Origin of Drastic Change of Fermi Surface and Transport Anomalies in CeRhIn5 under Pressure

0.19 MB·English

by  Shinji Watanabe

Letter
Origin of Drastic Change of Fermi Surface and Transport Anomalies in CeRhIn
5
under Pressure
Shinji Watanabe and Kazumasa Miyake
Division of Materials Physics, Department of Materials Engineering Science, Graduate School of Engineering Science,
Osaka University, Toyonaka, Osaka 560-8531, Japan
0 Themechanism of drasticchangeof Fermisurfaces as wellas transport anomalies nearP =
1 Pc ≈2.35 GPa in CeRhIn5 is explained theoretically. Thekey mechanism is pointed out tobe
0 theinterplayofmagneticorderandCe-valencefluctuations.Weshowthattheantiferromagnetic
2 statewith“small”Fermisurfaceschangestotheparamagneticstatewith“large”Fermisurfaces

n withhugeenhancementofeffectivemassofelectronswithkeepingfinitec-fhybridization.This
a explainsthedrasticchangeofthedeHaas-vanAlphensignals.Furthermore,itisalsoconsistent
J with the emergence of T-linear resistivity simultaneous with the residual resistivity peak at

5 P =Pc in CeRhIn5.
1
KEYWORDS: CeRhIn5, Fermisurface, non-Fermiliquid,localized-itinerant transition,valence fluctuation
]
l
e
- The mechanism of instability of electronic states sistivity ρ(T = 2.25 K) has a sharp peak at P = P ,
r c
t emerging when the magnetically-orderedtemperature is suggesting that the residual resistivity ρ is strongly en-
s 0
. suppressed to absolute zero by tuning material parame- hanced near P = P .7,11,12 The low-temperature resis-
t c
a tershasbeenoneofthe centralissuesincondensedmat- tivity also exhibits anomalous behavior: The T-linear
m terphysics.1–3 Aheavy-electronmetalCeRhIn hasbeen registivity ρ(T) T emerges most prominently near
5
∝
- extensively studied as the prototypical material. Accu- P = P with a wide-T range up to 10 K.7,11,12 This
d c ∼
mulated experiments, however, have revealed that the behavior is quite different from that of normal metals
n
o physics of CeRhIn5 seem to be beyond conventional un- described by the Fermi liquid ρ T2 as well as the
∝
c derstanding1–3 and require an essentially new concept. conventional quantum criticality scenario near the AF
[ CeRhIn undergoes an antiferromagnetic (AF) tran- quantum critical point (QCP) in three dimension (3D)
5
1 sition at TN = 3.8 K with an ordered vector Q = ρ T1.5.1–3 The origin and mechanism of these trans-
∝
v (1/2,1/2,0.297)at ambient pressure.4 When pressure is portanomaliesaswellasthedrasticchangeoftheFermi
8 applied under a magnetic field larger than the upper surface at P = P accompanied by huge mass enhance-
c
9
critical field, the AF order is suppressed at P = P ment have not been clarified so far.
5 c ≈
2.35 GPa.5–11 Interestingly, a drastic change of Fermi Recently, an important measurement has been per-
2
. surfaces was discovered at P = Pc by the de Haas-van formed under pressure and magnetic field.11 G. Knebel
1 Alphen (dHvA) measurement:9 For 0 P < P , the et al. have measured the T2 coefficient A in the resistiv-
0 ≤ c
maindHvAfrequenciesareingoodagreementwiththose ity in the T 0 limit at the magnetic field H = 15 T,
0
→
1 ofLaRhIn5 wherethe4felectronis absent.Onthe other and found that A increases as pressure increases toward
: hand, for P > P , the dHvA frequencies were identi- P =P .Theyhaveshownthatunderpressure,√Ashows
v c c
i fied to be approximately the same as those of CeCoIn5, anenhancementsimilartotheeffectivemassofelectrons
X where 4f electrons are itinerant, contributing to the for- m∗ of the β blanch obtained by the dHvA measure-
2
r mation of the Fermi surface. The electrons on a typi- ment,9 satisfying the √A/m∗ =const. scaling. This in-
a
cal Fermi surface, the cylindrical β blanch, shows the dicates that the mass enhancement is not caused by the
2
mass enhancement from m∗ = 6m at P = 0 to 60m quantum criticality of the AF spin fluctuations, since in
0 0
at P 2.2 GPa with m being a free-electron mass.9 the case of the 3D (2D) AF QCP, √A/m∗ is expected
0
≈
For P > P , the dHvA signal of the β blanch is lost to show the T−1/4 (T−1/2) divergence.1–3 Namely, the
c 2
probably because of a too-large effective mass close to √A/m∗ =const. scaling strongly suggests that the mass
100m , where the heavy-electron state is realized in the enhancement near P = P purely comes from the band
0 c
paramagnetic-metalphase.9 effect.
So far, it has been thought that this drastic change of On the basis of these observations, in this Letter, we
theFermisurfacesmightbeexplainedintermsofthe“lo- present a theoretical explanation for resolving this out-
calized” to “itinerant” transition of f electrons.12 How- standing puzzle in CeRhIn . We show that the drastic
5
ever, this conception encounters a serious difficulty in change of the Fermi surface from a “small” to a “large”
elucidating the experimentalfact that the effective mass one occurs at the phase transitionfrom AF to paramag-
of electrons is enhanced even at P = 0 with the Som- neticmetalwithhugemassenhancementunderpressure,
merfeld constant γ 56 mJmol−1K−2,5 which is about asobserved.Animportantpointhereisthattheseresults
≈
10 times larger than that of LaRhIn .13,14 areobtainedwithfinitehybridizationbetweenfandcon-
5
Furthermore,resistivitymeasurementsrevealedthata duction electrons, which overcomesthe difficulty of con-
striking anomaly emerges near P = P :7 The low-T re- ventional scenario that f electrons undergo “localized”
c
2 J.Phys.Soc.Jpn. Letter S.WatanabeandK.Miyake
to“itinerant”transition.12 Ourresultnotonlynaturally (a) χ (b)
explains the √A/m∗ =const. scaling, but also gives the 1.5 1 v
reasonwhytheT-linearresistivityemergesaswellasthe 0.8 Ufc = 1.0
residual resistivity has a peak in the vicinity of P =Pc. 0.6 0.5
Let us start our discussion by introducing a minimal 0.4
1 0.2 0.0
model, which describes the essential part of the physics
of CeRhIn5, in the standard notation: Ufc 0-1 0 1ε2 3 4
m f
H =Hc+Hf +Hhyb+HUfc, (1) 1 s
0.5
0.8
where H = ε c† c represents the conduction
c Pkσ k kσ kσ 0.6
band, H = ε nf + U N nf nf the f level 0.4
and onsitfe CoulfoPmbiσreipσulsionPfoir=1f eil↑ectir↓ons, H = 0.2 (c)
hyb
0
V Piσ(cid:16)fi†σciσ+c†iσfiσ(cid:17)thehybridizationbetweenfand -2 -1 ε0 1 2 -00.4-0.3ε5-0.3-0.25
conduction electrons, and H = U N nfnc the f f
Ufc fcPi=1 i i
Coulomb repulsion between f and conduction electrons,
Fig. 1. (color online) (a) Ground-state phase diagram in the
respectively. The H term is a key parameter for ex-
Ufc plane of Ufc and εf for paramagnetic and AF states (see text).
plainingtheanomaloustransportpropertiesofCeRhIn5; The first-order valence-transition line (solidline with triangles)
The ρ peak and ρ T observed in CeRhIn are quite terminates at the quantum critical point (filled circle). The va-
0 5
similar to the observ∝ations in CeCu Ge ,15 CeCu Si ,16 lence crossover occurs at the dashed line with open circles, at
2 2 2 2
and CeCu (Si Ge ) .17 The pressure dependence of which χv has a maximum, as shown in(b). The solidline with
2 x 1−x 2 filledsquaresrepresentstheboundarybetweentheAFstateand
thecoefficientA15–17 aswellastheCu-NQRfrequency18
the paramagnetic state (see text). (c) The AF order parameter
strongly suggest that these anomalies occur at the pres- ms vs. εf for Ufc = 0.5. All results in (a)-(c) are calculated for
sure where the valence of Ce changes sharply. The Ce- t=1,V =0.2,andU =∞atn=0.9.
valence transition is well known as the γ-α transition
in Ce metal. Band-structure calculation for Ce metal
showed that 4f- and 5d-electron bands are located at n¯f ≡Pihnfii/N and n¯c ≡Pihncii/N. By approximating
the Fermi level.19 Since both the orbitals are located mean fields and Lagrange multipliers as uniform ones,
at the same Ce site, the inter-orbital Coulomb repul- i.e., e=heii, pσ =hpiσi, andd=hdii, λ′ =λ′i,δλ=δλi,
sion Ufc has a considerable magnitude, causing the first- and λ = λi, respectively, the set of mean-field equa-
order valence transition.20,21 Although U is considered tions is obtained by optimal conditions ∂ H /∂x = 0
fc h i
to be rather moderate in Ce compounds, valence fluctu- for x=λ′,δλ,λ,e,d,p↑, and p↓:
ationsstillaffectphysicalquantitiessignificantlyevenin e2+p2↑+p2↓+d2−1=0,
suchavalence-crossoverregime:21–24Theoreticalcalcula- p2 p2 = 2 ′ f† f f† f ,
tionsbasedonthemodel(1)haveshownthatstrongCe- ↑− ↓ N Pkhh k↑ k+Q↑i−h k↓ k+Q↓ii
valence fluctuations cause the enhancement of the resid- p2↑+p2↓+2d2 = N1 P′kσhhfk†σfkσi+hfk†+Qσfk+Qσii,
ual resistivity25 as well as the T-linear resistivity.16 V ′ ∂Zσ f† c + f† c +λ′e=0,
Since the ρ peak and ρ T appear just at the N Pkσ(cid:0) ∂e (cid:1)hh kσ kσi h k+Qσ k+Qσii
0
boundary between the AF a∝nd paramagnetic phases, NV P′kσ(cid:0)∂∂Zdσ(cid:1)hhfk†σckσi+hfk†+Qσck+Qσii
P Pc, in CeRhIn5, the interplay of AF and the Ce- +(U +λ′ 2λ)d=0,
val≈ence fluctuation seems to be a key mechanism.8 To V ′ ∂Zσ f† c + f† c −
treat both effects on equal footing, we apply the slave- N Pkσ(cid:16)∂p↑(cid:17)hh kσ kσi h k+Qσ k+Qσii
boson mean-field theory26 to Eq. (1). We use the hy- +(λ′ λ δλ)p↑ =0,
− −
bridizationformVZiσfi†σciσ insteadofVfi†σciσ inEq.(1) NV P′kσ(cid:16)∂∂Zp↓σ(cid:17)hhfk†σckσi+hfk†+Qσck+Qσii
by introducing bose creation (annihilation) operators +(λ′ λ+δλ)p =0,
↓
e†(e ) and d†(d ) for the empty and doubly-occupied where Q is the AF-ordered vector.−Here, ′ denotes
stiateis, respecitiviely, and p† (p ) and p† (p ) for singly- the summation over the 1st Brillouin zone.PTkhe chemi-
i↑ i↑ i↓ i↓
occupiedstatesbyrequiringtheconstraintforcomplete- cal potential µ is determined so as to give the total fill-
inannegsdstPhceoinσLdλaiitgσiro(anfni†σgPfeiiσmλ−u′i(ltpei†i†ipσelpiiei+σrs−.p†iHd↑pe†idri↑ei,)+twhpiet†ih↓rpeλin↓′io+armndda†ildλiziiσa−tbio1en)- ithnfiogk†n+nsQaσ=rfek(s+n¯ofQlv+σeidn¯+cs)he/clf†k2-σcwcoknitσshiis+tn¯efhnc+t†kl+yn¯.Qcσ=ck+N1QPσi′k].σT[hhfek†sσefkeσqiua+-
factor for hybridization is defined as Z (1 d†d To clarify the mechanism of drastic change of the
p† p )−1/2(e†p +p† d )(1 e†e p†iσ ≡p )−−1/i2.iT−o Fermi surface of CeRhIn5, the most typical one, the
iσ iσ i iσ i−σ i − i i− i−σ i−σ two-dimensional-like Fermi surface, observed as the β
capture the essence of CeRhIn as noted later, we con- 2
5
blanch, we consider ε = 2t(cosk + cosk ) on the
sider the commensurate AF order on a bipartite lattice k x y
−
square lattice. Hereafter, the energy unit is taken as
and consider λ and λ corresponding to the two
iaσ ibσ
t = 1. To simulate the AF order in the heavy-electron
sublattice (ia and ib sites) instead of λ , which are ex-
iσ
state realized in CeRhIn , we consider the small hy-
pressedasλ =λ λ +δλ , λ =λ λ δλ , 5
ia↑ ib↓ i i ia↓ ib↑ i i
≡ ≡ − bridization V = 0.2 case near half filling n = 0.9, and
respectively. For H in Eq. (1), we employ the mean-
field decoupling asUfncfnc n¯ nc + n¯ nf n¯ n¯ with the AF order with Q = (π,π). Although we here cal-
i i ≃ f i c i − f c
J.Phys.Soc.Jpn. Letter S.WatanabeandK.Miyake 3
culated several U cases, we show the result for U = ,
∞
since the result is essentially unchanged even for finite
U, as far as U is larger than the bandwidth.
First, we show the ground-state phase diagram deter-
mined under the assumption of the paramagnetic states
with p = p and δλ = 0 in Fig. 1(a). The first-order
↑ ↓
valence transition line (solid line with filled triangles)
terminates at the QCP (filled circle) at (εQCP,UQCP)=
f fc
( 0.3623,1.1778).For U > UQCP, n¯ shows a jump as
− fc fc f
a function of ε , indicating the first-order transition be-
f
tween the paramagnetic metals with n¯ close to 1 and
f
n¯ < 1 in deep-ε and shallow-ε regions, respectively,
f f f
since large U forces electrons to pour into either the
fc
f level or the conduction band.20–22,24 At the QCP, va-
lence fluctuations diverge χ = ∂n¯ /∂ε = , and for
v f f
− ∞
U <UQCP,χ hasaconspicuouspeakatε represented
fc fc v f
bythedashedlinewithopencirclesinFig.1(a),indicat-
ing strong valence fluctuations, as shown in Fig. 1(b).
At the QCP, the characteristic energy scale of the sys-
tem, the so-called Kondo temperature is given by T
K ≡ Fig. 2. (color online) The contour plot of the energy band with
ε¯f µ=2.93 10−3 with ε¯f =εf +λ+Ufcn¯c. ↓ spin located at the Fermi level µ for t = 1, V = 0.2, U =
− ×
When the AF states are taken into account, the AF ∞, Ufc = 0.5, and n = 0.9 at h = 0.005: (a) εf = −0.40, (b)
order parameter defined as ms ≡ p2↑ −p2↓ decreases as εpfar=ts−ar0e.2r9e,pr(ecs)enεfte=db−y0w.2h8i,teanredgi(odn)s.εIfn=(a−)0a.n2d5.(bT)h,ethEekd↓as>heµd
ε increases, as shown in Fig. 1(c) for U = 0.5. When
f fc
line indicates the Fermi surface of the conduction band, εk for
the ground-state energies of this AF state and the para-
n¯c=0.8.
magnetic state are comapared, the level crossing occurs
at ε =εc. Then, this AF and paramagnetic phase tran-
f f
sition is identified to be of the first order. The phase only conduction electrons, we plot the Fermi surface of
boundary determined in this way is shown by the solid the conduction band ε at the filling n¯ = 0.8 as the
k c
line with filled squaresin Fig.1(a). We find thatthe AF dashed line in Fig. 2(a). This Fermi surface corresponds
orderterminates inthe vicinity ofthe first-ordervalence tothatwhenthehybridizationbetweenfandconduction
transition line and the valence-crossover line. These re- electrons is switched off, V = 0 in Eq. (1); Since f elec-
sults imply that the suppression of the AF order occurs trons for n¯ = 1 are located at the localized f level for
f
at the points with strong valence fluctuations. V =0,extraelectronsinn=0.9,i.e.,n¯ =0.8areinthe
c
These results are favorably compared with CeRhIn5. conduction band. We see that the Fermi surface of the
Applying pressuretoCe systems correspondsto increas- AF state for V = 0.2 is nearly the same as the “small”
ing εf, since negative ions approach the tail of 4f wave- Fermi surface represented by the dashed line. This cor-
function at the Ce site. Experimental fact of the sudden respondsto the experimentalfactthat the dHvAsignals
disappearance of the AF order at P = Pc and simulta- of CeRhIn5 are very similar to those of LaRhIn5 in the
Pneo=usPcemseeermgetnocebeowf ethlledeρs0crpibeaekdbays twheellreassuρlts∝shTownneainr AFT-horedesrhedappehasoef fotrheP ≤FePrmc.9i,13surface close to the
Fig.1:Thevicinityofεcf formoderateUfc(<UfQcCP)with “small” Fermi surface remains until εf reaches the AF-
well-developedχv seems to correspondto the vicinity of paramagnetic boundary εc = 0.283, as shown in
P =Pc in CeRhIn5. Fig. 2(b), which corresponfds to −CeRhIn5 at P < Pc.
To analyze the Fermi-surface change shown by the When ε exceeds εc, the Fermi surface drast∼ically
dHvA measurement in CeRhIn , we apply the magnetic f f
5 changes, as shown in Fig. 2(c) for ε = 0.280; The
field to Eq. (1) as h (Sfz +Scz). The dHvA effect9 f −
andAcoefficient11−werPemi eaisurediatH =12 17Tand folding of the lower hybridized band disappears and the
∼ “large” Fermi surface recovers, which is clearly different
H =15 T, respectively. The magnetic field of H =15 T
from the “small” Fermi surface shown in Figs. 2(a) and
is estimated to be h=0.0046t when the half bandwidth
(b). This “large”Fermi surface remains in the paramag-
of the conduction band 4t of Eq. (1) is compared with
netic phase, as shown in Fig. 2(d) for ε = 0.250.
f
thatofCeRhIn bytheband-structurecalculation,about −
5 To facilitate the comparisonwith the dHvA result,we
1.5eV.27 Then,weshowinFig.2thecontourplotofthe
plot the Fermi wave number k , defined by the distance
F
energy band of Eq. (1) with spin located at the Fermi
↓ betweenk=(0,0)andtheintersectionpointoftheFermi
level µ for U =0.5 at h=0.005.
fc surface and the line connecting k = (0,0) and (π,π), in
IntheAFstate,thelowerhybridizedbandofEq.(1)is Fig. 3(a). Here, for the AF-ordered state ε < εc, k in
folded,givingrisetotheholeregionemergingatthemag- f f F
the1stBrillouinzoneisplotted.Weseethatk isalmost
F
netic zone boundary connecting k=(0,π) and (π,0), as
unchangedasafunctionofε ,whichisnearlythesameas
f
shown in Fig. 2(a). Here, in order to make a compar- k oftheconductionbandatn¯ =0.8,kc representedby
ison with the “small” Fermi surface, which consists of thFesolidline.Then,k showsacnabruptFjumpatεc.This
F f
4 J.Phys.Soc.Jpn. Letter S.WatanabeandK.Miyake
3 creases.Thisis alsoconsistentwiththe field-dependence
(a)
10 AF of m∗ of the β blanch in CeCoIn ,29 which is expected
s20 2 5
e (b) to correspond to the paramagnetic state of CeRhIn for
t 5
kF2 kcF of sta 5 P T>hPeci.n23c,r3e0ase in D(µ) toward εcf, shown in Fig. 3(b),
y 10 suggests that total DOS at the 3D Fermi surface of
t
1 nsi 0-0.4 -0.35 -0.3 CeRhIn5 givesrisetothemeasuredpeakstructureofthe
AF para De AF εf para A coefficient at P = Pc.11 Our results clearly show that
the“small”FermisurfacecanbeobservedbydHvAmea-
0 0
surement even without switching off the c-f hybridiza-
-0.4 -0.35ε -0.3 -0.25 -0.4 -0.35 ε -0.3 -0.25
f f tion, which reminds us of the elucidation of metamag-
netism in CeRu Si .31–33
2 2
Fig. 3. (coloronline)(a)Fermiwavenumber kF vs.εf intheAF Recently, a possibility that the AF and paramagnetic
state (circles) and the paramagnetic state (triangles) for t=1, transitioniscontinuousunderpressurehasbeenreported
V =0.2,U =∞,Ufc=0.5,andh=0.005atn=0.9.Thesolid by the In-NQR measurement at H =0.34 To clarify the
line represents kF for the conduction band εk at n¯c = 0.8. (b)
detailed nature of the phase competition at the mag-
D(µ)vs.εf forthesameparametersas(a).Insetisenlargement
oftheAFphase. netic field smaller than the upper critical field, 10 T,
∼
existence of the superconductivity should be taken into
account, which is out of scope of the present study.
is quite consistentwiththe dHvAmeasurementthat the In summary, we have shown that the drastic change
dHvA frequencies, including the β blanch, keep almost of Fermi surface with huge mass enhancement as well as
2
constant for 0 P < Pc, and they suddenly jump at the transport anomalies in CeRhIn5 under pressure are
≤
P =P and remain constant for P >P . naturally explained from the viewpoint of the interplay
c c
We find that the measured enhancement of the ef- of the AF order and Ce-valence fluctuations.
fective mass of electrons from m∗ 6m at P = 0
0
∼
to m∗ 60m at P P 9 is also reproduced; The
0 c−
∼ →
ε dependence of the density of states (DOS) at the 1) T.Moriya,SpinFluctuationsinItinerantElectronMagnetism
f
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f
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