG. 3: The non-collinear angle φ in the dilute density
x x
withk /k = 0.02 << 1, η → 0 and L is defined as π . regime with k /k = 3.5 and η → 0. The period of the
The gRreenF dotted line is represented thFe upward tendenkFcy spiral angle isRL+s =F π/(kR+kF).
resulted from the Rashba effect. Notice that the lumps at
discontinuous points are come from numerical error.
regime k /k (cid:29) 1. In this regime, the angle φ (r)
R F x
ceases to oscillate and rotates gradually with the junc-
itcorrespondstotheRashbacouplingγR =8.91×10−12 tion width as shown in Fig. 3. At first glance, the spiral
eVm and the carrier density n2D =1.25×1012 cm−2. evolution is similar to the spin-precession argument in
The results are shown in Fig. 2. In the weak Rashba Datta-Das SFET. But, it is not exactly so. The semi-
regime kR/kF (cid:28)1, the spatial trends of the angle φx(r) classical spin precession is described by the length scale
oscillates near the vicinity of ±π/2 and is very similar L = 2π/(2k ). But, as Fig. 3 shows, the spiraling
s R
to the ordinary RKKY interaction. The main difference length in our numerics is L+ = 2π/(2k +2k ) < L .
s R F s
is that, in the presence of the spin-orbit coupling, there Only in the extreme strong coupling where k /k (cid:28) 1,
F R
is an upward background. The oscillatory behavior is the semiclassical picture will be completely correct.
characterized by L ≡ 2π/2k as in ordinary RKKY
F F It is important to mention that the momentum 2k +
R
oscillations. It is important to emphasize that the spa-
2k connects patches of the outer Fermi surface which
F
tialtrendisdrasticallydifferentfromthespin-precession
are related by time-reversal symmetry.The period of the
argument in Datta-Das SFET8 because the quantum in-
spiral angle indicates that the exchange coupling mainly
terferences from different patches of the Fermi surface
contribute by the interference of the time-reversal sec-
cannot be ignored.
tion of outside chirality. Since the mediated exchange
The RKKY oscillation with an upward trend can be
coupling is spiral-like, many phenomena, in this regime,
understood in a simple picture. Taking the asymptotic
relatedtotheRKKYexchangecouplingshouldberecon-
limit k r (cid:29) 1, the reduced spin susceptibility along the
F sidered. For example, a tunable non-local spin control
radial direction χ (r), where a,b = x,z, can be well
ab in a coupled quantum dot system via a gate voltage has
approximated as 1D Rashba system. Applying a lo-
beenbeautifullydemonstratedinthe2Delectrongassys-
cal gauge transformation25,33, U(r) = e−ikRrσy/2, the tem of semiconductor26,27,28. The authors showed that
Rashba Hamiltonian can be mapped into the 1D free
the Kondo effect is suppressed by a nonlocal RKKY-like
electron gas with the well-known RKKY spin suscepti-
interaction which can be used to control the quantum
bility. Since the local gauge transformation is nothing
dot spin. If the mediated exchange coupling is no longer
but the local rotation about the y-axis with the spiral
RKKY-like, it is interesting to see how the non-collinear
angle φ(r) = k r, the reduced susceptibility is approx-
R coupling will reshape the idea.
imately the usual RKKY oscillation twisted by a local
spiral transformation,
(cid:20) (cid:21) III. PATH INTEGRAL FORMALISM
cosk r −sink r
χ (r)≈ R R χRKKY(r), (9)
ab sink r cosk r cb
R R ac
In this section, we derive the effective exchange cou-
where the summation over the repeated index c = x,z pling analytically. Taking the asymptotic limit, the spa-
is implied. The gauge argument explains why our nu- tialtrendsofthenon-collinearexchangecouplingbecome
merical results resemble the RKKY oscillation but with rather clear. Within the path integral formalism, the ef-
a gradual spiral background. fective coupling is obtained by integrating out the itin-
By increasing the strength of Rashba coupling or low- erant carriers and retaining the action to the quadratic
ering the electron density, we enter the strong coupling order. After some algebra (details can be found in Ap-
4
pendix A), the effective Hamiltonian can be written as a H = ASzSz +BSzSx. The function A(R) represents
eff 1 2 1 2
contour integral25,34,35 the collinear exchange coupling and B(R) for the non-
collinear part. By computing these two functions, the
H = −J2 (cid:90) µ dω Im(cid:110)tr(cid:104)(cid:16)(cid:126)σ·S(cid:126) (cid:17)G(R,ω+iη) spiral angle can be determined.
eff 4π 1 Let us start with the weak Rashba coupling. In this
−∞
(cid:16) (cid:17) (cid:105)(cid:111) regime,wehaveq >k ,andbyignoringtheslightcontri-
(cid:126)σ·S(cid:126) G(−R,ω+iη) ,(10) R
2 bution of k <0, the Green’s functions can be simplified
(cid:15)
to be
where µ is the chemical potential, R = R − R and
1 2
η represents an infinitesimal imaginary energy, and tr G (cid:39)−im∗(cid:104)H(1)(qR+k R)+H(1)(qR−k R)(cid:105),(19)
means a trace over the spin degrees of freedom. In 2D 0 4 0 R 0 R
Rashbasystem,theretardedGreen’sfunctioninmomen- m∗ (cid:104) (cid:105)
G (cid:39) H(1)(qR+k R)−H(1)(qR−k R) . (20)
tum space takes the form, 1 4 1 R 1 R
(cid:26) (cid:20) k2 (cid:16) (cid:17) (cid:21)(cid:27)−1 With the asymptotic form of the Hankel functions,
G(k;(cid:15)+i0+)= (cid:15)+i0+− +γ k kˆ×zˆ ·(cid:126)σ .
2m∗ R (cid:114)
2
(11) Hn(1)(x)(cid:39) πxexpi(x−n2π−π4), (21)
Fouriertransformingitbacktothecoordinatespace25,35, we end up with
the retarded Green’s function along the x-direction is
m∗
G(R;(cid:15)+i0+)=G0(R;(cid:15))1+G1(R;(cid:15))σy (12) G0 (cid:39)−i√2πqR ei(qR−π4)cos(kRR), (22)
m∗
with G0 and G1 defined as G1 (cid:39) √ ei(qR−π4)sin(kRR). (23)
2πqR
G (R;(cid:15))=−im∗ (cid:26)(cid:18)1+ kR(cid:19)H(1)(cid:2)(q+k )R+i0+(cid:3) The effective Hamiltonian is thus greatly simplified in
0 4 q 0 R
the asymptotic limit,
(cid:18) (cid:19) (cid:27)
+ 1− kqR H0(1)(cid:2)(q−kR)R+i0+(cid:3) ,(13) HeRffKKY(cid:39)F1(R)[cos(2kRR)S1zS2z+sin(2kRR)S1zS2x],(24)
G (R;(cid:15))= m∗ (cid:26)(cid:18)1+ kR(cid:19)H(1)(cid:2)(q+k )R+i0+(cid:3) where the range function25
1 4 q 1 R
−(cid:18)1− kqR(cid:19)H1(1)(cid:2)(q−kR)R+i0+(cid:3)(cid:27),(14) F1(R)=−J82πm2∗sin(R2q2FR). (25)
Note that the range function is identically the same as
(cid:112)
where R = |R| and q ≡ k2+k2 with k2 = 2m∗(cid:15). the usual two-dimensional RKKY range function36, ex-
(cid:15) R (cid:15) (cid:112)
Hn(1) is the Hankel function of the first kind, defined as cept that kF is replaced by qF = 2m∗(cid:15)F +kR2. There-
H(1)(z)=J (z)+iY (z)withJ andY beingtheBessel fore, as clearly demonstrated in Eq. (24), the mediated
n n n n n
exchange coupling is similar to the usual RKKY with a
functions of the first kind and second kind. Notice that
non-collinear spiral background as obtained numerically
the Bessel or Hankel functions have a branch cut along
the negative real axis and the infinitesimal i0+ is impor- in Fig. 2.
Now we turn to the strong coupling regime. In this
tant in the spiral regime. We substitute the retarded
regime, because (q − k )R < 0, we need to be cau-
Green’s function into Eq. (10) and obtain the effective R
tiousabouttheanalyticpropertiesoftheHankelfunction
Hamiltonian along the x-axis,
H(1)(z). Onthecomplexplane,H(1)(z)hasabranchcut
n n
H =AS(cid:126) ·S(cid:126) +B(S(cid:126) ×S(cid:126) ) +CSySy, (15) onthenegativerealaxis. Thus, weneedtopayextraat-
eff 1 2 1 2 y 1 2
tention to the i0+ factor. Note that the Hankel function
where coefficients A, B, and C are of the first kind along the negative real axis can be con-
nected to the Hankel function of the second kind,
A(R)= −J2 (cid:90) µd(cid:15) Im(cid:2)G2+G2(cid:3), (16)
2π 0 1 H(1)(−x+i0+)=(−1)n+1H(2)(x+i0+), (26)
∞ n n
J2 (cid:90) µ
B(R)= 2π d(cid:15) 2Re[G0G1], (17) with the definition Hn(2)(z) = Jn(z)−iYn(z). By this
−∞ connection, we can rewrite the Eq. (13) and (14) as,
C(R)= 2Jπ2 (cid:90)−µ∞d(cid:15) 2Re(cid:2)G21(cid:3). (18) G0(cid:39)−im4∗kqR(cid:104)H0(1)(qR+kRR)+H0(2)(kRR−qR)(cid:105),(27)
The hard magnet is aligned along the z-axis, S(cid:126) = Szzˆ, mk (cid:104) (cid:105)
1 G (cid:39) R H(1)(qR+k R)−H(2)(k R−qR) . (28)
so that we can rewrite the effective Hamiltonian as 1 4 q 1 R 1 R
5
the numerical results are shown in Fig. 4. It is clear
"
that the trend belongs to the spiral type. However, pay
special attention to the length scale of the spiral. In
the strong coupling regime, the spiral length is roughly
"/2
L+. Right at the critical point (k = k ), the spiral
s R F
period is π/(2k ) = L /2, which follows the rule in the
! R s
x strongcouplingregime. Thisfactoroftwocanbeunder-
0
stood as the more general formula for the length scale
L+ = π/(k +k ) in the strong coupling limit and at
s R F
!"/2 the critical point. The analytic calculations can be done
by substituting q =k =k into Eq. (13) and Eq. (14).
F R
The asymptotic forms of the Green functions near the
!" R/L Fermi surface are
s
0 1 2 3 4
m
G0 =G1 (cid:39)−i √ ei(2kRR−π4), (35)
2 πk R
R
FIG.4: Thenon-collinearangleφ inthecriticalregimewith
x
k /k =1andη→0. Thespirallengthatthecriticalpoint and the effective Hamiltonian is
R F
is π/2k = L /2, only half of that in the strong coupling
R s
regime. Hecff ∝[cos(4kRR)S1zS2z+sin(4kRR)S1zS2x], (36)
with the spiral period 2π/4k = L /2, as we mentioned
R s
In the asymptotic limit, the second kind of Hankel func- above.
tions are
(cid:114)
2 IV. FINITE SPIN RELAXATION
Hn(2)(x)(cid:39) πxe−i(x−n2π−π4). (29)
So far, we have revealed the interesting transitions be-
Simple algebra leads to the following results,
tweenweak,strongandcriticalregimesassumingtheme-
m∗ (cid:18)k (cid:19) (cid:16) π(cid:17) diating carriers are perfectly ballistic. In realistic mate-
G (cid:39)−i√ R eiqRcos k R− ,(30) rials,thespinrelaxationrateisfiniteduetomanymech-
0 2πk R q R 4
R anisms. Thus, it is interesting to study how the spatial
G (cid:39) √ m∗ (cid:18)kR(cid:19)eiqRsin(cid:16)k R− π(cid:17). (31) trends of the mediated exchange coupling changes when
1 2πk R q R 4 the carriers are no longer perfectly ballistic. To fully
R
address this issue, one needs to rederive the effective ex-
Finally, the effective Hamiltonian takes the form, change coupling using a modified single-particle propa-
gator. While this is certainly an interesting direction
(cid:104) (cid:16) π(cid:17)
Hespffrial (cid:39)F2(R) cos 2kRR− 2 S1zS2z for future studies, we peek into the problem by intro-
(cid:16) π(cid:17) (cid:105) ducing a phenomenological spin relaxation ration within
+sin 2kRR− 2 S1zS2x , (32) the linear response theory. It is motivated by the obser-
vation that the inclusion of finite spin relaxation breaks
with the range function time-reversalsymmetryandshallsuppresstheRKKYin-
teraction. Therefore, we expect the non-collinear spiral
F (R) = J2m∗kR (cid:90) qF dq sin(2qR) (33) interaction should be enhanced and becomes dominant.
2 4π2 R q For convenience, we introduce the parameter k =
0 √ η
= J42πm2∗kRR Si(2qFR), (34) an2dmk∗Rη./kInF F=ig.05.0,2w,eacnhdoowseetchaenpasereamtehtaetrsthkηe/RkFK=KY1
pattern is indeed suppressed significantly due to the fi-
where Si(x)=(cid:82)x sintdt is the sine integral function and nite relaxation rate. Furthermore, as we move closer
0 t
is positive for x > 0. In the asymptotic limit, Si(x) to the critical regime ( but still in the RKKY regime)
reaches the constant value π/2 with minor oscillations. with k /k = 0.2, even a moderate spin relaxation
R F
Thus, the spatial trend of the mediated exchange cou- k /k =0.5willalmostwashoutthesignatureofRKKY
η F
pling is dominated by the spiraling part with minor os- oscillations and the spiral trend becomes rather robust,
cillations. This is drastically different from the results as shown in Fig 6. Therefore, it seems that the finite
in the weak Rashba regime. Again, our analytic results spin relaxation will suppress the RKKY oscillation and
agree with those shown in Fig. 3 obtained by numerical enhance the spiral interaction.
methods. In conclusions, we apply both analytic and numeric
The same calculation can be done for the critical case methodstostudytheexchangecouplingmediatedbythe
betweentheweakandthestrongcouplingregimeswhere itinerant carriers with Rashba interaction. We demon-
one of the Fermi surface disappears. For comparison, strate that the Fermi surface topology greatly alters the
6
APPENDIX A: DERIVATIONS FOR EFFECTIVE
" HAMILTONIAN
Within the path integral approach, we investigate the
"/2
effective coupling between two ferromagnets S(cid:126) and S(cid:126)
1 2
! mediated by itinerant carriers, as illustrated in Fig.1.
x
0 Consider the Zener model, but only included two im-
purity spin, the Hamiltonian can be represented as
!"/2
H =H +H , (A1)
0 I
where
!"
0 20 R(nm) 40 60 (cid:90) (cid:90)
(cid:88)
H = dDr dDr(cid:48) Ψ†(r)h (r,r(cid:48))Ψ(r(cid:48)), (A2)
0 0
α
FskpIiG/nk.r5e:=laSx1ua.ptiporne.ssitohneoRfKRKKYKYphoassceillwatiitohnskRin/ktFhe=pre0s.0en2caenodf HI =(cid:90) dDr(cid:88)Ψ†(r)(cid:20)JS(cid:126)I(r)· (cid:126)σ2(cid:21)Ψ(r(cid:48)). (A3)
η F I=1,2
Here we ignore the correction effects ofthe itinerant car-
riers, and represent the unperturbed Hamiltonian H as
0
"
twofermionicoperators. Ψisatwo-componentspinorfor
the itinerant carriers and S(cid:126) (r) = S(cid:126) δ(r−R ) denotes
I I I
theimpurityspinatthepositionR intheD-dimension,
"/2 I
and J represents the coupling between the itinerant and
the localized spin densities.
!
x 0 In order to write down the partition function in path
integral form, we introduce the Grassmann variables
φ, φ¯ for the itinerant carriers37 and the coherent state
!"/2 for the impurity spins38, which is denoted as S(cid:126)|Ωˆ(cid:105) =
SΩˆ|Ωˆ(cid:105), where spin orientation is denoted here by Ωˆ =
(sinθcosψ,sinθsinψ,cosθ) and the length of impurity
!"
0 50 R(nm) 100 spin vector by S. In path integral language, the parti-
tion function is represented as
FofIGsp.in6:reSluapxaptrieosnsiownitohfcRloKseKrYtoostchiellaptihoansseitnratnhseitiporneswenitche Z =(cid:90) D[Ω(cid:126)1,Ω(cid:126)2](cid:90) D[φ¯,φ] e−(cid:82)0βdτL(Ω(cid:126)1,Ω(cid:126)2;φ¯,φ), (A4)
k /k =0.2 and k /k =0.5.
R F η F
wheretheintegraldenotesthesummationofinfinitetime
section. The Lagrangian can be writte as
property of mediated exchange coupling in 2D Rashba (cid:90)
gas system. When the Rashba energy is much smaller L= dDr (cid:88) (cid:104)ΩˆI|∂τ|ΩˆI(cid:105)+φ¯(r,τ)∂τφ(r,τ)
than the Fermi energy, the mediated exchange coupling I=1,2
is almost the RKKY oscillation plus a slight upward +H(Ω(cid:126) ,Ω(cid:126) ;φ¯,φ), (A5)
1 2
trend. On the other hand, if the Rashba energy is larger
than the Fermi energy, the mediated exchange coupling
wherethefirstandsecondtermsarethedynamicsofthe
shows a spiral pattern. Apparently, the Fermi surface
impurity spins and itinerant fermions respectively. Inte-
topology plays a crucial part about the mediated ex-
grate out the itinerant carrier and arrive at an effective
change coupling. We suggest that, in the quantum-dot
description for localized spin,
systems26,27,28 with dilute carrier densities, the spiral-
like exchange coupling should be considered instead of (cid:90)
the usual RKKY interaction. Z = D[Ω(cid:126)1,Ω(cid:126)2] e−Seff(Ω(cid:126)1,Ω(cid:126)2), (A6)
We acknowledge support from the National Science
with the effective action
Council of Taiwan through grants NSC-96-2112-M-007-
004 and NSC-97-2112-M-007-022-MY3 and also support
(cid:90) β (cid:90)
fromtheNationalCenterforTheoreticalSciencesinTai- S = dτ dDr (cid:88) (cid:104)Ωˆ |∂ |Ωˆ (cid:105)−ln(cid:2)detG−1(cid:3).(A7)
eff I τ I
wan. 0 I=1,2
7
Here we use the notation, the effective action after Fourier transformation is
G−1(Ω(cid:126)1,Ω(cid:126)2) = G0−1+δG−1(Ω(cid:126)1,Ω(cid:126)2), (A8) S = J2 (cid:88)tr(cid:110)(cid:104)(cid:126)σ·S(cid:126) (cid:105)G (R ,R ;ik )
J (cid:16) (cid:17) eff 4 1 0 1 2 n
δG−1(Ω(cid:126)1,Ω(cid:126)2) = 2 Ω(cid:126)1+Ω(cid:126)2 ·(cid:126)σ, (A9) ikn (cid:104) (cid:105) (cid:111)
G−1 = ∂ 1−h . (A10) (cid:126)σ·S(cid:126)2 G0(R2,R1;ikn) .(A16)
0 τ 0
Byutilizingtherelation,ln[detM]=tr[lnM],thesecond Noticethatbyneglectingtheretardationeffect,thepoles
term of the effective action can be expanded, of the Green’s function, G (z), only show up on the real
0
ln(cid:2)detG−1(cid:3)= tr(cid:2)ln(cid:0)G−1+δG−1(cid:1)(cid:3), axis. This property let Eq. (A16) to be rewritten in a
0 retarded Green’s function form.
= tr(cid:2)lnG0−1(cid:3)+tr(cid:2)ln(cid:0)1+G0δG−1(cid:1)(cid:3), First, we can demonstrate the stationary RKKY
∞ effect between two impurity spins mediated by free
= tr(cid:2)lnG−1(cid:3)−tr(cid:88) 1 (cid:0)−G δG−1(cid:1)n.(A11) fermions from Eq. (A16). By Fourier transforming
0 n 0
n=1 Eq. (A16) to the momentum space and fixing the di-
rection of one impurity spin to the z-axis, we have
The zeroth order term does not contain any spin opera- (cid:104) (cid:16) (cid:17)(cid:105)
tors and can be neglected when computing the effective tr ((cid:126)σ·S1zzˆ) (cid:126)σ·S(cid:126)2 = 2S1zS2z, and thus the equation
spin-spin interaction. The first-oder term is linear in the can be rewritten as
spin operator and vanishes since the itinerant carriers
(cid:90) β
consider here are not polarized,
S = dτ H , (A17)
eff eff
tr(cid:2)G δG−1(cid:3) = J (cid:90) dτ(cid:90) dDr G (r,τ;r,τ +0+) 0
0 2 0 with H =J(R )SzSz and
× (cid:88) tr(cid:110)(cid:126)σ·Ω(cid:126) (r,τ)(cid:111)=0, (A12) eff 12 1 2
I=1,2 I J(R )= J2 (cid:90) dDq (cid:90) dDk eiq·R12
12 2 (2π)D (2π)D
where τ +0+ is for the creation operator.
1 (cid:88)
Thesecond-ordertermcontainstwospinoperatorsand β G0(k+q;ikn)G0(k;ikn). (A18)
is relevant for the effective exchange coupling, ikn
S(cid:48) ≡ 1tr(cid:2)G δG−1G δG−1(cid:3) Hereweusethetranslationinvariantpropertyoffreecar-
eff 2 0 0 riersandassumenospinpolarizationofthissystem. The
J2 (cid:90) β (cid:90) β (cid:110)(cid:104) (cid:105) summationofMatsubarafrequenciesforfermionscanbe
= dτ dτ(cid:48) tr (cid:126)σ·Ω(cid:126) (τ) G (R ,τ;R ,τ(cid:48))
4 1 0 1 2 computed and represented as the particle-hole propaga-
0 0
(cid:104) (cid:105) (cid:111) tion. After some calculation, we end up with
× (cid:126)σ·Ω(cid:126) (τ(cid:48)) G (R ,τ(cid:48);R ,τ) . (A13)
2 0 2 1
By truncating the expansion to second order, the ef- J(R12)= J22 (cid:90) (d2πD)qD eiq·R12 χ(q), (A19)
fective action for the two impurity spins is S =
eff
(cid:82)0βdτ(cid:82) dDr(cid:80)I=1,2(cid:104)ΩˆI|∂τ|ΩˆI(cid:105)+Se(cid:48)ff. The first term of where the particle-hole propagation is denoted as
thiseffectiveaction,knownasBerryphase,describesthe
dynamicsoftheimpurityspins. Hereweassumethatthe (cid:90) dDk nF(ξk+q)−nF(ξk)
dynamics of the itinerant spins is much faster so that we χ(q)= , (A20)
(2π)D ξ −ξ +iη
can ignore the retardation effect between two impurity k k+q
spins. Inotherwords, wetakethestationarylimitofthe
impurity spins Ω(cid:126) (τ) = S(cid:126) and ignore the Berry-phase where ξk is the eigenenergy of free fermions and nF(ξk)
I I is the Fermi-Dirac distribution function. As shown as
term.
the Eq. (A19), we obtain the mediated RKKY exchange
Now we would like to write down the effective action
coupling strength between two impurity spins.
in the Fourier space of the imaginary time31,
Next, we will replace the imaginary Green’s functions
G (τ −τ(cid:48))= 1 (cid:88) e−ikn(τ−τ(cid:48))G (ik ) (A14) in Eq. (A16) by the retarded ones. The connection be-
0 β 0 n tween these two types of Green’s functions can be seen
ikn in the complex plane31. The kernel of the Eq. (A16) in
where the summation is over all Matsubara frequencies the complex plane,
k = (2n+1)π for fermions. Making use of the identity
n β f(z) =tr(cid:110)(cid:104)(cid:126)σ·S(cid:126) (cid:105)G (R ,R ;z)
1 0 1 2
β1 (cid:90) βdτ ei(km−kn)τ =δmn, (A15) (cid:104)(cid:126)σ·S(cid:126)2(cid:105)G0(R2,R1;z)(cid:111), (A21)
0
8
Im[z] integral can be simplified to two line integrals very close
to the real axis,
(cid:26)(cid:90) ∞+iη dz
I = lim n (z)f(z)
η→0+ −∞+iη 2πi F
(cid:90) −∞−iη dz (cid:27)
+ n (z)f(z) , (A25)
Re[z] ∞−iη 2πi F
(cid:90) ∞ dω
= lim n (ω)[f(ω+iη)−f(ω−iη)].(A26)
η→0+ −∞ 2πi F
R
CombiningEq.(A23)andEq.(A26),weobtaintheuseful
identity,
(cid:88) (cid:90) β (cid:90) ∞ dω
f(ik )=− dτ n (ω)
n 2πi F
ikn 0 −∞
[f(ω+iη)−f(ω−iη)], (A27)
FIG. 7: Contours in the complex plane. The black points
denotestheMatsubarafrequenciesz=i(2n+1)π forfermions. where the first term of the right hand side is represented
β
by the retarded Green’s function and the second term
is denoted the advanced Green’s function. By using the
identity,
has poles on the real axis. If we consider a contour inte-
gral, tr[ABCD]∗ =tr(cid:2)B†A†D†C†(cid:3)=tr(cid:2)A†D†C†B†(cid:3),
(cid:90) dz (A28)
I = f(z)n (z)ez0+, (A22) and the relation between advanced and retarded Green’s
C 2πi F function,[G (R ,R ;ω+iη)]∗ =G (R ,R ;ω−iη),we
0 2 1 0 1 2
can obtain the second term as complex conjugate of the
where the contour path is shown in the Fig.7, it includes
first. Afterreshapingtheeffectiveaction,weendupwith
twoparts,theupperandlowerofthecomplexplane. The
the effective Hamiltonian of the Eq. (A17),
contour integral can be rewritten,
I = zR=eiskn(cid:104)f(z)nF(z)ez0+(cid:105)=−β1 (cid:88)ikn f(ikn).(A23) Heff = −4Jπ2 (cid:90)−∞∞dω(cid:16)nF(ω)(cid:17)Im(cid:110)tr(cid:104)(cid:16)(cid:126)σ·S(cid:126)1(cid:17) (cid:105)(cid:111)
G (R ,R ;ω+iη) (cid:126)σ·S(cid:126) G (R ,R ;ω+iη) .(A29)
HeretheFermi-Diracdistributionisrepresentasn (z)= 0 1 2 2 0 2 1
F
(eβz + 1)−1, and has poles at z = i(2n + 1)π/β, n =
···,−1,0,1,···, where the residues at these poles are At T =0, the Fermi function is a step function. We can
simply replace n (ω) by 1 and change the upper limit
F
Res [n (z)]= lim z−ikn =−1. (A24) of the integral to the chemical potential µ. Furthermore,
z=ikn F z→ikn eβz +1 β if the unperturbed system is translational invariant, the
effective Hamiltonian is just the function of the relative
Because in the |R|→ ∞ limit, the factor n (z)ez0+ ap- distance and we denote R =R −R . In the end, the
F 12 1 2
proaches zero whether Re(z) > 0 or not, the contour modified effective Hamiltonian comes to be the Eq. (10).
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