Enhancement of the Kondo effect through Rashba spin-orbit interactions

0.38 MB·English

by  Mahdi Zarea

Enhancement of the Kondo effect through Rashba spin-orbit interactions
Mahdi Zarea,1,2 Sergio E. Ulloa,2 and Nancy Sandler2
1Department of Chemistry, Northwestern University, Evanston, Illinois 60208
2Department of Physics and Astronomy, Nanoscale and Quantum Phenomena Institute,
and Condensed Matter and Surface Science Program,
Ohio University, Athens, Ohio 45701
(Dated: January 21, 2013)
We study a one-orbital Anderson impurity in a two-dimensional electron bath with Rashba
spin-orbit interactions in the Kondo regime. The spin SU(2) symmetry breaking term couples
the impurity to a two-band electron gas. A Schrieffer-Wolff transformation shows the existence
of the Dzyaloshinsky-Moriya (DM) interaction away from the particle-hole symmetric impurity
2 state. A renormalization group analysis reveals a two-channel Kondo model with ferro- and anti-
1 ferromagnetic couplings. The parity breaking DM term renormalizes the antiferromagnetic Kondo
0 coupling with an exponential enhancement of the Kondo temperature.
2
PACSnumbers: 72.10.Fk,71.70.Ej,72.80.Vp
n
a
J
The search for new materials and architectures appro- while some works supported Gainon and Heeger’s find-

1
priate for spintronic and quantum computing devices [1] ings, others reached opposite conclusions. Noting that
3
has renewed the interest on the study of Rashba spin- SO interactions preserve time-reversal symmetry, Meir

] orbit interactions (RSO) [2] in low-dimensional systems. andWingreen[20]showedthattheKondoregimeisunaf-
el For systems on surfaces, the natural lack of inversion fectedbySOinteractionsintheinfiniteHubbard-Ulimit.
- symmetry makes RSO ubiquitous, and particularly rel- More recently, however, a solution for a two-dimensional
r
t evant for studies of surface-related magnetic properties Kondo model in the presence of SO interactions predicts
s
. [3]. Research on this area has been driven by rapid that the Kondo temperature increases by a multiplica-
t
a developments of STM techniques that have made pos- tivefactorproportionaltotheSOcouplingconstant[21].
m sible the design and manipulation of atomic structures Thus, the question: what is the role of SO interactions
- on surfaces, and study charge and spin physics in two- in the Kondo regime for two dimensional systems? re-
d
dimensional (2d) systems. Among the many structures mainscontroversial. ThepurposeofthisLetteristopro-
n
of interest, several groups have focused on the physics of vide a definitive answer by presenting a solution to the
o
c isolatedmagneticimpuritiesonmetallicsubstratestoin- model of a one-orbital Anderson magnetic impurity [22]
[ vestigate signatures of Kondo physics in two dimensions embedded in a two-dimensional metallic host with RSO

[4–14]. While these studies have unveiled new physics, interactions. Weshowthatthepresenceoftheseinterac-
2
v they have not addressed the relevance of spin-orbit (SO) tions reduces the Anderson Hamiltonian to an effective
2 interactions possible on many metallic substrates serv- two-band Anderson model coupled to the impurity. By
2 ing as reservoirs for the magnetic impurity. The fact an appropriate Schrieffer-Wolff transformation [23], the
5
that these interactions can profoundly modify the spin Hamiltonian reduces to an effective two-channel Kondo
3
structureonasurfacewasdemonstratedinspin-polarized model plus a Dzyaloshinsky-Moriya (DM) [24] interac-
.
5 measurements of Mn impurities on W substrates [15]. tion term. The renormalization group analysis for this
0
Images and characterization of beautiful chiral spin or- effective Hamiltonian reveals that the impurity couples
1
dered structures are indeed understood in terms of the to the bath with ferro- and antiferromagnetic couplings,
1
: strong SO coupling on the surface. Moreover, recent in- with the one-channel Kondo model as the fixed point at
v
vestigations of magnetic impurities on graphene [16, 17], low energies. More surprisingly, as we show below, the
i
X the ultimate two-dimensional system, pose the question presence of DM interactions, which vanish at half-filling
r of the influence of the underlying graphene substrate on andattheHubbardU-infinitylimits,introducesanexpo-
a
measured magnetic properties. nential increase in the value of the Kondo temperature.
The model. The Anderson Hamiltonian of a two di-
The role of SO interactions in the Kondo regime of
mensional electron gas in the presence of RSO interac-
magneticimpuritiesembeddedinmetallichostshasbeen
tions is given by H = H +H +H +H . Here
atopicofdebatesincetheearlymeasurementsofelectri- 0 U hyb RSO
H = (cid:80) ε c† c + (cid:80) ε c† c describes free elec-
cal resistivity in Cu:Mn compounds doped with Pt, car- 0 ks k ks ks s d ds ds
tronswithmomentumk andspins, andonespherically-
ried out by Gainon and Heeger [18]. The reduced diver-
symmetric impurity level with energy ε . The Hub-
gence in resistivity, interpreted as due to the presence of d
bard interaction term is given by H = Un n with
SO scatterers, was taken as evidence for the suppression U d↑ d↓
of the Kondo effect. Much theoretical and experimen- nds = c†dscds as the impurity electron density. The hy-
talactivityfollowedwithratherinconclusiveresults[19]: bridization term H = (cid:80) V c† c +h.c. describes
hyb ks k ks ds
2
the coupling between the impurity and bath electrons in and 4, we find S to be given by [32]:
theabsenceoftheRSOinteraction. Weconsiderspin1/2
electronsforthebathandtheimpuritysiteand(without S =(cid:88)T (cid:20)(cid:16)c1/2(cid:17)†c +(cid:16)c−1/2(cid:17)†c (cid:21)−h.c.
loss of generality) V to be real and independent of the kh kh d↑ kh d↓
k
kh
electron spin. The RSO interaction is described by:
Tkh =Vk(ndh¯Gkh+gkh)
HRSO = (cid:88)λR(ky+ikx)c†k↑ck↓+h.c. Gkh = ε −1ε −U − ε 1−ε
k kh d kh d
(cid:88) 1
= λRke−iθkc†k↑ck↓+h.c. (1) gkh = ε −ε , (5)
kh d
k
wki=th|(cid:126)kθ|k. Tdehfiisnemdodbeylkaxllo=ws−onkesitnoθskt,ukdyy t=hekecffoescθtko,fatnhde wh¯h=ere−whe. dAefisniendtnhdeh¯u=sunadl↓c(nasde↑,)Sforinhv=olv1e(sht=he−G1)r,eeann’ds
functionsoffreeparticlesmovinginabaththatcontains
RSO with variable strength λ . Notice that the effect of
R
thelocalizedimpuritylevel. Noticethatthistransforma-
localRSOinteractions(actingonlyattheimpurityatom)
tionreducestothestandardformfortheSWT,writtenin
has been the topic of previous studies [25–31] but it is
thechiralbasis,whentheRSOcouplingiszero(λ =0).
not considered in the present model. An experimental R
The SWT reveals that the bath fermions relevant in the
setup where this model is realized consists for example
Kondo regime are:
of isolated magnetic impurities (Co, Mn, etc.) deposited
on the surface of materials with high RSO interactions √
c1/2 = (cid:0)c0 ±c1 (cid:1)/ 2
such as Au(111). k± k↑ k↓
(cid:16) (cid:17) √
Intheangularmomentumbasiselectronoperatorsare: c−1/2 = c−1±c0 / 2 (6)
k± k↑ k↓
m(cid:88)=∞ eimθk (cid:114) k (cid:90)
cks = √2πkcmks ; cmks = 2π dθke−imθkcks. emphasizing the conservation of the total angular mo-
m=−∞ mentum in the z-direction: a spin flip process is com-
(2) pensated by changes in the orbital angular momentum
The canonical transformation: channelsm=0,±1. Oncetherelevantmodesareidenti-
(cid:16) (cid:17) √ fied, it is more convenient to return to the original basis
cm+1/2 = cm +hcm+1 / 2 (3)
kh k↑ k↓ todescribeband-electronsandintroducestandardspinor
notation. The resulting effective Hamiltonian contains
diagonalizes the bath Hamiltonian in the presence of (cid:80)
two different terms: H = (H + H ). The
RSO. Here h = ±1 and j = m+h/2 are the chirality eff k,k(cid:48) K DM
first, equivalent to the standard Kondo Hamiltonian is:
and angular momentum quantum numbers, respectively.
These operators satisfy standard anticommutation rela- 1
tions: {(cm+1/2)†,cm(cid:48)+1/2}=δ δ δ . HK =Jkk(cid:48)(skk(cid:48) ·S− 4ρckk(cid:48)ρd), (7)
kh k(cid:48)h(cid:48) k,k(cid:48) h,h(cid:48) m+1/2;m(cid:48)+1/2
Thecorrespondingbandenergiesareε =ε +λ kh.
kh k R
After the transformation, the total Hamiltonian is: where skk(cid:48) = 21c†ksτss(cid:48)ck(cid:48)s(cid:48), ρckk(cid:48) = c†ksτ0ss(cid:48)ck(cid:48)s(cid:48). The
corresponding definitions for the impurity operators are:
H = (cid:88)ε (cid:16)cm+1/2(cid:17)†cm+1/2+H + (cid:88) V˜k√δm,0× S = 12c†dsτss(cid:48)cds(cid:48) and ρd = c†dsτ0ss(cid:48)cds(cid:48). Here (τ,τ0)
kh kh kh imp 2 are the standard Pauli matrices. The Kondo coupling
khm khm isgivenbyJ =−V V (G +G +G +G )/2,
(cid:20)(cid:16) (cid:17)† (cid:16) (cid:17)† (cid:21) kk(cid:48) k k(cid:48) k+ k− k(cid:48)+ k(cid:48)−
cmkh+1/2 cd↑+(−1)(1−2h) cmkh−1/2 cd↓+h.c. (4) itisaveragedoverthechiralityquantumnumberhandis
spin-independent. OnecanverifythatH reducestothe
K
standard Kondo Hamiltonian when λ = 0. In the an-
where H = (cid:80) ε c† c + H ; c is the operator R
imp s d ds ds U ds gula√rmomentumbasis(2)thisHamiltonianreadsHK ∼
fV˜okr=thVekl(cid:113)oca2klπo.rTbihtuasl,inthtehReSaOngtuelramr mproomdeuncteusmanbeaffsiescatinvde Jankkd(cid:48) ρkkkk(cid:48)(cid:48)(=s0kkc(cid:48)0k†s·τS0ss−(cid:48)c0k41(cid:48)sρ(cid:48),ckki(cid:48).ρed.,)iwnvitohlvisn0kgk(cid:48)o=nly21tch0k†seτmss(cid:48)c=0k(cid:48)s0(cid:48)
two-band(h=±1)Andersonproblemwiththeimpurity mode. In this expression the angular integration has al-
coupled to j =±1/2 channels in each band. ready been carried out.
The Kondo regime. To describe the Kondo regime we After some algebra [32], the expression for the second
perform a Schrieffer-Wolff transformation (SWT) [23]. term, H , is:
DM
As in the usual one-impurity Anderson model, the SWT
isobtainedbyrequiringtheeffectiveHamiltonianHeff = HDM =iλRkFC(k−k(cid:48))·(skk(cid:48) ×S) (8)
eSHe−S not to contain an H term to first order. The
hyb
resulting H contains H +H with renormalized pa- that corresponds to the Dzyaloshinsky-Moriya (DM) in-
eff 0 U
rameters, plus an exchange Hamiltonian. Using Eqs. 3 teraction [24]. Here k−k(cid:48) is a vector in the 2d plane,
3
s and S are the bath and impurity spin vectors re- Although the presence of RSO results on known
kk(cid:48)
spectively. The expression has been evaluated for scat- physics in the Kondo regime, its effect on the Kondo
teringprocessesneartheFermisurfacewhereε ≈0and temperature is quite dramatic. Away from particle-hole
kh
k ≈ k(cid:48) = k up to first order in λ , with C = C(ε ,U). symmetry conditions at the impurity, the DM term in-
F R d
It is important to remark that C vanishes in the limit of creasestheKondocouplingproducinganexponentialin-
U → ∞ and when the impurity state is particle-hole crease in the Kondo temperature given by:
symmetric, i.e. ε = −U/2, in agreement with gen-
d
eral time-reversal symmetry arguments [20, 32]. Simi- T (cid:18)T (cid:19)1−J1/J
K = 0 , (11)
lar terms were found in previous works [33], when the T D
0
hybridization coupling V is made to be spin-dependent,
k
i.e, Vks. However in these models the DM term does not where T0 is the Kondo temperature in the absence of
vanish at the particle-hole symmetric point as expected. RSO and D is the bandwidth cut-off [32]. We should
It is also instructive to write the DM term in the an- notethatthechangeinthecouplingJ producedbyRSO
gular momentum basis (using Eq. 2 and carrying out interactions is not compensated by changes in the effec-
the angular integration part), where it reads: H = tive density of states ρ((cid:15) ) at the Fermi level. One can
DM F
λ k C(sλ ·S−1ρλ ρ ),wheresλ = 1(c0†τss(cid:48)c2s(cid:48) + show that terms in the SWT that can renormalize ρ((cid:15) )
R F kk(cid:48) 4 kk(cid:48) d kk(cid:48) 2 ks k(cid:48)−s(cid:48) F
c2s† τss(cid:48)c0 ), ρλ =(c0†τss(cid:48)c2s(cid:48) +c2s† τss(cid:48)c0 ). Itis are of two types: i) those that are independent of λR,
k−s k(cid:48)s(cid:48) kk(cid:48) ks 0 k(cid:48)−s(cid:48) k−s 0 k(cid:48)s(cid:48)
clearthatthisinteractioncouplesthem=0andm=±1 and as such cannot eliminate the effect of the DM term
modes of band electrons (as c2s =c1 for s= 1, etc.). in J; or ii) those that depend on λR via the renormal-
k−s k↓ 2
ized value of this coupling. As this last renormalization
Renormalization group analysis. To understand the
is mainly due to the presence of the Hubbard term U,
effect of the DM terms in the Kondo regime we perform
and the DM corrections depend on both U and ε , the
a renormalization group (RG) analysis, assuming that d
correction on ρ((cid:15) ) can not generically compensate the
Vk ≈ VkF and Jkk(cid:48) ≈ JkF. The RG flow reveals the change in J. TheFse effects can be substantial and signif-
existence of an additional term involving higher energy
bands: H = γ(sγ .S − 1ργ ρ ) introducing a new icantly enhance TK. Figure 1 illustrates TK as function
γ kk(cid:48) 4 kk(cid:48) d of the RSO parameter λ , changing in a clearly super-
coupling constant γ. In the angular momentum basis F
(afterangularintegration),theexpressionforsγ reads: linear fashion, unlike the linear increase expected from
kk(cid:48)
sγ = c2s† τss(cid:48)c2s(cid:48) and ργ = c2s† τss(cid:48)c2s(cid:48) . The density of state effects [21]. For a Co atom adsorbed on
kk(cid:48) k−s k(cid:48)−s(cid:48) kk(cid:48) k−s 0 k(cid:48)−s(cid:48) graphenewithSiO assubstrate[16],forexample,T has
resulting coupled equations are: 2 0
been measured to be ≈ 14K; changing the substrate to
J˙ = J2+λ2/4 Au/Ni has been shown to enhance the RSO strength to
F
λ (cid:39)0.2 eV [39]. We estimate λ /J ≈0.3, which would
γ˙ = γ2+λ2/4 R F
F result in a 20% increase for T ≈ 17K. We emphasize
K
λ˙F = (J +γ)λF , (9) that this strong enhancement holds in the generic situa-
tion away from particle-hole symmetry at the impurity,
where λ =2Cλ k . These equations become
F R F suggesting that it should be observed in quite general
situations.
J˙ =J2, J˙ =J2, (10)
1 1 2 2 Conclusion. Insummary,wehaveanalyzedtheKondo
(cid:113) (cid:113) regime of an Anderson impurity model with RSO inter-
whereJ1 = 12(J++ J−2 +λ2F);J2 = 12(J+− J−2 +λ2F); actionsina2delectrongas. BecauseofthebrokenSU(2)
J = J ±γ and the ratio λ /J = constant. Eq. 10 spin symmetry in the presence of RSO, the coupling be-
± F −
describes the RG flow of two decoupled Kondo Hamilto- tweenimpurityandbandelectronsoccursviaacombina-
nians with couplings J and J . Considering an initial tion of different angular momentum modes in the bath.
1 2
condition with γ = 0 and an antiferromagnetic Kondo The resulting two-band model produces a two-channel
coupling J >0, the equations render J >0 and J <0. Kondoregimewithferro-andantiferromagneticcoupling
1 2
Therefore, as temperature is lowered, J grows to the constants, having a standard one-channel Kondo physics
1
strongcouplingregimewhileJ goestozero,hencereduc- as fixed point. We have shown, in agreement with previ-
2
ing to the standard one-channel Kondo problem. There ous studies, that theeffects of the RSOinteractionsvan-
is also another possible initial condition that has J < 0 ish for an impurity in the particle-hole symmetry point.
(ferromagnetic Kondo Hamiltonian). In this seemingly Away from this point, however, RSO interactions intro-
unfavorablecaseforthedevelopmentofaKondoregime, ducedramaticmodificationstotheKondoregimebygen-
however, (10) predicts the appearance of an antiferro- erating a DM term that has been missed in previous
magneticcouplingatlowenergiesforJ whileJ remains studies. The DM term is responsible for an exponen-
1 2
<0, becoming eventually zero. Note that in both cases, tial increase in the Kondo temperature. It is reasonable
the final Kondo state is formed by the impurity coupled to expect that these effects can be observed in experi-
toacombinationofm=0and±1modesfromthebath. mentscarriedonwithmagneticatomsplacedonsurfaces
4
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