Magnetic Ordering of Nuclear Spins in an Interacting 2D Electron Gas

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by  Pascal Simon

Magneti
Ordering of Nu
lear Spins in an Intera
ting 2D Ele
tron Gas
1,2 1 1
1 Pas
al Simon , Bernd Braune
ker , and Daniel Loss
Department of2Physi
s, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
Laboratoire de Physique et Modélisation des Milieux Condensés,
CNRS and Université Joseph Fourier, BP 166, 38042 Grenoble, Fran
e
(Dated: February 1, 2008)
We investigatethemagneti
behaviorof nu
learspinsembeddedina 2Dintera
ting ele
tron gas
8 usingaKondolatti
emodeldes
ription. Wederiveane(cid:27)e
tivemagneti
Hamiltonianforthenu
lear
0 spinswhi
hisoftheRKKYtypeandwheretheintera
tionsbetweenthenu
learspinsarestrongly
0 modi(cid:28)edbytheele
tron-ele
tronintera
tions. Weshowthatthenu
learmagneti
orderingat(cid:28)nite
2 temperature relies on the (anomalous) behavior of the 2D stati
ele
tron spin sus
eptibility, and
n thusprovidesa
onne
tionbetweenlow-dimensionalmagnetismandnon-analyti
itiesinintera
ting
a 2D ele
tron systems. Using various perturbative and non-perturbative approximation s
hemes in
J ordertoestablishthegeneralshapeoftheele
tronspinsus
eptibilityasfun
tionofitswaveve
tor,
1 we show that the nu
lear spins lo
ally order ferromagneti
ally, and that this ordering
an be
ome
1 globalin
ertainregimesofinterest. Wedemonstratethattheasso
iated Curietemperatureforthe
nu
learsystemin
reases with theele
tron-ele
tron intera
tions upto themillikelvinrange.

]
l PACSnumbers: 71.10.Ay,71.10.Ca,71.70.Gm
l
a
h
- I. INTRODUCTION toextendthespin de
aytimebyoneorderofmagnitude
s
e through polarization of the nu
lear spins, a polarization
m ofabove99%isrequired,14 quitefarfromthebest result
In the last de
ade, the (cid:28)eld of spintroni
s has seen re- 19
. 1,2,3 so far rea
hed in quantum dots, whi
h is around 60%.
t markable progress. Among them, the possibility of
a A
ommon point to the aforementioned approa
hes is

on(cid:28)ning ele
tron spins in quantum dots opens the door
m their aim at mitigating nu
lear spin (cid:29)u
tuations by ex-
to quantum spintroni
s. This is based on the possibil-
- ternal a
tions. Re
ently, the possibility was raised of
d ity of
ontrolling and manipulating single ele
tron spins anintrinsi
polarizationofnu
learspinsat(cid:28)nitebut low
n in order to build devi
es able to a
hieve operations for
temperatureinthetwodimensionalele
trongas(2DEG)
o quantum information pro
essing. The most promising 20

on(cid:28)ned by the GaAs heterostru
ture.
c and
hallenging idea is the use of spins of
on(cid:28)ned ele
-
[ trons in quantum dots to realize quantum bits.4 Within The nu
lear spins within the 2DEG intera
t mainly
2 the last years, all the ne
essary requirements for spin- via the Rudermann-Kittel-Kasuya-Yosida (RKKY)
21
v based quantum
omputation have been realized experi- intera
tion, whi
h is mediated by the
ondu
tion
4 mentally, going from the
oherent ex
hange of two ele
- ele
trons (the dire
t dipolar intera
tions between the
6 tron spins in a double dot5 to the
oherent
ontrol of nu
lear spins are mu
h weaker, see below). An intrinsi
1
a single ele
tron spin, in
luding the observation of Rabi nu
lear spin polarization relies on the existen
e of a
0 os
illations.6 These a
hievements have be
ome possible temperature dependent magneti
phase transition, at
.
9 be
ause ele
tron spins in semi
ondu
tor quantum dots whi
h a ferromagneti
ordering sets in, thus de(cid:28)ning a
0 are relatively weakly
oupled to their environment and nu
lear spin Curie temperature.
7
therefore long lived quantities, quite robust against de-
0 The(cid:28)rstestimateofsu
haCurietemperaturewasob-
:
ay. Indeed, longitudinal relaxation times in th1esseec sys- tainedforthree-dimensional(3D)metalli
samples,using
v tems have been measured to be of the order of .7,8,9
a Weiss mean (cid:28)eld treatment by Fröhli
h and Nabarro
i
X A lower bound on the spin de
oheren
e time for an en- more than sixty years ago.22 They determined the nu-
sembleof ele
tronspinsin GaAsquantum dots hasbeen
r 100ns
lear spin Curie temperature to be in the mi
rokelvin
a measured to be typi
al larger than ,10 while a
o-
µs range or less for 3D metals. A Weiss mean (cid:28)eld treat-
T
heren
e timein asinglequantumdotex
eeding 1 has c
ment also gives a nu
lear spin Curie temperature in
5
been re
ently a
hieved using spin-e
ho te
hniques. It is
the mi
rokelvin range for a typi
al 2DEG made from
by now well established that one of the major sour
es 20
GaAs heterostru
tures, yet a more detailed analysis
of de
oheren
e for a single ele
tron spin
on(cid:28)ned in a
is desirable for at least two reasons. First, a Weiss
quantum dot is the
onta
t hyper(cid:28)ne intera
tion with
mean (cid:28)eld analysis does not take into a

ount properly
11
the surrounding latti
e nu
lear spins.
the dimensionality of the system, and se
ond ignores
One possibility to lift this sour
e of de
oheren
eis the ele
tron-ele
tron (e-e) intera
tions. In two dimensions
23
development of quantum
ontrol te
hniques whi
h e(cid:27)e
- (2D), the Mermin-Wagnertheorem states that thereis
tively lessen or even suppress the nu
lear spin
oupling no phase transition at (cid:28)nite temperature for spin sys-
5,12,13
to the ele
tron spin. Another possibility is to nar- tems with Heisenberg (isotropi
) intera
tions, provided
14,15,16
row the nu
lear spin distribution, or dynami
ally that the intera
tionsare short-rangedenough. However,
11,14,17,18,19
polarize the nu
lear spins. However, in order RKKY intera
tions are long-ranged and stri
tly speak-
2
ing, the Mermin-Wagner theorem does not apply, al- but pointin oppositedire
tionsat the s
aleofthe Fermi
though a
onje
ture extending the Mermin-Wagner the- wavelength(roughlytwoordersofmargn<itu1delargerthan
s
oremforRKKYintera
tionsdue tonon-intera
tingele
- the nu
lear latti
e spa
ing at small ∼ )χ. D(qe)pending
s
tron systems has been re
ently formulated (and proved on the general non-perturbative shape of (whi
h
T r
24 s
in some parti
ular
ases). mayhavea
omplexdependen
eon and ),wedis
uss
In Ref. 20, we started from a Kondo latti
e des
rip- thepossibleorderedphasesandtheirasso
iatedmagneti
tion for the system
omposed of nu
lear spins and ele
- properties.
trons,thenderivedarathergenerale(cid:27)e
tiveHamiltonian Theoutlineofthepaperisasfollows: InSe
. II,wefor-
for nu
lear spins after integrating out ele
tron degrees mulateaKondolatti
edes
riptionofourproblemwhere
of freedom, and (cid:28)nally performed a spin wave analysis thenu
learspinsareplayingaroleanalogoustomagneti
arounda ferromagneti
groundstate(whi
h weassumed impurities embedded in an ele
tron liquid. We then de-
to be the lowest energy state). We indeed showed that riveagenerale(cid:27)e
tive magneti
Hamiltonian fornu
lear
T = 0
c for non-intera
ting ele
trons in agreement with spins where the intera
tion is
ontrolled by the ele
tron
the latter
onje
ture. However, taking into a

ount e-e spin sus
eptibility in 2D. In Se
. III, we
al
ulate the
intera
tions
hanged drasti
allythis
on
lusion. It turns ele
tronspinsus
eptibilityin anintera
ting2DEGusing
out that e-e intera
tions modify the long range nature various approximation s
hemes for both, short-ranged
of the 2D RKKY intera
tions (whi
h are dire
tly re- andlong-rangedintera
tions. Parti
ularattentionispaid
lated here to the stati
ele
tron spin sus
eptibility) and to renormalization e(cid:27)e
ts in the Cooper
hannel whi
h
therebyallowsomeorderingofthe nu
learspinsat(cid:28)nite turn out to be important. Se
. IV is devoted to the
temperature.20Furthermore,weshowedthatthetemper- magneti
properties of the nu
lear spins depending on
atures
aleatwhi
hthisorderingtakespla
eisenhan
ed the general wave-ve
tor dependen
e of the ele
tron spin
by e-e intera
tions.20 sus
eptibility. We dis
uss two di(cid:27)erent phases: A ferro-
magneti
phase and a heli
al phase with a period of the
The study of thermodynami
quantities in intera
ting
order of the ele
tron Fermi wavelength. Finally, Se
. V
ele
tron liquids (espe
ially in 2D) has attra
ted some
25,26,27,28,29,30,31,32,33,34 35
ontainsasummaryofourmainresultsandalsoperspe
-
theoreti
al and experimental
tives. AppendixA
ontainssomedetailsofthederivation
interestre
entlywiththegoalto(cid:28)nddeviationsfromthe
of the e(cid:27)e
tive nu
lear spin Hamiltonian and of the re-
standard Landau-Fermi liquid behavior. It is therefore
du
tion to a stri
tly 2D problem.
quite remarkable that the ma
ros
opi
magneti
proper-
tiesofnu
learspinsina2DEG,andthustheir(cid:28)nitetem-
perature ordering, are dire
tly related to the
orre
tions
tothethestati
ele
tronspinsus
eptibilityindu
edbye- II. MODEL HAMILTONIAN
eintera
tions. Theymaythereforebeasso
iatedwithan
indire
tsignatureofFermiliquidnon-analyti
ities. Nev- A. Kondo latti
e des
ription
ertheless, it turns out that the temperature dependen
e
χ (T)
s
of the ele
tron spin sus
eptibility is rather intri- Inordertostudyanintera
tingele
trongas
oupledto

ate. Ontheonehand, fromperturbative
al
ulationsin nu
learspinswithinthe2DEG,weadoptatight-binding
se
ondorderinχt(hTe)short-rangedintera
tionstrengthone representationin whi
hea
hlatti
esite
ontainsasingle
s 29,30,31
obtains that | | in
reases with temperature. nu
lear spin and ele
trons
an hop between neighboring
The same behavior is reprodu
ed by e(cid:27)e
tive supersym- sites. A general Hamiltonian des
ribing su
h a system
32
metri
theories. On the other hand, non-perturbative reads

al
ulations, taking into a

ountrenormalizatione(cid:27)e
ts,
χs(T) 1 Nl
(cid:28)forustnddet
hraeats|es with| theamspaernaotunr-em.3o3n,3o4toTnhi
isblaethtaevriobrehaanvd- H = H0+ 2 Ajc†jστσσ′cjσ′ ·Ij + viαjβIiαIjβ
iorisinagreementwithre
entexperimentson2DEGs.35 Xj=1 Xi,j
= H +H +H ,
0 n dd
Inviewofthesere
ent
ontroversialresults,wewantto (1)
re
onsider the question of a (cid:28)nite temperature ordering H
0
oe(cid:27)fen
ut
sleoafrthspeinstsabtiy
tsapkininsguisn
teoptaib

iloituyntχsre(qn)o,rmwhaelirzeatqioins wHHhnertehe eled
etnroonte-snut
hleea
ronsdpuin
thioynpeerle(cid:28)
nteroinntHeraam
tiilotonniaannd,
dd
the wave ve
tor, and therefore going beyond Ref. 20. It theHgeneral dipolar intera
tion between the nu
lear
0
turns out that, a priori, di(cid:27)erent nu
lear spin orderings spins.
an be rather general and in
ludes ele
tron-

an o

ur, depending on temperature and other sample ele
tron (e-e) intera
tions. In Eq. (1), c†jσ
reates an
r σ = ,
j
parameters su
h as the intera
trion strength, measured τele
tron at the latti
e site with spin ↑ ↓, and
s
by the dimensionless parameter (essentially, the ratio repreIsen=ts(Ithx,eIyP,aIuzl)i matri
es. We have also intro-
between Coulomb and kineti
energy of the ele
trons). du
ed j j j j the nu
lear spin lo
ated at the
r A
j j
We
onsider at least two possible ordered phases in the latti
e site , and the hyper(cid:28)ne
oupling
onstant
r
36 j
nu
lear system: a ferromagneti
ordering but also a between the ele
tron and the nu
lear spin at site .
α,β = x,y,z
heli
al spin ordering where the nu
lear spins align ferro- Summation over the spin
omponents is
r
j
magneti
ally at the s
ale of the nu
lear latti
e
onstant implied. The ele
tron spin operator at site is de(cid:28)ned
3
by Sj = ~c†jστσσ′cjσ′ (for
onvenien
e we normalize the B. Derivation of an e(cid:27)e
tive magneti
Hamiltonian
N
l
spin operator here to 1). denotes the total number of
latti
esites. >Fromhereon,weassumethatAj =A>0, We(cid:28)rst go to Fourierspa
eand rewriteHn in Eq. (1)
whi
hmeansweassumethehyper(cid:28)neintera
tionisanti- as
ferromagneti
and thesameforall atomsthat
onstitute A
H = S I ,
the heterostru
tures (typi
ally Ga and As and their iso- n 2N q· q (2)
l q
topes). X
where Iq = jeiq·rjIj and Sq = je−iq·rjSj are the
I S
j j
The nu
lear spins are also
oupled via the dipolar in- Fouriertra~ns=Pfo1rmsof aAnd ,respPe
tively. (Fromnow
tera
tiontoothernu
learspins,whi
harenotembedded on we set .) Sin
e is a small energy s
ale in our
inthe2DEG.Takingintoa

ountthisintera
tionaswell
ase,we
anperformaS
hrie(cid:27)er-Wol(cid:27)(SW)transforma-
A
makes the problem of the magnetism of nu
lear spins in tion in order to eliminate terms linear in , followed by
GaAsheterostru
turesanapriori3Dtremendously
om- integratingouttheele
trondegreesoffreedom. Further-
pli
ated one. Nevertheless, it turns out that the dipolar more,we
anredu
eourinitial3Dmodeltoagenuine2D
E
dd
intera
tion energy s
ale is the smallest one. It has problem. The main steps of these
al
ulations are given
E 100 nK
dd 37
been estimated to be ≈ . In parti
ular, inAppendixA. Weareleftwithane(cid:27)e
tiveHamiltonian
k T E T H
B dd eff
≫ , where is the temperature of a typi
al ex- for the nu
lear spins in a 2D plane:
periment. In the rest of the paper, we negle
t all dire
t A2 1
H = Iα χ (q) Iβ ,
dinipgoelnaerrianltsemraa
ltleiorntshabnettwheeeinndthireev
nαtuβi
nletea0rras
ptiionns,,washwi
ehwairlel eff 8nsN Xq q αβ −q (3)
see. Therefore, we assume that ij ≈ in Eq. (1). This
assumption is important sin
e it allows us to fo
us only where
onthosenu
learspinswhi
hliewithinthesupportofthe χ (q,ω)= i ∞dt e iωt ηt [Sα(t),Sβ ] ,
ele
tron envelope wave fun
tion (in growth dire
tion). αβ −Na2 Z0 − − h q −q i (4)
is a general 2D ele
tron spin sus
eptibility tensor,
χ (q) = χ (q,ω = 0) q = q N
The general Hamiltonian in Eq. (1) is theHw0ell-known αβ αβ and |a|. is the number
Kondo latti
e Hamiltonian (KLH), though
ontains of latti
e sites in the 2D plane and denotes the latti
e
...
also e-e intera
tions. The KLH is one of the most stud- spa
ing for nu
lear spins. Note that h i means aver-
ied models in
ondensed matter theory due to its large age over ele
tron degrees of freedom only. We have nor-
χ
variety of appli
ations. The KLH has been used to de- malized su
hthatit
oin
ideswith thedensity-density
38
s
ribe the properties of transition metal oxides, heavy Lindhard fun
tion (see below) in the isotropi
and non-
39,40
fermions
ompounds, more re
ently also magneti
intera
ting limit.
semi
ondu
tors (or semi-metals) in the series of rare The only assumptionswe make aretime reversalsym-
earth substanG
aes,41,ManndAsd,iluted magneti
semi
ondu
- metry of H0, as well as translational and rotational in-
1 x x 42,43
tors su
h as − to list only a few. The varian
e. The e(cid:27)e
tive Hamiltonian in Eq. (3) is there-
nu
lear spins play a role analogous to magneti
impuri- forequitegeneralanddoesnotdependonthedimension-
ties in the Kondo latti
e problem. The regime in whi
h ality of the system. Note that Eq. (3) is also valid when
we are interested
orresponds toAthe wEeak KondoE
ou- ele
tron-ele
tron intera
tions are taken into a

ount. It
F F
pling regime in the sense that ≪ , where is is worth emphasizing that the SW transformation ne-
the Fnermi energy. Furthermore, the nu
lear snpin den- gle
ts retardation e(cid:27)e
ts. This is appropriate sin
e the
s e
sity is far larger than the ele
tron density . It is the nu
lear spin dynami
s is slow
ompared to the ele
-
wpeorratthurneotTi
Kin≈g tDhaetxpth(−e EsiFng/lAe)n(uw
lietharDspbineiKngontdhoe teelem
-- tfaro
tntohnaet(Ain≪terEmFs)o.fTenheerrgeyfosr
ea,leelse
tthriosniss sreeelataendatlomtohset
tronbandwidth)isextremelysmall
omparedtoallother stati
nu
learspinba
kground,andtheadiabati
approx-
energy s
ales. We are therefore far away from the so- imation (for the
ondu
tion ele
trons) is well justi(cid:28)ed.
44

alled
ontroversial exhaustion regime where the indi- Inthe
aseofferromagneti
semi
ondu
tors,su
han ap-
vidual s
reening of the impurity
ompetes with indire
t proximationbreaksdownandretardatione(cid:27)e
tsmustbe
magneti
ex
hange between the nu
lear spins. taken into a

ount.45 If we also assume spin isotropy in
χ (q,ω 0) =δ χ (q) χ (q)
αβ αβ s s
the 2DEG, then → , where
is the isotropi
ele
tron spin sus
eptibility in the stati
In this low ele
tron density regime, the ground state limit.
ofthemagneti
system(herethenu
learspins) hasbeen In real spa
e, the e(cid:27)e
tive nu
lear spin Hamiltonian
shown to be ordered ferromagneti
ally in 3D using var- reads
ious treatments that go beyond mean (cid:28)eld theory and H = 1 Jαβ IαIβ,
whi
h notably39in
lude spin wave modes (but negle
t e-e eff −2 r,r′ r−r′ r r′ (5)
intera
tions). X
4
where Let us now in
lude ele
tron-ele
tron intera
tions. It
Jαβ = (A2/4n )χ (r), is
onvenient to introdu
e a relativisti
notation with
|r| − s αβ | | (6) p¯ ≡ (p0,p) being the (D+1)-momentum where p0 de-
p D
I
r notes the frequen
y and the -dimensional wave ve
-
is the e(cid:27)e
tive ex
hange
oupling. The nu
lear spins D = 2
tor (here ). In a zero-temperature formalism, the
arethereforeintera
tingwithea
hother,thisintera
tion
46
sus
eptibility
an be written diagrammati
ally as
being mediated by the
ondu
tion ele
trons. This is just
21
the standard RKKY intera
tion, whi
h, however, as i
we shall see,
an be substantially modi(cid:28)ed by ele
tron- χs(q¯)=−LD σσ′Gσ(p¯1−q¯/2)Gσ(p¯1+q¯/2)Λp¯1σσ′(q¯),
ele
tron intera
tions
ompared to the free ele
tron
ase. p¯1X,σ,σ′
σ,σ = L = aN1/D (10)
′
with ±, and where is the system
III. ELECTRON SPIN SUSCEPTIBILITY IN A Gσ(p¯1)
length, is the exa
t single-parti
le Green's fun
-
2D INTERACTING ELECTRON GAS Λ(q¯)
tion and is the exa
t vertex fun
tion, whi
h
an be
Γ(q¯)
expressedintermsoftheexa
ts
atteringamplitude
Themainresultofthepreviousse
tionisthatthemag- 46
as follows:
neti
ex
hange intera
tion between the nu
lear spins is
mediatedbytheele
trongas. Therefore,thekeyquantity Λp¯1σσ′(q)=δσσ′ − (11)
governing the magneti
propertiesχosf(qth)e nu
lear spins LiD Γpσ¯1σ,σp¯2′σ′(q¯)Gσ′(p¯2+q¯/2)Gσ′(p¯2−q¯/2).
is the ele
tron spin sus
eptibility in two dimen-
Xp¯2
sions. The
al
ulation of this quantity in an intera
ting
2DEG has been the subje
t of intense e(cid:27)orts in the last This s
attering amplitude plays a
ru
ial role as we see
Γ
de
ade in
onne
tion with non-analyti
ities in the Fermi next. isforagenerals
atteringeventafun
tionoffour
liquid theory.25,26,27,28,29,30,31,32,33,34 On a more funda- spin variables σ1,σ1′,σ2,σ2′. Nevertheless, one
an use a
mental level, in
orporating e-e intera
tions in the
al
u-
onvenientparametrizationwhi
hensuresrotationalspin
lations of thermodynami
quantities has been an impor- invarian
e,46
tant area of
ondensed matter theory over the last (cid:28)fty
yioerarosf. thInermpaordtiy
nualamri,
tqhueasnttuitdiyesoafnndonsu-asn
eaplyttibi
ilibtieehsavin- Γσp¯11,σp¯1′2σ2σ2′(q¯)=Γ+p¯1,p¯2δσ1σ1′δσ2σ2′ +Γ−p¯1,p¯2τσ1σ1′ ·τσ2σ2′,
ele
tron liquids has attra
ted re
ent interest, espe
ially τ (12)
ifnor2tDhe.2f5o,2l6lo,2w7,i2n8g,29is,30t,h31e,3r2e,
33e,n3t4(cid:28)Onfdipnagrstib
yulCarhuimbupkoorvtaann
de twrhi
eerse(τxis,τayv,τe
zt)oarnwdhτosσe1σ
1′om·τpσo2nσe2′n=tsaa=rex,tyh,zeτPσa1aσu1′lτiσam2σa2′-.
Maslov29, nχasm(qe)ly that the stati
non-uniform spin sus- NotethatΓ± isspin-independentand
orPrespondstothe

eptibilityq = q depenqds liknFearly on the wave vqe2
tor
harge and spin
hannels, respe
tively. Following Ref.
modulus | | for ≪ in 2D (while it is in
3D),withkF theFermimomentum. Thisnon-analyti
ity 4Γ6−, we next write the Bethe-Salpeter (BS) equation for
(
orresponding to the spin
hannel) as follows:
arises from the long-range
orrelations between quasi-
parti
les mediated by virtual parti
le-hole pairs, despite Γ−p¯1p¯2(q¯)=(Γ−irr)p¯1p¯2(q¯)+ (13)
1
the fa
t that e-e intera
tions was assumed to be short-
ranged. LD (Γ−irr)p¯1p¯′′(q¯)Rp¯′′(q¯)Γ−p¯′′p¯2(q¯),
p¯′′
Letus (cid:28)rstre
allthe
aseofnon-intera
tingele
trons. X
χ
s
In this
ase,
oin
ides with thχeLu4s6ual density-density where (Γ−irr)p¯p¯′(q¯) is the exa
t irredu
ible ele
tron-hole
(or Lindhard) response fun
tion ,
s
attering amplitude in the spin
hannel, and
1 n n
k,σ k+q,σ
χL(q)= Na2 ǫk,σ ǫ−k+q,σ+iη, (7) Rp¯(q¯)=−2iG(p¯+q¯/2)G(p¯−q¯/2) (14)
k,σ −
X
47
is the ele
tron-hole bubble.
n ǫ
k k
where is the ele
tronnumber operator, the disper- One
an exa
tly solve, at least formally, the BS equa-
η >0
sion relation,and is an in(cid:28)nitesimal regularization tion (13) using a matrix notation where the matrix in-
p¯ R
parameter. This Lindhard fun
tion
an be evaluated ex- di
es run over . Within this notation is a diagonal
46
a
tly and reads in 2D matrix. We (cid:28)nd that
1
χL(q)=−Ne 1−Θ(q−2kF)pq2−q 4kF2 !, (8) Γ−p¯1p¯2 =Xp¯′′ (Γ−irr)p¯1p¯′′(cid:18)1−Γ−irr(q¯)R(q¯)(cid:19)p¯′′p¯2. (15)
N =n /E
e e F This enables us to derive an exa
t and
losed expression
where is the ele
tron density of states (per
N = m /π m
e ∗ ∗ for the spin sus
eptibility, given by
spin). Note that where is the e(cid:27)e
tive
1 1
ele
tron mass in a 2DEG. It follows from Eq. (8) that χ (q¯)= R(q¯) .
δχL(q)≡χL(q)−χL(0)=0 for q ≤2kF. (9) s LD Xp¯,p¯′(cid:18) 1−Γ−irr(q¯)R(q¯)(cid:19)p¯p¯′ (16)
5
In general, Γ−irr
annotbe
al
ulatedexa
tly and some result has been
on(cid:28)rmed by using a supersymmetri
ef-
32
approximations are required. The approximationwe use fe
tive theory of intera
ting spin ex
itations.
inthefollowing
onsistsinrepla
ingtheexa
tirredu
ible Ontheotherhand,re
entexperimentsonalowdensity
ele
tron-hole s
attering amplitude (Γ−irr)p¯,p¯′ by an aver- intera
tingele
tronsgasinSili
onMOSFETshavefound
χ (T)
s 35
aged value
al
ulated with respe
t to all possible values that | | de
reases at low temperature in apparent
p p
′
of and near the Fermi surfa
e. This is equivalent to
ontradi
tionwithperturbative
al
ulations. Thisrather
the following approximation: puzzlingsituationdemandsanon-perturbativeapproa
h.
(Γ−irr)p¯,p¯′(q¯)≈Γ−irr(q¯) ∀ p,p′. (17)
q = 0 q 2. Beyond lowest order perturbation theory:
0 0
We now assume and suppress the -argument renormalization e(cid:27)e
ts
inwhatfollowssin
eweareinterestedinthestati
prop-
erties of the spin sus
eptibility. The previous
al
ulations give a spin sus
eptibility
whi
hisquadrati
intheba
ks
atteringamplitude. How-
ever,itisknownthatatlowenoughenergytheba
ks
at-
A. Short-ranged intera
tion
tering amplitude be
omes renormalized. One should
thereforeinstead
onsidersometypeofrenormalizedper-
q
In this se
tion, we
onsider a -independent short- turbation theory approximation (RPTA). Shekhter and
ranged intera
tion potential, whi
h
orresponds within Finkel'stein33 argued re
ently that the strong renormal-
our notations to Γ−irr(q)=−U. This approximation
on- izationofthes
atteringamplitudeintheCooper
hannel
δχ (T)
siderably simpli(cid:28)es the BS equation (13) and the formal mayexplainthesignof s intheexperimentbyPrus
χ χ (q)
expression of s in Eq. (16). The derivative of s et al.35
q Γ(θ,T)
with respe
t to
an be expressed in a simple
ompa
t Let us introdu
e , the two-parti
le s
attering
θ
form: amplitude at a parti
ular s
attering angle and tem-
T θ = π
∂χs(q)= ∂Π(q) 1 , perature . The CoΓocp(Ter)
=hanΓn(πel,T
o)rresponds to
∂q ∂q (1+UΠ(q))2 (18) and we denote by the
orresponding
two-parti
le s
attering amplitude. In the Cooper
han-
Π(q) = R (q)/LD q k nel the two parti
les that s
atter have exa
tly opposite
where p¯ p¯ . In the low ≪ F limit, Γc
Π(q) momenta. We (cid:28)rst expand in Fourier harmoni
ssu
h
one
an approxPimate the term in tχhe(d0)en=omiNnator that
L e
ofTEhqe. (m18u)ltbipylii
tastinvoen-fain
tteorra
1t/in(1g−vaUluNe e)2 in Eq−. (1.8) Γc(T)= (−1)nΓc,n(T), (20)
n
signals the onset of the ferromagneti
Stoner instability X
UN
e n
when approa
hes unity. The Stoner instability is
r 20 where isanintegerrepresentingtheangularmomentum
s
supposed to o

ur for very large ∼ a

ording to
r 10 quantum.
48 s
Monte Carlo results. For smaller ≤ , we are still
In order to analyze the temperature dependen
e of
Γ (T) Γ
far from the Stoner instability. Though this multipli
a- c,n c
r , it is enough to write the BS equation for
s
tive term does not play a signi(cid:28)
ant role at small , it
r in a way fully analogous to Eq. (13). In the Cooper
s
in
reases with , showing the tenden
y.

hannel, due to the fa
t that the momentum of the
en-
ter of motion of the two s
attering parti
les is zero, the
integrationovertheele
tron-holebubblegivesrisetolog-
1. Perturbative
al
ulation
arithmi
infrared(IR)divergen
es. Thesedivergen
esare
Π(q) duetoparti
le-holeex
itationsaroundtheFermiseaand
The
orre
tions to the polarization bubble are already o

ur in the ladder approximation (i.e. by
on-
dominatedGby(pt)he (cid:28)rst bubble
orre
tion to the self- sidering only the bare short-rangedintera
tion potential
σ 29
energy of . TheUse
orre
tionsqhave been
al
u- in the BS equation). We refer the reader to referen
es
latedinse
ondorderin inthesmall limitinRef. 29, 49 and 33 for more details. The logarithmi
IR diver-
with the result gen
es
an be absorbed by a res
aling of the s
attering
Γ(π)2 amplitudes in the Cooper
hannel su
h that:
δΠ(q)=Π(q) Π(0) 4qχ | | ,
− ≈− S 3πkF (19) Γ (T)= Γc,n ,
c,n
1+Γ ln(E /k T) (21)
c,n F B
χ = χ (0) Γ(π) Um /4π (2k )
S s ∗ F
where | | and ∼ − is the
UN 1 Γ =Γ (E )
e c,n c,n F
ba
ks
attering amplitude. When ≪ , we re
over where isthebarevalue. Thelogarithmi
δχ (q) = δΠ(q)
s 29 50,51
from Eq. (18) the known result . This fa
torsgiverisetothewellknownCooperinstability.
δχ (q)=αq
s
perturbative
al
ulationthereforegivesthat They are just the 2D equivalent of the one found in the
α < 0 χ (q)
s 52
with , or equivalently that | | in
reases with dis
ussionofKohn-Luttingersuper
ondu
tivity. InEq.
q q
36
at low . Finite temperature
al
ulations along the (21), thelogarithmi
divergen
eis
ut-o(cid:27)bythetemper-
δχ (T)=γT γ <0
s 30,31
same line imply that with . This ature at low energy.
6
Γ (q)
c
Another way of re
overing Eq. (21) is to regard the more spe
i(cid:28)
ally of . Following Ref. 49, one
an
Γ Γ (ξ)
c,n c
ba
ks
attering amplitudes as some energy s
ale de- writeaBSequationatzerotemperaturefor , where
Γ (Λ) Λ ξ
c,n
pendent
oupling
onstants of the 2DEG ( is is an energy s
ale in the vi
inity of the Fermi energy
T E ξ E ξ =v q
F F F
some running energy s
ale, the equivalent of in Eq. . By linearizing around su
h that , one
(21)). The next step is to write renormalization group immediately infers from (21) that
equations (RG) for these
ouplings. These
ouplings are Γ Γ
c,n c,n
Γ (q)= .
marginal and the RG equations at lowest order read: c,n 1+Γ ln(E /v q) ≈ 1+Γ ln(k /2q)
c,n F F c,n F
dΓ
c,n = (Γ )2.
c,n (23)
dln(Λ/E ) − (22)
F
Notethatat(cid:28)nitetemperatureandforarealisti
system,
Λ max v q,k T,∆
F B
Γ (Λ) the infrared
ut-o(cid:27) is repla
ed by { }.
c,n
The solutions of the RG equations for are given
T Λ Thisresultisalsoobtainedfromarenormalizationgroup
33
by Eq. (21) where essentially is repla
ed by .
Γ (T) approa
hby workingdire
tlyin momentumspa
eatlow
c,n k
The low energy behavior of the strongly de- F
energy where the energy
an be linearized around .
Γ (θ,q)
pends on the bare s
attering amplitudes being repulsive c
Γ > 0 Γ < 0 At low enough momentum, is dominated by the
c,n c,n n
( ) or attra
tive ( ). >From Eq. (21), 0
Γ > 0 Γ (T) 0 harmoni
su
h that
c,n c,n
we immediately infer that when , →
Γ
aΓtc,nlow<e0n,erΓgcy,n/(tTem)preernaotrumrea.lizOesnttohethoethsetrrohnagn
do,uwplhinneng Γc(q)≈(−1)n01−|Γc−,n|0|cl,nn(0k|F/2q). (24)
0
regime. Assuming there exists at least one harmoni
Γ < 0
su
h that c,n0 implies that there exists a tempera- Byrepla
ingthebarevalueofthes
atteringampδlχitu(dqe)in
s
teunrteirebleylodwomwihnia
thedthbeys
Γact,tne0r.3in3gin the Cooper
hannelis −Eq4.qN(1e9|)Γ3(bππyk,qF)t|h2earlseonoar
mqualiirzeesdaonnoe,nw-teriv(cid:28)inadl qt-hdaetpenden
≈e.
This reasoningrelieson thenftah
tthat at leastonebare In parti
ular, using Eq. (24), we obtain that
s
attering amplitude in the harmoni
is negative.
dχ 4N
Thisassumption
anbe furthersubstantiatedwith some s e (Γ2 2Γ3 ),
dq ≈−3πk c − | c| (25)
expli
it perturbative
al
ulations of the irredu
ible s
at- F
52,53,54
tering amplitudes. By
omputing the lowestorder Γ (q) >1/2
c

orre
tioΓns
ontributingtotheirredu
ibles
atteringam- whi
h is positive when∆| | .
plitude c,irr,(thereforegoingbeyondtheladderapprox- Letusassumethat ,theaforementionekdinTfra<red∆
ut-
B L
imation in the BS equation whi
h leads to Eq. (21)) one o(cid:27), is
lose to the CoopΓer(iqn)stability, i.e. ∼ .∆In
c
t
haantaΓ
tcu,nal<ly0p.r4o9vethatthereexisthigherharmoni
ssu
h sUus
ihnga1
/aΓsec,,nt0h=e (cid:29)lno(wEoFf/kBTL)in, oEnqe.o(b2t4a)iniss
tuhtato(cid:27) by .
Note that these
al
ulations imply that a super
on- ( 1)n0+1
Γ − for q <∆/v
Tdu
t<∼ingTLinst=abiElitFyexsph(o−ul1d/|Γdce,vne0l|o)/pkBat.a tTehmispeirsatuenre- c ≈ ln(kB∆TL) ∼ F (26)
tirely analogous to the Kohn-Luttinger me
hanism for
super
ondu
tivity.52,55 Nevertheless, in a typi
al 2DEG, is renormalizedqtowa1rd0qlarge valueqs. =ThkisTim/pvlies that
0 L L B L F
disorderorsomeotherme
hanismmayprovideanatural there dexδiχsts(qa) > 0∼, q <(wqhere d δχ (q) < 0) su
h
infrared
ut-o(cid:27) preventingth∆is>supTer
ondu
tinginstabil- qth>atqdq s ∀ 0, and dq s when
L 0
itytoberea
hed. Letus∆
all thisinfrared
ut-o(cid:27). ∼ .
It is worth noting that should not be too large either We must point out that the RG equation in Eq. (25)
Γ Γ n =
in orderto let c,n0 (cid:29)ow to relative large values of order assumes that all other s
attering amplitudes n for 6
n
0
one for this me
hanism to be relevant.
an be negle
ted below some low energy. In order to
As shown in Ref. 33 this me
hanism, if it takes pla
e, go beyond this approximation, onehas to solve(numeri-
leads to a non-monotoni
behavior of the temperature
ally) a set of RG di(cid:27)erential equations instead.
∆
dependen
e of the ele
tron spin sus
eptTibility and more k TThis s
enario relies on a (cid:28)neTtun<ing∆o/fk
<om10pTaredto
0 B L L B L
sdpχes
>i(cid:28)
0ally to the existen
e of a s
ale , below whi
h (typi
ally this demands ∼ δχ (q)).<I0f
dT (note that we use a di(cid:27)erent sign
onvention for this
onditionisnotmet,onemaythenexpe
t s
χ (0)
s

ompared to Ref. 33). By (cid:28)tting the experimen- in a

ordan
e with lowest order perturbative
al
ula-
tal data of Prus et al. with su
h a theory, the estimate tions. Nevertheless, one should mention that an alter-
Γ 0.25 0.3 δχ (T)>0 T
c,n0 ∼ − wasobtainedinRef. 33, whi
himplies native theory, also giving s at low , has been
T 10 K
0 34
∼ , a surprisingly large temperature s
ale of the put forward. This alternative s
enario applies for van-
E 40 K
F
order ∼ in this experiment. It is worth em- ishingCooperamplitudes. Insu
ha
ase,theanomalous
phasizingthat su
has
ale, being dependent on thebare temperaturedependen
eofthespinsus
eptibilityisdom-
Γ
value of c,n0, is non-universal and therefore sample de- inated by non-analyti

ontributions from parti
le-hole
34
pendent. res
attering with small momentum transfer. Whether
This RPTA also raises a similar issue
on
erning the su
h a s
enario, not
onsidered here, also implies that
χ (q) δχ (q)>0
s s
behavioroftheele
trostati
spinsus
eptibility and is an interesting but open question.
7
0 2 4 q/k
q0 q1 F
The aforementioned
onsiderations immediately raise
q
0
the issue about the amplitude of in a typi
al inter-
a
ting 2DEG. In order to des
ribe the temperature de-
χ (T)
s
penden
eof ,ShekhterandFinkel'steindetermined
Γ
n0 su
hthattheexperimentalbehaviorofRef. 35isre-
produ
ed. This(cid:28)xesthevalueofthisparameterandalso 1 (a)
k T E
the s
ale B 0 to the order of F. One may therefore − (b)
q k
expe
t 0 to be of the order of F for a similar 2DEG. (c)
Another independent way of substantiating these esti- χ¯s
mates is to go beyond the ladder approximation in the
49
BS equation. First, as we mentioned before, this al-
n
Γlocw,ns0o<ne0t.oSseh
oowndtq,htahtisth
earne geixviestussaavnaleusetim0ataetfworhiT
Lh Fχ(asI)G(,q.)(/b1|)χ:sa(n0Td)|h(r
e)e foprostshibelenoqrmuqaa/llikitzFaetdivesuss
heappteibsilidtyenoχ¯tse(dq)b=y
0 as a fun
tion of (dashed lines)
ompared
and therefore for . Indeed, in Ref. 49 the Cooper q0
to the non-intera
ting value (thin full line). Here, and
instabiklBityTLhas bEeFeen−e1s6tEiFm/a(rts3eξd) to set inξ at ∆the temper- q1 arethepositions of theextremafor
urves (a) and (b). In
ature ∼ , where ∼ plays the

ontrast,thelo
al(cid:28)eldfa
torapproximationdis
ussedinSe
.
role oξ/fEthe inf1rared
ut-o(cid:27) in Ref. 49, and the
ondi- IIIB results in a monotoni
in
rease of χ¯s(q) (not sket
hed
F
tTion ∆ ≪E was assurmed. Consisqten
ykrequires that in the (cid:28)gure), being always larger than the non-intera
ting
L F s 0 F
∼ ∼ at large , implying ∼ . This is in value.
agreement with our previous estimate.
B. Long-ranged Coulomb intera
tions
The previous
al
ulations also give us information
χ (q)
s
aizbaotuiotnthine pthosesibCloeosphearpe
hsafnonrel is i.mpWorhteanntr,enwoermoabl-- In tqhe pre
eding se
tion, we repla
ed Γ−irr(q) by an al-
tain at least one extremum around some wave ve
tor most −independent
onstant operator, assuming that
q O(k ) q
0 ∼ F . Furthermore, at large , we should re- the Coulomb intera
tion was s
reened and, therefore,
χ (q)
χ
ov(eqr)then0on-inqtera
tingbehaviorχan(qdt)h<ere0fore s ∼ short-ranged. Let us
onVsi(dqe)r=in2πthei2s/qse
tion thee bare
L → for → ∞. Be
ause s 0 , we expe
t 2D Coulomb intera
tion, , where is the
q > q
another extremum around a value 1 0. Sin
e the ele
tron
harge.
q 2k
F
non-intera
ting behavior is re
overed for ≫ , one
q O(2k )
1 F
may suspe
t ∼ . From the previous
onsider-
1. Lo
al Field Fa
tor Approximation
ations we therefore
on
lude that there exist (at least)
χ (q)
s
twoextrema for the ele
tronspin sus
eptibility . It
is worth emphasizing that this double-extremum stru
- One ofthemostsu

essfulapproximationsforthe
al-
ture is a dire
t
onsequen
e of the nontrivial renormal-
ulation of ele
tron response fun
tions is the lo
al (cid:28)eld
ization of the s
attering amplitude in the Cooper
han- fa
tor approximation (LFFA). It improves the random
46
nel. We have s
hemati
ally drawn in Fig. 1 the possible phase approximation for whi
h the e(cid:27)e
tive (cid:28)eld seen
χ¯ (q) =
s
qualitative shapes denoted by (a) and (b) of by an ele
tron is the (cid:28)eld that would be seen by a
las-
χ (q)/χ (0) q/k
s s F
| | as a fun
tion of and
ompared it to si
al test
harge embedded in the ele
tron gas. The idea
χ (q)/N T = 0
L e
the (normalized) non-intera
ting at . oftheLFFA to
orre
ttherandom phaseapproximation
χ (q ) > χ (0)
s 2 s
In the
ase denoted by (a), we
hoose , and to better a

ountfor the
orrelationsexisting in the
whereas in the
ase denoted by (b) the absolute value ele
trongas,istorepla
etheaverageele
trostati
poten-
q 2k
2 F
of the sus
eptibility at ≃ is
hosen to ex
eed the tialbyalo
al(cid:28)elde(cid:27)e
tivepotentialseenbyanele
tron
χ (q )<χ (0) σ
s 2 s
stati
value, i.e. . The previous
onsider- withspin whi
hispartofthe2DEG. (We referto Ref.
G
ations donot allowus to dis
riminate betweenthesetwo 46forareview). Thelo
al(cid:28)eldfa
tor −
anbede(cid:28)ned
χ (q) r
s s
possible shapes of . Furthermore, by in
reasing , as follows:
χ
s

an evolve from one shape to another. G−(q)V(q)=χ−s1(q)−χ−L1(q), (27)
On the other hand, if the renormalization in the χs
or, equivalently, the stati
spin sus
eptibility
an be
Cooper
hannel does not take pla
e, e.g. when it is
ut
written as
o(cid:27)bydisorder,thentheperturbative
al
ulationsatlow-
δχ (q) < 0 χ (q)
s L
est order apply and give instead that at low χ (q)= .
q.29,30,31Apossibleshapeforχs(q),
onsistentwiththese s 1+V(q)G (q)χL(q) (28)
−

al
ulations, has been drawn in Fig. 1 and
orresponds
G (q) q
to label (
). We should note, however, that the e(cid:27)e
t The pre
ise determination of − for all is an open
of res
attering of a pair of quasiparti
les in all di(cid:27)erent problem. However, the asymptoti
regimes, parti
ularly
q 0
34

hannels should be
arefully examined and may still the → limit, are quite well established be
ause they
46
lead to shapes (a) or (b) in Fig. 1. arestrongly
onstrainedbysum rules. Inthiswork,we
8
useasemi-phenomenologi
alinterpolationformulagiven given in Eq. (29). In su
h a modi(cid:28)ed lo
al (cid:28)eld fa
tor
g
1
in Ref. 46 approximation(MLFFA), is approximately given by
q
G (q) g .
− ≈ 0q+g0(1−χP/χS)−1κ2 (29) g1 ζ1 = ζ1 χS 1 ,
≈ r (α 1) r χ − (32)
(gµ ) 2χ µ s − s (cid:18) P (cid:19)
B − P B
Here, isthePaulisus
eptibility( theBohr
magneton and χP > 0), χS = |χs(q0)=| t0heκren=orkmarliz√ed2 with ζ1 some numeri
al
onstant of order one.
2 F s
value of the spin sus
eptibility at , g We shouldmention that someother more
ompli
ated
0
is the 2D Thomas-Fermriw=av0eve
tor, and is the pair- analyti
al (cid:28)ts of the QMC data have been obtained in

orrelation fun
tion at , des
ribing the probability Ref. 61. Neverδtχhe(lqe)ss, wqe2note that the (cid:28)ts used in that
of (cid:28)nding two ele
trons (of opposite spins) at the same paper lead to s ∼ for 2D, whi
h is in
ontradi
-
posiGtionintheele
trongas. Thisphenomenologi
alform tionwithallpreviousapproximations. Itseemsdesirable
for − has been modi(cid:28)ed from the one originally pro- to test Eq. (32) with more detailed QMC
al
ulations.
56
posed by Hubbard in orderto satisfy exa
tly the
om-
46,57
pressibility sum rule. The main weakness of this ap-
G
proa
h is the arbitrariness ofχth/eχ
ho=sen1 form for −.
For non-intera
ting ele
trons P S . An approxi- C. Comparison of the various approximation
g
0 s
hemes
mateform for givingagoodagreementwith quantum
Monte Carlo (QMC)
al
ulations has been proposed re-
58

ently by Gori-Giorgi et al. and reads: If we summarize the various approximation s
hemes
g0(rs)≈(1+Ars+Brs2+Crs3)e−Drs/2. (30) presentedinthepreviousse
tions,whi
hareperturbative
or semi-phenomenologi
al, we
an
learly as
ertain that
δχ (q) q q q
A = 0.088, B = 0.258, C = s ∝ for ≪ F. Nevertheless, the sign of the
The parameters
0.00037, D = 1.46 proportionality
onstant depends on the approximation
are (cid:28)tting parameters reprodu
ing
g0 58 s
heme we used.
QMC results for in a 2DEG. This approximation
Lowest order perturbation theory in the intera
tion
yields dχ (q)/dq < 0 q
s 29,30
q+g κ α strength leads to at low . However,
0 2
χs(q) Ne for q <2qF, within the RPTA, renormalization e(cid:27)e
ts in the Cooper
≈− q+g κ (α 1) (31)
0 2 33,49
−
hannel are important and
hange the pi
ture given
α = χ /(χ χ ) by lowest order perturbation theory. In this latter
ase,
S S P dχ (q)/dq
where −
an be regarded as a Fermi s
q the RPTA yields an opposite sign for below
q
liquid parameter. The low- semi-phenomenologi
al ap- 0
some wave ve
tor . The LFFA we used implies that
dχ (q)/dq > 0 q
proximation for the edlχes
(tqr)on>sp0i,n squs
eptibility given in s for all and therefore a monotoni
be-
Eq. (31) results in dq ∀ , in
ontrast to the havior (not shown in Fig. 1), whereas the RPTA leads
29,30,30
lowest order perturbative
al
ulations. to a non-monotoni
behavior (see Fig. 1). Establish-
G (q)
Notethatadire
testimateof q− byre
entQMCin ingami
ros
opi

onne
tionbetweenthesetwodi(cid:27)erent
a2DEGgivesaqn<al2mkostlinearin behavioruptorather approa
hesis obviouslyarather di(cid:30)
ult and open issue.
F
largevaluesof ∼ ,follow2kedbyamore
omplexnon- The LFFA is a semi-phenomenologi
al approximation
F
monotoni
qbehav4io6,r59a,6r0ound , anqd (cid:28)nally diverges in iGnw(hq)i
hananalyti
alexpressionforthelo
al(cid:28)eldfa
tor
the large- limit. This large- limit is not repro- − is (cid:16)guessed(cid:17) with the
onstraints that the asymp-
du
ed by Eq. (29). This is not a serious drawba
k sin
qe toti
behavior should reprodu
e some known results in-
most quantities of interest are dominated by the low- ferredfromexa
tsumrules. Theunknownparametersof
G (q)
regime. q = g κ (α 1) − are(cid:28)xedfroma(cid:28)ttoQMCdata.61Onemaywon-
∗ 0 2
However, the s
ale − de
reases expo- derwhetherone
anextra
tsomeinformation about the
nentially with rs a

ording to Eq. (30). This would im- possible shapes of χs(q) dire
tly from the original QMC
G (q)
pqly an almost
onstantbehaviorfor − ex
ept at low data. The QMC data showsGa (rqa)ther
omp2lik
ated stru
-
. When we
ompare this behavior with available QMC ture with two extrema for − around F (see Ref.
46,59
data, we (cid:28)nd that there is a manifest
ontradi
tion. 59 or Ref. 46 (p. 244) ). Though it might be tempting
Therefore, this raises some doubt about the presen
e of torelatethedouble-extremumstru
tureobtainedby the
g
0
(a short distan
e quantity) in Eq. (29). RPTA to the QMC results, it turns out to be not possi-
χ (q)
s
ble to extra
tthe behaviorof from availableQMC
G (q)
dataχfo(rq) − . NewGQM(qC)
al
ulationsdire
tly
omput-
2. Modi(cid:28)ed Lo
al Field Fa
tor Approximation ing s instead of − are thus highly desirable.62
q
g If we insteagdκre(pαla
e1)g0=ignrEq√.2((α29)1b)yka pa2rakmeter δχFs(iqn)allmy,imwie
shtahvee tseemenpetrhaatturtehedelopwendend
eepeonfdeδnχ
se(To)f
1 1 2 1 s F F
, su
h that − − ≫ , we whi
h is in agreement with the experiment by Prus et
G (q) T
35
have
he
kedthatthesQMCd2aktafor − aremu
hbet- al. at low . These experimentalfeaturesmay provide
′ F
ter reprodu
ed for q up to than by the expression another,thoughindire
t,
onsisten
y
he
koftheRPTA.
9
IV. MAGNETIC PROPERTIES OF THE 1. Magnetization and Curie temperature of the nu
lear
NUCLEAR SPINS spins
65
Weassumeinthisse
tionthatsomenu
learspinorder- >From standard spin waves analysis , the dispersion
ing a
tually takes pla
e at low enough temperature and relationofthespinwavesintheferromagnetsimplyreads
A2
analyzehowthis orderingisdestroyedwhen the temper- ω =I(J J )=I a2(χ (q) χ (0)),
q 0 q s s
ature is raised. − 4 − (34)
J J
q r
where is the Fourier transform of de(cid:28)ned in Eq.
(6).
At this stage, we already see that the stability of
A. Mean (cid:28)eld approximation δχs(q) =
the ferromagneti
ground state demands that
χ (q) χ (0) > 0
s s
− . We
an therefore
on
lude that the
Sin
etheintera
tionbetweennu
learspinsisofRKKY se
ond order
al
ulation implies that the ferromagneti
36
dtyp≪e,ktF−h1e (inthteerala
rtgioenqisbefehrarvoimoragonfeχtis
(qa)tisshoorntlydiwsteaank
lye gthroeurnednosrtmataeliziesdaplweratyusrbuantsitoanblteh.eorOynaptphreoxoimthaetriohnandde-,
modi(cid:28)ed by e-e intera
tions). Furthermore, many mean veloped in Se
. IIIA2 shows that it is ne
essary to go
(cid:28)eld
al
ulations performed for the 3D Kondo latti
e at beyond lowest order perturbation theory.
low ele
tron density (negle
ting e-e intera
tions though) When renormalizatione(cid:27)e
ts arenotimportantin the
39,64
predi
t a ferromagneti
ordering. Assuming a low RPTA, the lowest order perturbative results are re
ov-
temperature ferromagneti
ordering of the nu
lear spins ered,and the ferromagneti
groundstateseems unstable
seems therefore a reasonableassumption. (though renormalization e(cid:27)e
ts in all
hannels must be
We (cid:28)rst re
all the mean (cid:28)eld results for
arefully taken into a

ount as des
ribed in Ref. 34).

ompleteness.20 The Weiss mean (cid:28)eld theory gives When renormalization e(cid:27)e
ts in the Cooper
hannel are
a Curie temperature important, we expe
t the two possible shapes χde(nqo)ted
s
by (a) and (b) for the stati
spin sus
eptibility at
T =0
I(I+1) A2 (see Fig. 1). If the
ase (b) is favored, then there
TcMF =− 3k 4n χL(q =0), (33) existsavalueofqatwhi
hωq <0signalinganinstability
B s
of the ferromagneti
ground state. >From this perspe
-
I tive,
ases(b) and (
) are similar. Another groundstate
where is the nu
lear spin value. must then be assumed and a subsequent analysis is re-
TMF
In 2D this mean (cid:28)eld theory yields for c a depen- quired. This will be detailed in Se
. IVD.
n /n
e s
den
e on the ratio . For a metal with about one On the other hand, if the shape of the sus
eptibility
n /n 1
e s

ondu
tionele
tronpernu
learspin,theratio ∼ , denoted by (a) is favored, the ferromagneti
assumption
and we re
over the result derived more than sixty years isself-
onsistent. The RPTA predi
ts that thereexists a
T dχ /dq < 0 q
ago by Fröhli
h and Nabarro for a 3D bulk metal.22 temperature 0 above whi
h s at low . This
T T
1 0
kFBorTca=2DI(Im+et1a)lAt2h/e12WEeFiss mean (cid:28)eld theory then gives implies that there exists aqn1other temperature ≤
. Fora2Dsemi
ondu
tor,how- at whi
h the minimum in tou
hes the horizontal axis
T
c
ever, the smaller Fermi energy is
ompensated by the signaling an instability. If the Curie temperature is
ne/ns 1 T1
smaller ratio ≪ . With typi
al values for GaAs larger than , then there exists a temperature regime
heterostru
tures, I = 3/2, A ∼ 90 µeV and a ∼ 2Å,14 (typi
ally for T > T1) where the ferromagneti
ground
T 1 µK
c
we estimate ∼ , whi
h is very low. (For su
h state be
omes unstable and a di(cid:27)erent ordering may be
T
c
low 's,ignoringnu
leardipole-dipoleintera
tionsfrom favored. This
ase will be analyzed in Se
. IVD. On
T
c
the start would not be legitimate.) However, this esti- the other hand, if the Curie temperature is smaller
T
1
mateisjustbasedonthesimplestmean(cid:28)eldtheoryand, than , theferromagneti
groundstateisself-
onsistent
T
c63
moreover, does not in
lude the e(cid:27)e
t of e-e intera
tions. below . Su
has
enarioisina

ordan
ewiththeone
T
c
It still leads to a (cid:28)nite under whi
h the nu
lear spins obtainedfrom LFFA.Let usthereforeanalyzethis latter
order ferromagneti
ally.
ase.
m
The magnetization per site for a ferromagnet at (cid:28)-
T
nite is de(cid:28)ned by
1 1 1
m(T)=I n =I ,
B. Spin wave analysis around ferromagneti
− N q q − N q eβωq −1 (35)
X X
ground state
n
q
where isthemagnono

upationnumberandthesum-
Weshallnowgobeyondtheabovemean(cid:28)eldapproxi- mationisoverthe(cid:28)rstBrillouinzoneofnu
learspins. In
mation and perform a spin waveanalysis. The
olle
tive the
ontinuum limit this be
omdeqss 1
low-energy ex
itations in a ferromagnet are then given m(T)=I a2 .
− (2π)2eβωq 1 (36)
by spin waves (magnon ex
itations). Z −
10
T
c
We de(cid:28)ne the Curie temperature as the temperature istheCurietemperature. Notethatwiththeseestimates
at whi
h the magneti
order is destroyed by those spin one has
waves. Thispro
edureisequivalentto the Tyablikovde- T∗ aq∗
= 1 .
66

ouplings
heme. AnotherwayofdeterminingtheTCurie Tc 2 3I/π ≪ (42)
c
temperature is to analyze at whi
h temperature the
sTpcinmwayavbeeantahleynsisdeb(cid:28)renaekdsbdyowmn(.TTc)he=Cu0r,iewtheim
hpe
raantubree Su
ωhq a dceq(cid:28)nitionqof Tcphas been obtained assuming
that ≈ for all . This approximation has two ma-
written as
jor aspe
ts: First it regularizes naturally the integral in
a2 dq 1
1= . Eq. (37) in tωhe UV limit, se
ondonly thelowenergyde-
I (2π)2eωq/kBTc 1 (37) penden
e of q istaken into a

ount, whi
his
onsistent
Z −
with a spin wave approximation.
χ (q) χ (0)=0
s s
For non-intera
ting ele
trons in 2D, −
q < 2k k
F 46 F
for , where is the Fermi wave ve
tor.
Tc = 0 2. Alternative UV regularization s
hemes
The spin wave analysis, therefore, gives . This
is in agreement with a re
ent
onje
ture extending the ω cq
Mermin-Wagner theorem23 for RKKY intera
tions to a Inthqe previousse
tion, wehaveassumedthat q ≈
non-intera
ting 2D system.24 For intera
ting ele
trons, for all . ωOn the othqer hand, one
an assume we know
q
however,thelongrangede
ayofthe RKKY intera
tions eqx<pliπ
/italy for all in the (cid:28)rst Brillouin zoδχne(iq.)e. for
s

an be altered substantially and no
on
lusion
an be despωiteonlytheasymptoti
limitsof q (and
q
drawn from the Mermin Wagner theorem or its exten- therefore of ) are well established. At large , we ex-
sions. pe
t ele
tron-ele
tron intera
tions to play a minor role
Let us now in
lude ele
tron-ele
tron intera
tions (ob- and the ele
tron spin sus
eptibility to be well approxi-
tainedeitherbytheRPTAorLFFA).Allapproqximationqs m1/aqt2ed by its non-intera
ting value, wδχhsi
(qh)de
reχass(e0s)a=s
iim.e.pltyhathtaωtqth≈ecmq,awgnhoenredispersion is linear in at low , χS at(lsaeregEeqq.an(8d))t.haTthtihseiaminptelig0ersatlhinatEqs. (36≈)o−r(37)are
a
tually diverging when → . If we adopt su
h a pro-
A2 ∂χ (q)
c=I a2 s ,
edure,theintegralinEq. (37)isfullydominatedbythe
4 ∂q (38)
(cid:12)q 0 short-distan
e modes, i.e. by the UV
ut-o(cid:27) (and there-
(cid:12) →
(cid:12) fore independent of any e-e intera
tions). Su
h a regu-
(cid:12)
67 larization s
heme is not very satisfying and furthermore

an be regarded as the spin wave velo
ity. Su
h lin-
even in
onsistent for a spin-wave approximation whi
h
ear spin wave behavior is usually asso
iated with anti- T
c
relies on the long-ranged modes. Note that the we
ferromagnetswhile onewouldexpe
t aquadrati
disper-
obtain with su
h pro
edure is similar (up to a prefa
tor
sion for ferromagneti
ally ordered states like
onsidered
oforderunity)totheCurietemperatureobtainedwithin
here. Thissomewhatunexpe
tedlineardispersion
omes
the mean (cid:28)eld theory in Eq. (33).
purely from ele
tron-ele
tron intera
tions.
Another regularization s
heme
onsists in
utting o(cid:27)
Theperturbative
al
ulationsortheirextensionstoin- q < 2ζ k ζ
F F F
the integral in Eq. (37) to with a
on-

ludetheCooperpairinstabilityallowsustoextra
tonly 1
q δχ (q)
s stant larger than . This
an be justi(cid:28)ed by integrat-
thelow asymptoti
behaviorof . MonteCarlore-
sults,however,seemtoindi
atethatthelo
al(cid:28)eldfa
tor ing out fast modeχs(qd)ire
tly at the 1H/aqm2 iltonqian lekvel
G (q) q q O(2kF)46 in Eq. (3) sin
e de
reases as for ≫ F.
− is almost linearωin ucpq to q∼<q O.(k W) e will
q ∗ F Su
h a reasoning is equivalent in real spa
e to a de
i-
therefore assume that ≈ for ∼ .
T <T
This implies that for ∗, where (mζataiokn)pr1o
ed(uζreakinw)h1i
hasquareplaquette
ontaining
F F − F F −
× nu
lear spins is repla
ed by an-
T =cq /k ,
∗ ∗ B otherplaquettewithasingleaveragespin. Sin
eatshort
(39)
distan
e, the RKKY intera
tion is mainly ferromagneti
m
the integral determining in Eq. (36) is entirely domi- this is equivalent to a mean (cid:28)eld pro
edure. The long
natedbythelineardispersionbehavior. Sin
efastmodes distan
e intera
tion is not substantially modi(cid:28)ed. The
q q
w
oerr
easnpqoenadsiilnyg
otompu≫te it∗aassruemexinpgonωeqnltiinaellayrsinupqpfroersstehde, Umπ/Vaain
ue(cid:27)t-eo
(cid:27)toinftEhiqs.in(t3e7g)raitsionnowoveorffoarsdtemro2dkeFsiisnsthteaatdthoef
whole range (extending the upper integration limit to . Although su
hTa pro
ωedure does not allow fo2rkan
c q F
in(cid:28)nity). We obtain exa
t
al
ulationof sin
e isnotknownaround ,
m(T)=I 1 (T/T )2 for T <T , this
onsiderably boost the Curie temperature by orders
c ∗
− (40) of magnitudes
ompared to the previous regularization
(cid:2) (cid:3) s
heme and in the same range as the Curie temperature
where determined from Eq. (41).
T = 2c 3I = A2I 3I ∂χs(q) In the folloTwcing, we will therefore use Eq. (41) as our
c k a π 2k πn ∂q (41) de(cid:28)nition of . This has the advantage of providing Tus
B r Br s (cid:12)(cid:12)q→0 with a simple
losed formula. Furthermore, su
h a c
(cid:12)
(cid:12)