Emergent p-wave Kondo Coupling in Multi-Orbital Bands with Mirror Symmetry Breaking

0.96 MB·English

by  Jun Won Rhim

Emergent p-wave Kondo Coupling in Multi-Orbital Bands
with Mirror Symmetry Breaking
Jun Won Rhim1,∗ and Jung Hoon Han2,3
1School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
2Department of Physics and BK21 Physics Research Division,
Sungkyunkwan University, Suwon 440-746, Korea
3Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea
(Dated: January 17, 2013)
We examine Kondo effect in the periodic Anderson model for which the conduction band is of
3
multi-orbital character and subject to mirror symmetry breaking field imposed externally. Taking
1
p-orbital-based toy model for analysis, we find the Kondo pairing symmetry of p-wave character
0
emerges self-consistently over some regions of parameter space and filling factor even though only
2
the on-site Kondo hybridization is assumed in the microscopic Hamiltonian. The band structure

n in the Kondo-hybridized phase becomes nematic, with only two-fold symmetry, due to the p-wave
a Kondocoupling. Thereducedsymmetry shouldbereadily observable inspectroscopic ortransport
J measurements for heavy fermion system in a multilayer environment such as successfully grown

6 recently.
1
PACSnumbers: 71.27.+a,73.20.-r,75.30.Mb

]
l
e
- Introduction.-State-of-the-arttechniquetogrowheavy
r
st fermion compounds with atomic thickness precision has 2D multi-orbital Kondo px, py γ pz
. beenachievedinrecentyears[1,2]. Heavyfermionphase lattice system
t
a andsuperconductivitywereidentifiedinCeCoIn5layerof Surface
m lessthanonenanometerthicknessgrowninthis manner,
z
- prompting speculations of exotic new phases in atomi- y
d
cally thin heterostructure [2]. Apart from the novelty of Bulk x
n Bulk
reduced dimensionality, the alternate stacking of heavy
o
c fermion layer with normal metal YbCoIn5 layer realized
[ inthegrowthprocessislikelytogiverisetonewphysical FIG. 1: (Color online) Schematic picture of Kondo lattice

phenomena by virtue of the mirror symmetry breaking modelsubjecttomirrorsymmetrybreakingatthesurface or
1
v (MSB) at the interface. Aspects ofnon-centrosymmetric interface. Hopping processes between, for instance, pz and
4 superconductivity have been exhaustively studied since px,y otherwise forbidden in the bulk become possible due to
thesymmetrybreakingfieldas shownin theupperright car-
6 its original discovery in the heavy-fermion compound
toon.
5
CePt Si [3]. The heavy fermion multilayer structure,
3 3
on the other hand, differs from the conventional non-
.
1 centrosymmetric material in that the violation of sym-
0 electron dynamics. In this new picture Rashba interac-
metryismacroscopicallyimposed(notmicroscopicallyat
3 tion becomes an electrostatic effect with its magnitude
thelevelofcrystalstructure)and,withrecentlyavailable
1 dictated by the degree of orbital asymmetry about the
: techniques such as electrolyte gating, controllable. Due
v mirror plane (see Fig. 1) [5–7].
to the thin layer thickness, such MSB is likely to infect
i
X the entire heavy fermion band structure and therefore In fact Kondo lattice behavior takes place in an
orbital-rich environment with the extended p- or d-
r the nature of emerging Kondo and/or superconducting
a orbital states forming the conduction bands while the
pairing as well. Some consequences of the MSB in the
localized moments arise from f-orbitals [8, 9]. Using
heavyfermionmultilayerhavebeenstudiedtheoretically
our recent experience with the Rashba phenomena in
by Maruyama et al. quite recently [4], where the main
non-Kondo materials as a guide, we present a faithful
focus was on the novel superconducting pairing symme-
modeling of Kondo lattice behavior where multi-orbital
triesandotherthermodynamicpropertiesinducedbythe
degrees of freedom in a mirror symmetry-broken en-
phenomenological Rashba term.
vironment are explicitly embedded in the conduction
IthasbeencustomaryinthepasttoregardtheRashba band Hamiltonian. The use of Rashba interaction as a
interaction as a relativistic effect manifesting itself in a purely phenomenological addition, as is often done in
mirrorsymmetry-brokenenvironment. Severalrecentpa- the current literature, is avoided. Our approach differs
pers[5–7],however,haveshownthatmulti-orbitaldegen- as well from the large-N generalization of the one-band
eracy, in addition to MSB, is an essential pre-requisite Kondo lattice model, where N identical copies with the
for the manifestation of strong Rashba coupling in the same orbital symmetry are adopted as a way to justify
2
0.1 0.12
the mean-field approximation [9]. In reality the MSB
field - usually the electric field acting perpendicular to III III
the interface or the surface - influences orbitals which
are extended along the field direction more than those Txy
which extend in the orthogonal direction. Conventional TB0.05 K 0.06
k
II
one-band or large-N approach assuming featureless II
orbitals will not capture this effect properly.
Tz
K
0(a) I I 0(b)
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 1.2
The model.- Choosing the degenerate p-orbital states γ γ
hopping on a square lattice as a toy example, we write
down the Hamiltonian [6, 7]
FIG. 2: (Color online) Phase diagrams with (a) α = 0 (no
SOC), nc = 2 and (b) α = 0.5 (SOC), nc = 1.5. Phase I
denotesallthreep-orbitalshybridized(hx,hy,hz 6=0). Phase
σ σ σ
II represents only px, py orbitals hybridized (hxσ 6=0,hyσ 6=0,
hz = 0). Phase III is trivial with no Kondo hybridization
σ
HMSB =Xkλσ(cid:16)ǫλkp†λkσpλkσ+3γisλp†zkσpλkσ+h.c.(cid:17) e(ahnnλσcdo=Tunx0tye).r.PtIawnroa(mad)e,itsetarisnscsuthsoeKwdontnobdoyobattearmiendpbevoreatrhttuipcrheasalsalearrbdoeiwaleg,drawmbeysmaTarKyze
+ µffk†σfkσ (1) (V1,V2K,JK)=(2,0.65,3). Fermisurfacestructuresatthefour
Xkσ points indicated by stars are shown in Fig. 3.
forthedispersivep-orbital(pλkσ)bandsandasinglepair
of localized Kramers doublet (fkσ) with on-site energy J bλ
µ . The orbital index λ ∈ {x,y,z} spans the three p- H = K σ¯σp† f −hλp† f +h.c.
f K 2 (cid:18) 2 λiσ iσ¯ σ λiσ iσ (cid:19)
orbital states of the conduction band and σ = +(−) is Xiλσ
the spin up(down). One can show, employing the two J J
+ K Γpf† f + K Γfp† p . (2)
Slater-Koster parameters V1, V2 for σ- and π-bonding 2 σ iσ¯ iσ 2 σ λiσ¯ λiσ
of p-orbitals, that ǫλ = −2(V + V )c + 2V (c + c ) Xiσ Xiλσ
k 1 2 λ 2 x y
with cx(y) = coskx(y), sx(y) = sinkx(y) and cz = sz = 0 In addition to the Kondo hybridization parameters
for the assumed two-dimensional bands. Mirror symme- bλσσ′ = hfi†σpλiσ′i and hλσ = bλσσ + bλσ¯σ¯/2, we also
try breaking is reflected in the second term of Eq. (1), allow same-orbital spin-flip processes encoded by
proportionalto the MSB parmaeter γ [7]. They give the Γp = hp† p i and Γf = hf† f i. The spin-flip
σ λ λiσ λiσ¯ σ iσ¯ iσ
hybridizationofpz-orbitalwithpx-andpy-orbitals;these paramePters Γp, Γf, and bλ can be nonzero due to the
σ σ σσ¯
areprocessesnormallyforbiddenbysymmetryifamirror spin-orbit interaction. We restrict ourselves however to
planeexisted. Thiskindofsymmetrybreakingisrealized non-magnetic, singlet Kondo coupling in the mean-field
naturallyonthesurfaceofabulkmaterialorbyapplying scheme and avoid the possibility of Kondo pairing be-
vertical electric field to the planar system as illustrated tween remote sites [10], which is another known avenue
in Fig. 1. to obtain non-s-wave Kondo pairing. The fact that our
model can exhibit the p-wave Kondo phase despite the
Secondly the Kondo hybridization of local and itin-
on-site-only Kondo hybridization Hamiltonian H em-
erant spins follows from the Heisenberg-type exchange K
phasizes the emergent nature of unconventional pairing.
interaction H = J S ·S , with spin-1/2 oper-
K K i,λ iλ if As demonstrated in Refs. [5–7], Rashba interaction is
ators made out of pλP-orbital (Siλ) and f-orbital (Sif), another emergent property of the band that can be
respectively. Finally H = (α/2) L ·σ involving
SOC i i i derived from the underlying microscopic Hamiltonian
the p-orbital angular momentum opPerator (L = 1) Li consisting only of H and H .
andthespin-1/2operator(1/2)σ ateachsiteigivesthe MSB SOC
i
spin-orbitcoupling(SOC)withinthe p-orbitalbands. In
Phase diagram.- All the hybridization parameters can
typicalheavyfermionmatter suchasCeCoIn the heavy
5 be determined self-consistently, along with µ to ensure
element In playing a main role in the conduction band f
h f† f i = 1, and the overall chemical potential µ,
structurewillalsocarrysubstantialamountofspin-orbit σ iσ iσ
energy α. foPr various input band structure parameters V1,V2,γ,
electron filling factor n = h p† p i, at various
c λσ λiσ λiσ
ThetotalHamiltonianH =HMSB+HSOC+HKcanbe temperatures T. Generic phasPe diagrams resulting from
handledwithintheframeworkofrenormalizedmean-field ourmodelHamiltonianareplottedinFig. 2inthe space
theory [10–12] which decomposes the Kondo exchange of temperature T and MSB parameter γ with and with-
interaction as (σ =−σ) out the SOC. We observe two distinct Kondo-coupled
3
π M With a proper choice of tight binding parameters and
filling factors we can identify cases in the self-consistent
band structure wherein the two lowest-energy spin-orbit
ky 0 Γ X coupledp-orbitalbands andthe two Kramersdoublet f-
electron bands occur in the vicinity of the Fermi energy
whileallothersarefurtherremovedfromit. Insuchcases
−π(a) (b) onecandescribethe Kondophysicswithin the projected
M
Hilbert space of those four bands associated with the
π
following operators:
k 0 6
y c˜nk = vλnkσpλkσ and f˜kσ =fkσ+ unkσc˜nk. (3)
Xλσ nX=3
−π(c) (d) Here, n=1−6 labels the six p-orbitalbands of H +
−π 0 π −π 0 π H , the state at k being denoted |vni = c˜† |0Mi SwBith
k k SOC k nk
x x the coefficients vn . The localized f-electron operator
λkσ
ismodifiedthroughKondohybridizationwiththehigher
FIG. 3: (Color online) Band dispersions at the Fermi level energybands, unkσ =hvkn|HK|fkσi/(µf−εcnk), n=3−6,
for thevarious phases marked byred stars (⋆) in Fig. 2. (a), where εc is the energy of n-th band of H +H
nk MSB SOC
(IIbI)inanFdig(.c)2(aar)ertehspeeFcteirvmeliy,suarnfdac(eds)ocfortrheesppohnadssestoI,thIeI satnadr and|fkσi=fk†σ|0i. TheeffectiveHamiltonianwithinthe
in Fig. 2(b). The reduced symmetry in the Kondo-coupled reducedHilbert space of {|vk1i,|vk2i,|f˜k+i,|f˜k−i}, where
phases(a), (b),and(d) duetothep-waveKondopairing are |f˜kσi=f˜k†σ|0i, is
apparent against the (kx,ky) axes in red dashed lines.
Heff = εcnkc˜†nkc˜nk+ εfσkf˜k†σf˜kσ
phases labeled I in Fig. 2, where all three p-orbitals nX=1,2 Xσ
honyblyridthizee iwni-tphlafn-eorpbxi,taply(horxσb,ihtyσal,shzσhy6=br0id)i,zean(dhzσII =whe0r)e. +n=X1,2,σ(cid:16)Vknσc˜†nkf˜kσ+h.c.(cid:17). (4)
The high-temperature non-Kondo phase with hλ = 0
σ
is labeled III. Both transitions, I→II and II→III, are The f-level dispersion εf and the effective Kondo cou-
σk
second-order. Phase I becomes unstable and gives way pling Vknσ =hvkn|HK|fkσi are given by
to phase II as showninFig. 2(b) asthe strengthofSOC
is increased. Although the addition of SOC complicates
the microscopic Hamiltonian, fortunately, many of the εf =µ + 6 |hvkn|HK|fkσi|2,
hybridization parameters were found to be vanishing σk f nX=3 µf−εcnk
Γp = Γf = bz = bx = by = 0 even with finite SOC.
σ σ σσ σσ¯ σσ¯ J 1
Features of the phase diagram are dependent on the Vknσ = 2K (cid:18)2bzσσ¯vznkσ¯ −hσx¯vxnkσ−hyσ¯vynkσ(cid:19). (5)
fillingfactorn ofp-electrons. PhaseIisobservedwithin
c
a small range of nc and, as indicated in Fig. 2(a), may Some properties of the self-consistently obtained hy-
even exhibit some re-entrant behavior with increasing bridization parameters of Kondo-coupled phases such as
γ. While the details regarding the extent of each Kondo Γp = Γf = bz = bx = by = 0 are used to derive
σ σ σσ σσ¯ σσ¯
phase or the Kondo temperature are sensitive to the the above effective Hamiltonian. The energy dispersion
occupation nc, the effect of SOC on them was found to obtainedfromHeff matchesthefulldispersionalmostex-
be much less significant by comparison. actly. CharacteristicsoftheKondo-coupledbandscanbe
discussed faithfully now in terms of the effective Hamil-
EffectiveHamiltonianforp-waveKondopairing.-Most tonian, H .
eff
interestingly, the two Kondo-coupledphases I and II ex- The effective Kondo coupling Vnσ appearing in H
k eff
hibit the band dispersion with the loss of symmetry un- has the explicit momentum dependence despite the fact
der the reflection (k ,k ) → (−k ,k ) or (k ,k ) → that the original Hamiltonian only contained the on-site
x y x y x y
(k ,−k ), as well as the 90◦ rotation, as shown in Fig. Kondointeraction. The symmetry ofthe Kondopairing,
x y
3. Reduction in the band structure symmetry only oc- self-consistently determined to be p-wave as illustrated
curs in the Kondo phase implying that it can only be in Fig. 4, also dictates that of the hybridized bands,
the consequence of p-wave Kondo coupling. Below we as shown in Fig. 3. The symmetry propertes of Vnσ,
k
show rigorously how such unconventional pairing occurs in turn, follow from those of the wave functions vn as
λkσ
for MSB-infected bands. outlined in Table I.
4
k wave function the M point meet the conditions assumed in deriving
(kx,ky) {vxnk+,vynk+,vznk+,vxnk−,vynk−,vznk−} the effective Hamiltonian Heff. Here one can show vλnkσ
(−kx,ky) {−vxnk∗+,vynk∗+,−vznk∗+,−vxnk∗−,vynk∗−,−vznk∗−} near k = (π,π) is vx1,yk+ ≈ −3γsx,y/2(V1 + V2) and
(kx,−ky) {vxnk∗+,−vynk∗+,−vznk∗+,−vxnk∗−,vynk∗−,vznk∗−} vx1,yk− = 0, and vice versa for n = 2. Turning off SOC,
(−kx,−ky) {−vxnk+,−vynk+,vznk+,vxnk−,vynk−,−vznk−} all spin-mixing mean field parameters bzσσ¯ vanish. We
canconcludeonlytwoequivalentgroundstateswithreal-
(−ky,kx) {−vynk+,vxnk+,vznk+,−ivynk−,ivxnk−,ivznk−} valued Kondo parameters hx = hy or hx = −hy are
σ σ σ σ
TABLE I: Relation between eigenstates of HMSB+HSOC at possible, giving out
several symmetry-related momenta. vλnkσ in the first row
refers to the amplitude of pλ-orbital with spin σ for the 3iJ hxγ
eigenstate at k = (kx,ky); σ = +(−) corresponds to spin Vk ≡Vk1+ =Vk2− = K (sinkx±sinky) (6)
4(V +V )
up(down). Eigenstates at other momenta are related to this 1 2
one as shown in thesubsequentrows.
and V1− =V2+ =0 for hx =±hy, respectively. The ef-
k k σ σ
fective Kondo coupling indeed has the p-wave symmetry
(a) α=0 (b) α=0.5 withanodealongthek =+k ork =−k line. Figure
π 0 .1 π 0 .3 x y x y
4(b) shows the plot of |V1+| for the case of hx = −hy
k σ σ
where the node extends along k = −k . It is also seen
x y
0.2 thatp-waveKondocouplingisnotpredicatedonSOCat
0 ky0 all.
Instead, the p-wave Kondo coupling is strictly a
0.1
consequence of nonzero MSB, γ 6= 0. When γ = 0,
Hamiltonian matrices of H + H at (k ,k ),
MSB SOC x y
−π−π 0 π 0 −π−π 0 π 0 (−kx,ky), (kx,−ky) and (−kx,−ky) are equal to each
kx kx other [13]. As a result, those momenta have identical
eigenvectors and, according to Eq. (5), the same
FIG. 4: (Color online) Contour plots of |Vk1+|, the absolute Kondo coupling Vknσ. In addition the p-wave character
value of the effective Kondo hybridization, in the Brillouin arises only when both hx and hy are nonzero. For
σ σ
zone. (a) α=0 (no SOC) and (b) α=0.5. phases I and II such condition is indeed achieved with
|hx| = |hy|. The p-wave Kondo states are favored away
σ σ
from half-filling regime such as considered in the phase
Transformation rules for the wave functions are de- diagram of Fig. 2. At half filling nc = 3 without SOC,
duced by examining whether the Hamiltonians at a pair on the other hand, we often find the ground phase with
of symmetry-related momenta can transform into each only hz non-vanishing. In this case, the effective Kondo
σ
other by some kind of unitary or anti-unitary opera- coupling is inthe formVknσ =−JKhzσ¯vznkσ/2≈iJKhσz¯/2,
tors such as Kσ ⊗ R (π), KI ⊗ R (π), σ ⊗ R (π) whichiss-waveasintheordinaryheavyfermionmatter.
z x y z z
and e−i34πDz1/2(3π/2)⊗Rz(π/2). We denote Rλ(θ) for
the SO(3) rotation matrix, λ = x,y,z, D1/2(θ) for the Discussion.- As the angstrom-scale deposition tech-
z
niqueofthinfilmsdevelopsintomaturity,theroleofsur-
SU(2) rotation matrix, and K for the complex conjuga-
facestateswithbrokenmirrorsymmetrytakesongreater
tion. From the Table and Eq. (5) one can show that
practical significance. Such MSB field effect leads to in-
Vnσ = −σVnσ , and therefore a unitary transfor-
−kx,−ky kx,ky terestingconsequenceswhenactingonamulti-orbitalen-
mation σ ⊗I between H (k) and H (−k) exists, en-
z eff eff vironment such as the emergence of Rashba effect [6, 7],
suringtheequivalenceoftheenergiesbetweenkand−k.
orbital-dependentspintransfertorque[14],andasargued
On the other hand Vnσ , Vnσ and Vnσ are not
−kx,ky kx,−ky −ky,kx inthis work,ofun-conventionalKondopairing. Conven-
relatedtoVnσ inanywayaccordingtotheTableI. For
kx,ky tional one-band Kondo model with the phenomenolog-
example,Vnσ =J (−bz vn∗ /2+hxvn∗ −hyvn∗ )/2
−kx,ky K σσ¯ zkσ¯ σ¯ xkσ σ¯ ykσ ical Rashba term could very well miss this feature due
does not even have the same magnitude as Vkn,σ in gen- to the inadequate treatment of the multi-orbital char-
eralbecause the wave function vλnkσ are complex-valued. acter. The asymmetric band structure we found in the
Referring to Eq. (5), neither the reflection about the x- Kondo-coupled phase should be readily detected in the
orthey-axisnorthe90◦rotationink-spaceisguaranteed spectroscopic study such as ARPES, or in the transport
to yield unitary-equivalent Heff. measurements through their directional responses. Al-
It is further possible to find an explicit expression of though proposals of non-s-wave Kondo pairing are al-
the effective Kondo coupling to prove its p-wave nature ready available [10–12], they tend to trace its origin to
for models without SOC. Given the parameters such as thenon-localcharacterofKondocouplingbetweenlocal-
V = 2, V = 0.65, γ = 0.3 and the filling of conduc- ized and itinerant moments [10], or the orbital symme-
1 2
tion electrons n = 1.5, the band dispersions around try of the f-orbitallocalized moments [12]. By contrast,
c
5
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[7] Jin-Hong Park, Choong H. Kim, Jun-Won Rhim, and
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