D-XY Critical Behavior in Cuprate Superconductors

6 Pages·0.29 MB·

by  T. Schneider

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0
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0 D-XY Critical Behavior in Cuprate Superconductors
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n T. Schneider and J. M. Singer
a
J
Physik-Institut, Universit¨at Zu¨rich,
9
Winterthurerstr. 190, CH-8057 Zu¨rich, Switzerland
1

] Weoutlinetheuniversalandfinitetemperaturecriticalpropertiesofthe3D-XY model,extendedtoanisotropic
n extreme type-II superconductors, as well as the universal quantum critical properties in 2D. On this basis we
o
review: (i) the mountingevidence for 3D-XY behavior in optimally doped cuprate superconductors and the3D
c
to 2D crossover in the underdoped regime; (ii) the finite size limitations imposed by inhomogeneities; (iii) the
-
r experimental evidence for a 2D-XY quantum critical point in the underdoped limit, where the superconductor
p
toinsulator transition occurs; (iv) the emerging implications and constraints for microscopic models.
u
s
.
t
a The starting point of the phenomenological a normal to neutral superfluid transition, which
m
theory of superconductivity is the Ginzburg- is one of the best understood continuous phase
- Landau Hamiltonian transitions with unparalleled agreement between
d
n D ~2 2π 2 theory, simulations and experiment [1]. In ex-
co H = Z dDR(cid:18)Xj=1 2Mj (cid:12)(cid:12)(cid:18)i∇j+ Φ0Aj(R)(cid:19)Ψ(cid:12)(cid:12) tprleinmgettoypvee-cItIosruppoetrecnotnidaulcfltuocrtsu,ahtoiwonesvearp,ptheearcsotuo-
(cid:12) (cid:12)
[ − r|Ψ|2+ u|Ψ|4+ |r(cid:12)otA|2 , j =(x,y,z).(cid:12)(1) be weak [2], but nonetheless these fluctuations
1 2 8π (cid:19) drive the system – very close to criticality – to a
v charged critical point [3,4]. In any case, inhomo-
8 D is the dimensionality of the system, the com- geneities prevent cuprate superconductors from
5 plex scalar Ψ(R) is the order parameter, M the
entering this regime, due to the associated finite
2 effectivemassofthepairandAthevectorpoten-
size effect. For these reasons, the neglect of vec-
1 tial. The pair carries a non-zero charge in addi-
0 torpotential fluctuations appears to be a reason-
tiontoitsmass. Thecharge(Φ =hc/2e)couples
0 0 able starting point. In this case the vectorpoten-
the order parameter to the electromagnetic field
0 tial in Hamiltonian Eq. (1) can be replaced by
/ via the first term in H.
at If Ψ and A are treated as classical fields, the its most probable value. The critical properties
at finite temperature are then those of the 3D-
m relative probability P of finding a given configu-
ration [Ψ,Ψ∗,A] is then XY model, reminiscent to the lamda transition
-
d in superfluid helium, but extended to take the
n P[Ψ,Ψ∗,A] = exp(−βH[Ψ,Ψ∗,A]), effective mass anisotropy into account [5,6].
o Theuniversalpropertiesofthe3D-XY univer-
β = 1/(k T).
B
c sality class are characterized by a set of critical
:
v The free energy F follows from exponents, describingthe asymptotic behaviorof
i the correlation length ξ±, magnetic penetration
X exp − F =Z = D[A]D[Ψ]D[Ψ∗]P[Ψ,Ψ∗,A], depth λi, specific heat Ai±, etc., in terms of
r (cid:18) kBT(cid:19) Z
a
(2) ξi± =ξi±,0|t|−ν, λi=λi,0|t|−ν/2, C = Aα±|t|−α, (3)
where the partition function on the right hand
side corresponds to an integral over all possible where3ν =2−α. Asusual,intheaboveexpres-
realizations of the vector potential A, the order sion±refertot=T/T −1>0andt<0,respec-
c
parameter Ψ and its complex conjugate Ψ∗. Set- tively. Thecriticalamplitudesξ± ,λ2 ,A±,etc.,
i,0 i,0
ting e = 0, the free energy reduces to that for are nonuniversal, but there are universal critical
2
2.22
2.22
2.21
2.20 2.18
T
C/ 2.19 T
C/ 2.14
2.18
2.17 2.10
2.16
91.9 92.0 92.1 92.2 2.06
T −4 −3 −2 −1
Figure 1. Specific heat coefficient C/T log |t|
10
[mJ/(gK2)]versusT [K]ofYBa2Cu3O7−δ (sam- Figure 2. Specific heat coefficient C/T
ple YBCO3, [12]). The two arrows mark Tc ≈ [mJ/(gK2)] versus log10|t| for YBa2Cu3O7−δ
92.12K and TP ≈91.98K,respectively. (sample YBCO3, [12]) for Tc =92.12K.
amplitude relations, including [6]
specific heat, magnetic torque, diamagnetic sus-
(k T )3 = Φ20 3 ξx−,0ξy−,0ξz−,0 ceptibility, melting line, etc. It should be recog-
B c (cid:18)16π3(cid:19) λ2x,0λ2y,0λ2z,0 nized that the universal relations, Eq. (4), also
Φ20 3 R− 3 imply constraints on, e.g., isotope and pressure
= ,
(cid:18)16π3(cid:19) A−λ(cid:0)2x,0λ2y(cid:1),0λ2z,0 coefficients.
(R±)3 = A±ξx±,0ξy±,0ξz±,0. (4) XYAluthnoivuegrhsatlhiteyreinisthmeocuunptriantgese[v5i–d9e,n?c,e11fo]rit3aDp--
The singular part of the free energy density pears impossible to prove that unambiguously.
adopts in an applied magnetic field H the scal- Indeed, due to inhomogeneities, a solid always is
ing form [7]
homogeneous over a finite length L only. In this
k TQ± case, the actual correlation length ξ(t) ∝ |t|−ν
fs= ξ±Bξ±ξ3±G±3(Z), G±3(0)=1, (5) cannot grow beyond L as t → 0, and the tran-
x y z
sition appears rounded. Due to this finite size
where effect, the specific heat peak occurs at a tem-
perature T shifted from the homogeneous sys-
H = H(cos(φ)sin(δ),sin(φ)sin(δ),cos(δ)), P
tem by an amount L−1/ν, and the magnitude
1
Z = Φ0qHx2ξy2ξz2+Hy2ξx2ξz2+Hz2ξx2ξy2. (6) oLfαt/hν.e pTeoakqulaonctaitfeydthatistpemoipnterwateursehoTwPinscaFliegs.a1s
R± and Q±3 are universal numbers, and G±3(Z) the measured heat coefficient of YBa2Cu3O7−δ
is an universal scaling function of its argument. [12] around the peak. The rounding and the
Provided that this scenario applies to cuprate shape of the specific heat coefficient clearly ex-
superconductors,theimplicationsinclude: (i)the hibit the characteristic behavior of a system in
universalrelationsholdirrespectiveofthedopant confined dimensions, i.e., rod or cube shaped in-
concentration and of the material; (ii) given the homogeneities [13]. A finite size scaling analy-
nonuniversalcriticalamplitudesofthecorrelation sis [14] reveals inhomogeneities with a character-
lengths,ξi±,0,andtheformoftheuniversalscaling istic length scale ranging from 300 to 400˚A, in
function G±(Z), propertieswhichcanbe derived the YBa Cu O samples YBCO3, UBC2 and
3 2 3 7−δ
fromthefreeenergycanbecalculatedclosetothe UBC1 of Ref. [12]. For this reason, deviations
zerofieldtransition. Thesepropertiesincludethe from 3D-XY critical behavior around T do not
P
3
1.0 0.10
0.0
0.8 0.08
-1.0
δ)H(s00..46 00..0046 bH(δ) +-QdG/dz1-2.0 YLHag--1-21212340, ,1 , TTTccc === 938348...358550 KKK,,, γγγ === 712.805..45
-3.0 Hg-1201, T = 95.65 K, γ = 29.0
c
0.2 0.02 La-214, Tc = 21.60 K, γ = 46.0
-4.0 La-214, T = 19.35 K, γ = 51.0
c
0.0 0.00 -4.0 -3.0 -2.0 -1.0 0.0 1.0
8877 8888 8899 9900 9911 9922 9933 z sgn(T/T - 1)
c
δδ Figure 4. Scaling function dG±(Z)/dZ derived
3
from the angular dependence of the magnetic
Figure3. H∗(δ)forBi Sr CaCu O withT =
2 2 2 8+δ c
torque for YBa Cu O , La Sr CuO ,
84.1K for T = 79.5K (◦) and T = 82.8K ((cid:4)). 2 3 6.93 1.854 0.146 4
HgBa CuO , HgBa CuO ,
Experimental data are taken from [16], the solid 2 4.108 2 4.096
La Sr CuO and La Sr CuO .
lines are derived from the respective 2D (H , 1.914 0.086 4 1.920 0.080 4
s
‘slab’, —) and 3D (H , ‘bulk’ with γ = 77, - -
b
-) scaling forms.
signal the failure of 3D-XY universality, as pre-
tion depth. Using A+ = 8.4·1020cm−3, derived
viously claimed [12], but reflect a mere finite size
fromthedatashowninFig. 2forsampleYBCO3
effectatwork. Indeed, fromFig. 2it is seenthat
with T = 92.12K, λ = 1153˚A, λ = 968˚A
the finite size effect makes it impossible to enter c a,0 b,0
and λ = 8705˚A, derived from magnetic torque
the asymptotic critical regime. To set the scale c,0
measurements on a sample with T = 91.7K [7],
we note that in the λ-transition of 4He the criti- c
as well as the universal numbers A+/A− = 1.07
cal properties can be probed down to |t| = 10−9
and R− ≈ 0.59, we obtain T = 88.2K. Hence,
[1,15]. In Fig. 2 we marked the intermediate c
the universal 3D-XY relation (4) is remarkably
regime where consistency with 3D-XY critical
well satisfied.
behavior, i.e., with C/T = A±10−αlog10|t| +B±
Another difficulty results from the pronounced
for α = −0.013 and A+/A− = 1.07, can be ob-
e e anisotropyofthe cuprates: a convenientmeasure
served. TheupperbreanchecorrespondstoT <Tc is the effective mass parameter γ = Mk/M⊥,
and the lower one to T > Tc. The open circles which depends on the dopant conpcentration.
closer to T correspond to the finite size affected Eventhoughthe strengthofthermalfluctuations
c
region, while further away the temperature de- grows with increasing γ, they are slightly away
pendence of the background, usually attributed fromTcessentiallytwo-dimensional. Accordingly,
to phonons, becomes significant. Hence, due to the intermediate 3D-XY critical regime shrinks,
andthecorrectionstoscalingareexpectedtobe-
the finite size effect and the temperature depen-
comesignificant. Anexperimentaldemonstration
dence of the backgroundthe intermediate regime
of the temperature driven dimensional crossover
is bounded by the temperature region where the
is shown in Fig. 3 in terms of the angular de-
data depicted in Fig. 2 fall nearly on straight pendence of the onset field H∗, where a measur-
lines. To provide quantitative evidence for 3D- able resistance is observed [16]. H∗(δ) follows
XY universality in this regime, we invoke the from Z(H∗) = Z∗. For superconducting sheets
universal relation (4) and calculate Tc from the of thickness ds, corresponding to 2D, the argu-
critical amplitudes of specific heat and penetra- ment of the scaling function G±(Z) is given by
2
4
140
120
100
80
K] 108 λ−2(T=0) [A−2]
T [c 60 0.0 0.2 0.4|| 0.6 0.8
10
8
40
K] 6
20 T [c 4
2
0
0
0 1 2 3 4 5 6 7 8 9
Figure 5. T and γ = M /M versus dopant
c ⊥ k 105 λ−2(T=0) [A−2]
concentration x for La2−pxSrxCuO4. ||
Figure 6. T versus λ−2(T → 0) as obtained
c k
from µSR measurements for La Sr CuO (N),
2−x x 4
YBa Cu O ,Bi Sr Ca Y Cu O (•),and
[17,18] 2 3 7−x 2 2 1−x x 2 8+δ
Tl Ba Ca Cu O ((cid:7)). Data taken from [23].
2 2 2 3 10
Z = H (ξ±)2|cos(δ)|+ H2(ξ±)2d2sin2(δ) 1/2. The straight line marks Tc = 3·108λ−k2 with Tc
(cid:18)Φ0 k Φ20 k s (cid:19) in [K] and λk in [˚A]. Inset: Tc vs. λ−k2(T = 0)
(7) of Bi Sr CuO ((cid:4)); the straight line is a
2+x 2−x 6+δ
guide to the eye. Data taken from [24].
Noting that Eqs. (6) and (7) lead to distinct
bell-shaped (3D) and cusp-like (2D) behavior
aroundδ =90o,respectively,thesemeasurements
clearly illustrate the temperature driven dimen- a strict 3D- to 2D-XY crossover occurs. By ap-
sional crossover. Indead, as seen in Fig. 3, at proaching the underdoped limit x = xu ≈ 0.05,
T = 79.5K H∗(δ) mirrors a 2D film behavior, γ = M⊥/Mk becomes very large (see Fig. 5)
while closer to Tc at T = 82.8K 3D bulk be- and Tpc vanishes. Here the materials correspond
toastackofindependentsheetsofthicknessd As
havior appears. To illustrate the difficulties as- s
there is a phase transition line with an endpoint
sociated with this crossover we show in Fig. 4
T (x = x ) = 0, one expects a doping driven
estimates for the derivative of the universal scal- c u
2D-XY insulatortosuperconductortransitionat
ingfunctionG±(Z)derivedfrommagnetictorque
3 T = 0. For such a transition the scaling theory
measurements [11]. Even though the qualitative
of quantum critical phenomena predicts [19–21]
behavior is the same for all samples, the devi-
ations increase systematically with increasing γ. lim 1 (T (δ))2λ2(δ,T=0)λ2(δ,T=0)=
This systematics cannot be attributed to the ex- δ→0d2s c x y
perimentaluncertaintiesofabout40%. Itismore 1 Φ20 2
= (8)
likely that it reflects the 3D to 2D crossoverand Q22,0 (cid:18)16π3kB(cid:19)
the associated reduction of the 3D-XY fluctua-
to be universal. δ = (x−x )/x is the control
tion dominated regime, requiring corrections to u u
parameter, Q is an universal number and
scaling. Indeed, in the derivation of the scaling 2,0
function from the experimental data, both, cor- T ∝δzν, λ2(δ,T=0)∝δ−ν. (9)
c i
rections to scaling and finite size effects have not
been considered. z is the dynamic critical exponent and ν the ex-
Inmaterials,suchasLa Sr CuO ,wherethe ponent of the correlation length. In Fig. 6 we
2−x x 4
underdoped regime is experimentally accessible, depict experimental data in terms of T versus
c
5
1/λ2(T = 0). As T approaches the underdoped 1. A. Singsaas and G. Ahlers, Phys. Rev. B30
k c
limit, the data appear to merge on the solid line. (1984) 5103.
In this context it should be emphasized that the 2. D.S.Fisheretal,Phys.Rev.B43(1991)130.
data, with the exception of Bi Sr CuO , 3. I. F. Herbut and Z. Tesanovic, Phys. Rev.
2+x 2−x 6+δ
are rather far away from the asymptotic regime Lett. 76 (1996) 4588.
where Eq. (8) is expected to apply. Moreover, 4. A. K. Nguyen and A. Sudbo, cond-
d is known to adopt material dependent values mat/9907385.
s
[22]. Nevertheless, the data collected in Figs. 5 5. T. Schneider and D. Ariosa, Z. Phys. B89
and 6 clearly point to a quantum phase transi- (1992) 249.
tion in D = 2 at x = x where T vanishes and 6. T.SchneiderandH.Keller,Int.J.Mod.Phys.
u c
λ2(T =0)tendstoinfinity. Hence,thereisstrong B8 (1993) 487.
k
evidencefora2D-XY quantumphasetransition, 7. T. Schneider et al, Eur. Phys. J. B3 (1998)
where Eq. (8) applies and dT /d(1/λ2(T =0)) is 413.
c k
not universal, as suggested by Uemura et al [23], 8. M. A. Hubbard et al, Physica C259 (1996)
but depends on d . 309.
s
To summarize, there is mounting evidence for 9. S. Kamal et al, Phys. Rev. Lett. 73 (1994)
intermediate D-XY critical behavior, a 3D- to 1845; Phys. Rev. B58 (1998) R8933.
2D- crossover as the underdoped limit is ap- 10. V. Pasler et al, Phys. Rev. Lett. 81 (1998)
proachedandfortheoccurrenceofaquantumsu- 1094.
perconductortoinsulatortransitionattheunder- 11. J. Hofer et al, Phys. Rev. B60 (1999) 1332;
dopedlimitinD =2. Emergingimplicationsand Preprint, 1999.
constraintsfor microscopicmodels include: (i) in 12. M. Charalambous et al, Phys. Rev. Lett. 83
theexperimentallyaccessibletemperatureregime (1999) 2042.
andclosetooptimumdoping,thereisremarkable 13. E. Schultka and E. Manusakis, Phys. Rev.
consistency with 3D-XY universality; (ii) close Lett. 75 (1995) 2710; cond-mat/9602085;
to criticality the symmetry of the order parame- cond-mat/9702216;cond-mat/9811251.
teriseitherd-waveors-wave;(iii)thedecreaseof 14. T. Schneider and J. M. Singer, cond-
T in the underdoped regime mirrors the dimen- mat/9911352.
c
sional crossover, enhancing thermal fluctuations 15. S.Methaetal,J.LowTemp.Phys.114(1999)
and the competition with quantum fluctuations 467.
which suppress superconductivity at the under- 16. E. Silva et al., Physica C214 (1993) 175.
dopedlimit;(iv)theenhancedthermalandquan- 17. T.SchneiderandA.Schmidt,Phys.Rev.B47
tum fluctuations reduce the single particle den- (1993) 5915.
sity of states at the chemical potential. This re- 18. T. Schneider and J. M. Singer, cond-
duction leads to a pseudogap above T ; (v) these mat/9911411.
c
fluctuations imply the existence of phase uncor- 19. Min-Chui Cha et al, Phys. Rev. B44 (1991)
related pairs above T and invalidate mean-field 6883.
c
treatments, including the Fermi liquid approach 20. K.KimandP.B.Weichman,Phys.Rev.B43
in the normal state; etc. (1991) 13583.
For a more elaborate review of the D-XY be- 21. T. Schneider, Acta Physica Polonica A91
haviorincupratesuperconductorswerefertoRef. (1997) 203.
[25]. 22. T. Schneider and J. M. Singer, Physica C313
The authors are grateful to J. Hofer for very (1999) 188.
useful comments and suggestions on the subject 23. Y.J.Uemuraetal,Phys.Rev.Lett.59(1987)
matter. 1045; Phys. Rev. B38 (1988) 909.
24. E. Janod. PhD thesis, CEA Grenoble, 1996.
REFERENCES 25. T. Schneider and J. M. Singer, Phase Tran-
sition Approach To High Temperature Super-
6
conductivity, ImperialCollegePress,London,
2000.