Theoretical understanding of the quasiparticle dispersion in bilayer high-$T_{c}$ superconductors

0.23 MB·English

by  Jian-Xin Li

Theoretical understanding of the quasiparticle dispersion in bilayer high-T
c
superconductors
Jian-Xin Li1,2,3, T. Zhou1, and Z. D. Wang1,2
1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
2Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
5
3The Interdisciplinary Center of Theoretical Studies, Chinese Academy of Sciences, Beijing, China.
0
(Dated: February 2, 2008)
0
2
The renormalization of quasiparticle (QP) dispersion in bilayer high-Tc cuprates is investigated

n theoretically by examining respectively the interactions of the QP with spin fluctuations (SF) and
a phonons. Itisillustratedthatbothinteractionsareabletogiverisetoakinkinthedispersionaround
J the antinodes (near (π,0)). However, remarkable differences between the two cases are found for

5 thepeak/dip/humpstructureinthelineshape,theQPweight,andtheinterlayercouplingeffecton
1 thekink, which are suggested to serve as a discriminance to single out thedominant interaction in
the superconducting state. A comparison to recent photoemission experiments shows clearly that

] thecoupling to thespin resonance is dominant for the QParound antinodes in bilayer systems.
n
o PACSnumbers:
c
-
r
p The elucidation of many-body interactions in high- they differ remarkably in the following aspects. (a) The
u T superconductors (HTSC) is considered as an essen- lineshape arising from the spin resonance coupling ex-
c
s tial step toward an insightful understanding of their hibits a clear PDH structure, while the phonon coupling
.
t superconductivity. Angle-resolved photoemission spec- wouldleadtoareversedPDHstructure,namelythepeak
a
m troscopy(ARPES)hasprovidedapowerfulwaytoprobe isinalargerbindingenergythanthehump. (b)Thefor-
the coupling of charge QPs to other QPs or collective mer coupling causes a rapid drop of the QP weight near
-
d modes. Recent ARPES experiments unveil several in- the antinodal region with underdoping, but the latter
n triguing features in the dispersion, the QP weight, and has only a very weak effect. Moreover, the correspond-
o the lineshape: (i) A kink in the dispersion was observed ing coupling constant for the fermion-SF interaction is
c
in both the nodal and antinodal regions [1, 2, 3, 4]. reasonable consistent with ARPES data, in contrast to
[
(ii) The QP weight around the antinodal region de- the much smaller value for the fermion-phonon interac-

1 creases rapidly with the reduction of dopings [5], while tion. (c)The bilayercouplingplaysapositive roleinthe
v changes a little around the nodal direction [6]. (iii) Af- occurrenceofthekinkinthecaseofthefermion-SFinter-
6
ter disentangling the bilayer splitting effect, an intrinsic action, but has a negative effect on the formation of the
5
peak/dip/hump (PDH) structure was seen around the kink for the fermion-phonon interaction. These results
3
antinodalregionbothinthebonding(BB)andantibond- suggest that the SF coupling be a dominant interaction
1
0 ing(AB)band[2,3]. (iv)Thekinkaroundtheantinodal involved in the antinode-to-antinode scattering.
5 regionseemly showsadifferentmomentum, temperature Wewillconsiderseparatelytheinteractionsoffermions
0 and doping dependence from that around the nodal di- with the SF and phonons. Let us start with the bilayer
t/ rection [2]. (v) The antinodal kink is likely absent (or t t′ J model with the AF interaction included,
a very weak) in the single-layered Bi2201 [4]. These fea- − −
d-m tcuorlleesc,tievsepemcoiadlley. (Sio),faimr,ptlwyothcaotlletchteivQePmoisdecsouofpl4e1dmtoeVa H = −<ij>X,α,α′,σtα,α′c(iσα)†c(jασ′)−<ijX>′,α,σt′c(iσα)†c(jασ)
on bspeiennrseusgogneasntceed,[b7u,t8]wahnicdh∼on3e6ismaeVkepyhfoanctoonr[r9e,sp10o]nhsiabvlee −h.c.+ X Jα,α′S(iα)·S(jα′), (1)
c for the kink is stillmuchdebatable [1,2, 3, 4, 9,10, 11]. <ij>α,α′
: So, it raises an important question as if the electronic
Xiv ipnetresriaocnt,ioonratloonwehisatreesxptoennstibtlheefoarntthineoindtarlig(uniondgaQl)PQdPis’s- wJhiefreαα==1α,′2, doetnhoetrewsisteh,etlaαy,αe′r i=ndetxp,/2tα,J,αα′,α=′ t,=JαJ,α⊥′/=2
and i = j. Other symbols are standard. In the
r properties are determined by the strong electronic inter-
a slave-boson representation, the electron operators c
action. jσ
are written as c = b†f , where fermions f carry
jσ j jσ iσ
In this Letter, we not only answer the above impor- spin and bosons b represent the charge. Using the
i
tant question but also present a coherent understanding mean field parameters χ = < f† f >= χ ,
ij Pσ iσ jσ 0
ontheabovefeaturesbystudyingindetailtherespective
∆ =< f f f f >= ∆ , and setting b √δ
ij i↑ j↓ i↓ j↑ 0
effectsofthefermion-SFinteractionandfermion-phonon − ± →
with δ the doping density (boson condense), we can de-
interaction on the QP dispersion based on the slave-
coupletheHamiltonian(1). ItsFouriertransformationis
bosontheoryofthebilayert t′ J model. Wefindthat
− − givenby,H = ε f(α)†f(α) ∆ (f(α)†f(α)†+
though both couplings are able to give rise to the kink m Pkσα k kσ kσ −Pkα k k↑ −k↓
structureneartheantinodalregionintheQPdispersion, h.c.) + [δt eikzcf(1)†f(2) + h.c.] + ε , with ε =
Pkσ ⊥k kσ kσ 0 k
2
2(δt+J′χ )(cosk +cosk ) 4δt′cosk cosk µ ,
0 x y x y f
−∆ = 2J′∆ (cosk cosk ),−ε = 4NJ′(χ2 +−∆2),
t k = t (cosk0 coxs−k )2/4 y[12] a0nd J′ = 3J/80. Diag0o-
⊥k p x y
−
nalizing the Hamiltonian, we get the AB and BB bands
with the dispersion ǫ(A,B) = ε δt . Then, the bare
k ⊥k
±
normal (abnormal) Green’s functions of fermions (A,B)
s
( (A,B)), and the bare spin susceptibility χα,α′(αG,α′ =
w
G
A,B) are obtained. The physical spin susceptibility
is given by χ = χ+cos2(q c/2)+ χ−sin2(q c/2), with
0 z 0 z
χ+ =χAA+χBB and χ− =χAB +χBA.
0 0
The slave-bosonmean-field theory underestimates the
AF correlation [13, 14], so we need to go beyond it and
include the effect of SF through the random-phase ap-
proximation (RPA), in which the renormalized spin sus-
ceptibility is,
χ±(q,ω)=χ±(q,ω)/[1+(αJ J )χ±(q,ω)/2] (2)
0 q± ⊥ 0
FIG. 1: The MDC dispersion of fermions due to the inter-
However, in the ordinary RPA (α = 1), the AF corre- action with spin fluctuations (a),(b) and (c), and phonons
lation is overestimated as indicated by a larger critical (d). Figs.(a),(c) and (d) are the results at (kx,π) (antinodal
doping density δ 0.22 for the AF instability than the region), and Fig.(b) at (kx,kx) (nodal direction). The scat-
≈
experimental data δ = 0.2 0.5 [15]. Thus, we use the tered points in Fig.(c) are derived from theEDC [16].
c
∼
renormalized RPA in which the parameter α is deter-
mined by setting the AF instability at the experimental
value δ . The fermionic self-energy coming from the SF kink. In contrast, Fig.1(b) shows that no kink is present
c
coupling is given by, in the nodal region. This can be understood from the
conservationofthe momentumin the scatteringprocess,
1
Σ(sA,w,B)(k) = ±4Nβ X[(Jq+J⊥)2χ+(q)Gs(,Aw,B)(k−q) ntraamnsefleyrrtehdemnoodmael-nttou-mnodthalanscaQtte=rin(gπ,iπn)vowlvheesreatshmeaAlleFr
q
spinfluctuationpeaks. Thus,wewillfocusmainlyonthe
+(J J )2χ−(q) (B,A)(k q)], (3)
q− ⊥ Gs,w − antinodal region in the following discussion. In Fig.1(c)
where, the +( ) sign is for the normal (abnormal) we replot the MDC dispersion together with the EDC
self-energy Σ −, and the symbol q represents an ab- deriveddispersion[16]. Nearandbelowtheregionwhere
s(w)
breviation of q,iω . The renormalized Green’s func- the kink appears in the MDC dispersion, the EDC dis-
m
tion is G(A,B)(k,iω) = [(G(A,B)(k,iω))−1 + (∆ + persion breaks into a two-part structure, a peak and a
s k
hump. Therefore, the appearance of the antinodal kink
Σ(A,B))2G(A,B)(k, iω)]−1 with G(A,B) =[iω ǫ(A,B)
w s − s − − hasanintimaterelationtothePDHstructureintheEDC
Σ(sA,B)]−1, and the spectral function is obtained via plot, being in agreement with experiments [18].
A(k,ω) = (1/π)Im[G(k,ω + iδ)]. To determine the
In fact, the kink may also be expected if fermions
−
QP weight z, we first write the Green’s function as
are predominantly coupled to other collective modes.
G(k,iω) = M/(iω +Σ )+N/(iω Σ ) and get z via
α β A hotly discussed mode responsible for the antinodal
−
z = M/[1+∂Σ /∂ω(ω )], with ω the pole of G. The
α 0 0 kink is the out-of-plane out-of-phase O bucking B
parameters we choose are t = 2J, t′ = 0.7t, J = 120 1g
phonon [9, 10]. To take into account this kind of mode,
−
meV and J = 0.1J [15]. Except in Fig.4, the doping
⊥ we may have the total Hamiltonian by including the fol-
density is set at δ =0.2.
lowing interaction in the slave-boson mean field Hamil-
Fig.1 (a) and (b) display the calculatedQP dispersion
tonian H ,
m
for the BB band obtained from the momentum distribu-
tioncurve(MDC)[16]intheantinodalandnodalregions, 1
respectively. One can see from Fig.1(a) that the antin- Hep = √N X g(k,q)fk†,σfk+q,σ(d†q+d−q) (4)
odalkink appears insome rangeof parameterst andα, k,q,σ
p
such as t = 2.0J and α = 0.43. The RPA correction
p
factor α = 0.34,0.43 and 0.55 correspond to the criti- where d† and d are the creation and annihilation opera-
cal doping density of the AF instability δc = 0.02,0.05 torsforphonons,g(k,q)=g0[Φx(k)Φx(k−q)cos(qy/2)−
and 0.09, while δ = 0.02 0.05 is within the experi- Φ (k)Φ (k q)cos(q /2)]/ cos2q /2+cos2q /2 and
c y y x p x y
∼ −
mental range [15]. Meanwhile, the ARPES experiment the detailed form of Φ ,Φ can be found in Ref. [9, 10].
x y
indicated that t (corresponding to δt , δ 0.2) is We note that the vertex g(k,q=0) cos(k ) cos(k )
⊥,exp p x y
∼ ∼ −
44 5meV [17], i.e., t =1.6 2.0J here. Therefore, in and vanishes for all k at q = (π,π) [9, 10]. Thus, the
p
± ∼
the reasonableparameter range,the interactionbetween fermions near (π,0) are strongly scattered by this inter-
fermions and SF reproduces well the observed antinodal action. The corresponding fermionic self-energy is given
3
by,
1
Σs,w(k) = X g(k q,q)2
±βN | − |
q
D (q)[ (A)(k q)+ (B)(k q)]. (5)
0 Gs,w − Gs,w −
where the Green’s function of phonon is D (q) =
0
2ω /[(iω)2 (ω )2], and a dispersionless optical phonon
q q
(B ) will −be taken as ~ω = 36meV [9, 10]. Unlike
1g q FIG. 2: (a) Imχ(q,ω) vs ω at (π,π). (b) and (c) show the
the fermion-SF interaction, the coupling constant here bare dispersion of the normal state quasiparticle for tp =2J
is not available now. We have tried various values and and 1.0J, respectively.
found that the well-established kink can be obtained
wheng 0.2J 0.4J ift =1.7J asshowninFig.1(d).
0 p
≈ ∼
This resultisquite similartothatforthe SFinFig.1(a).
So, a mere reproduction of the QP kink is not sufficient
to single out the main cause of the renormalization, and
we must resort to other features.
A meaningful difference between the effects of these
two interactions on the dispersion can be seen clearly
fromacomparisonofFig.1(a)and(d),namely,theinter-
layer coupling t has opposite impacts on the antinodal
p
kink. It enhances the kink feature for the fermion-SF
interaction; as shown in Fig.1(a), when t decreases to
p
1.7J,noantinodalkinkisobservedevenforthestrongest
AF coupling α=0.55which is in fact beyondthe exper-
imentally acceptable value. In contrast, the kink feature
is weakened by the interlayer coupling for the fermion-
phonon interaction; as seen in Fig.1(d), the kink disap-
pears when t increases to 2J from 1.7J with g =0.2J.
p 0
Recent ARPES experiments revealed that a more pro-
nounced kink is presentin the multilayeredBiSrCaCuO, FIG. 3: The lineshape of fermions at different k points.
insharpcontrasttothecaseinthesingle-layeredone[4], Fig.(a) is obtained when fermions is coupled to spin fluctua-
which can be considered as an indication to favor the tions. Fig.(b) shows the result arising from the coupling to
fermion-SFinteractioninthebilayersystem. Becausethe the B1g phonons. The inset in Fig.(a) shows the lineshape
for the bonding and antibonding bands at (0,π), separately.
spin susceptibility in the bilayer system involves scatter-
The inset in Fig.(b) is thelineshape for thebonding band at
ingsbetweenlayers,theself-energy[Eq.(3)]hasafeature
k=(0,π) and (0.2π,π).
that the fermions in the BB (AB) band is scattered into
AB (BB) band via the SF in the odd channel χ−. The
so-called spin resonance (a sharp peak in Imχ) appears
around Q both in the odd χ− and even χ+ channels band, the decrease from the BB band overcompensates
(see Fig.2(a)). However,it is more prominent in the odd that increase, and thus the self-energy decreases with tp
channel, which is in agreement with very recent exper- on the whole.
iments [19], mainly due to the larger vertex αJ J In Fig.3, we show the lineshape for different k points
Q ⊥
−
(J = 2J)[Eq.(2)], comparedto αJ +J inthe even from (0,π) to (0,0.2π). For the fermion-SF interaction,
Q Q ⊥
−
channel. On the other hand, the AB band (and its as- both the BB and AB spectra near (0,π) consist of a low
sociated flat band near Q) is much close to the Fermi energy peak, followed by a hump, and then a dip in be-
surface compared to the BB band, as shown in Fig.2(b). tween, though the intensity of the AB hump is much
Asaresult,thefermionicself-energyforBBbandislarge, weaker than its peak intensity [inset of Fig.3(a)]; both
andconsequentlytherenormalizationisstrong. Wehave develop their own PDH structure near (0,π). When
indeed observed that the AB band is much less affacted moving from (0,π) to (0.2π,π), one will see that the
by this coupling, so no kink is present in this band (not intensity of the BB peak increases, while the AB peak
shown here). As increasing t , the splitting between the decreases rapidly. Eventually, only the BB peak can be
p
ABandBBbandpushestheflatportionoftheABband seennear(0.2π,π). Thesefeaturesareingoodagreement
to be more close to the Fermi level [Fig.2(b) and (c)], so with ARPES experiments [17]. However, the lineshape
enhances the scattering [20]. However, for the coupling caused by the phonon-coupling displays a striking con-
tophonons,the ABandBBbandscontributetotheself- trast to those shown in Fig.3(a) and in experiments [17]
energy in the same way[Eq.(5)]. In this case, though , namely, the peak is far below the dip and the dip is
the increaseoft increasesthe contributionfromthe AB below the hump [Fig.3(b)]. This is because the renor-
p
4
slowly, and it is still 0.7 even at δ = 0.08. This ex-
hibitsthehighlyanisotropicinteractionbetweenfermions
and SF and is well consistent with what is inferred from
ARPES [5] and a recent argument based on the analysis
of experimental data [21]. Moreover, the coupling con-
stant obtained using z = 1/(1+λ) shows a reasonable
fit to experimental data [3][Fig.4(b)], while that at the
nodal direction is much smaller than the experimental
data [6] [inset of Fig.4(b)]. This consistence is signifi-
cant, because we use the well established parameters in
the t t′ J model with only one adjustable parameter
− −
α being fixed by fitting to experiments. In contrast, the
weightdecreasesmuchslowlyfor the fermion-phononin-
teractionasshowninFig.4(c),andthecouplingconstant
λisnearly3 5timessmallerthanexperimentalvalues.
∼
Finally,wewishtomaketwoadditionalremarks. First,
FIG. 4: The quasiparticle weight z of fermions and its cou-
becausethespinresonancecontributeslittletothenode-
pling constant λ at the antinodes arising from the coupling
to-node scattering, a kink structure in the nodal direc-
to spin fluctuations (a) and (b), and to phonons (c) and (d).
tion should be caused by the other mode coupling, such
Theinsetsshowthoseatthenodes. Thescatteredpointsare
as an in-plane Cu-O breathing phonon [1]. The dif-
experimental data from Ref. [3] and Ref. [6].
ferent momentum, temperature and doping dependence
between the nodal and antinodal kink [feature (iv)] sup-
malization to the QP peak in the BB band due to the ports this point of view. Second, since the interlayer
phononsisratherweak,sothepeakis almostunchanged coupling plays opposite roles in the antinodal kink re-
comparing to the bare one. When moving from (0,π) to spectively for the fermion-SF and fermion-phonon inter-
(0.2π,π), the QP moves to be near the Fermi level, an actions, it is expected that, even though a rather weak
ordinary PDH structure is recovered [inset of Fig.3(b)] fermion-phonon coupling may be present and lead to a
becausethehumparisingfromthephononcouplingdoes weak antinodal kink-like behavior in the single-layered
not change in the process. cuprates, as possibly seen in the experiment for Bi2201
Figure 4presents the QPweightz offermions andthe [4], the coupling is too weak to affect significantly the
coupling constantλ. Notice that the weightof the phys- lineshapewhichis mainlydeterminedbythe SFandwas
ical electron is δz due to the condensation of holons in shown to exhibit the peak/dip/hump structure in the
the slave-bosonapproach[13]. For the fermion-SF inter- single-layeredcase [14].
action, the weight decreases rapidly with underdoping, We are grateful to F.C. Zhang for many useful discus-
from nearly 0.42 and 0.39 at doping δ = 0.24 (the bare sions. The workwas supported by the NSFC (10474032,
value is 0.5) to be below 0.075 and 0.15 at δ = 0.08 10021001,10429401,10334090),the RGC of Hong Kong
for the BB and AB band, respectively. On the other (HKU 7050/03P),the URC fund of HKU, and partly by
hand,theweightalongthenodaldirectiondecreasesvery RFDP (20030284008).
[1] A.Lanzara et al.,Nature 412, 510 (2001). [13] P. A. Lee, Physica C 317-318, 194 (1999); J. Brinck-
[2] S.V.Borisenkoetal.,Phys.Rev.Lett.90,207001(2003); mann and P.A. Lee, Phys. Rev.Lett. 82, 2915 (1999).
A.D. Gromko et al.,Phys. Rev.B 68, 174520 (2003). [14] J. X. Li, C. Y. Mou, and T. K. Lee, Phys. Rev. B 62,
[3] T.K. Kim et al.,Phys. Rev.Lett. 91, 167002 (2003). 640 (2000).
[4] T. Sato et al.,Phys. Rev.Lett. 91, 157003 (2003). [15] A. P. Kampf, Phys.Rep. 249, 219 (1994).
[5] A. Damascelli, Z. Hussain and Z. X. Shen, Rev. Mod. [16] Besides MDC, another intensity plot in ARPES is the
Phys.75, 473 (2003). energy distribution curve (EDC). Due to the abnormal
[6] P.D.Johnsonetal.,Phys.Rev.Lett.87,177007(2001). lineshapeofEDC[5],MDCismoreoftenusedtoanalyze
[7] H.F. Fong et al.,Phys. Rev.Lett. 75, 316 (1995). the dispersion.
[8] P.Bourges et al.,Science 288, 1234 (2000). [17] D. L. Feng et al.,Phys. Rev.Lett.86, 5550 (2001).
[9] T. Cuk et al.,Phys.Rev.Lett. 93, 117003 (2004). [18] M. R.Norman et al.,Phys. Rev.B 64, 184508 (2001).
[10] T. P. Devereaux, T. Cuk, Z. X. Shen, and N. Nagaosa, [19] S. Pailhes et al.,cond-mat/0308394 (2003).
Phys.Rev.Lett. 93, 117004 (2004). [20] Notice that theinterlayerAFcouplingJ⊥ also enhances
[11] M. Eschrig and M. R. Norman, Phys. Rev. Lett. 89, the coupling to spin resonance [see Eq.(3)].
277005 (2002). [21] D. E. Sheehy, T. P. Davis, and M. Franz,
[12] O. K. Anderson et al., J. Phys. Chem. Solids 56, 1573 cond-mat/0312529 (2003).
(1995).