Huge metastability in high-T_c superconductors induced by parallel magnetic field

5 Pages·0.15 MB·English

by  R. G. Dias

Huge metastability in high T superconductors induced by parallel magnetic field
c
R. G. Dias
Departamento de F´ısica, Universidade de Aveiro,
3810 Aveiro, Portugal.
3
0
0 J. A. Silva
2 Instituto Polit´ecnico de Portalegre,
7301 Portalegre Codex, Portugal.
n
(Dated: February 2, 2008)
a
J Wepresentastudyofthetemperature-magneticfieldphasediagramofhomogeneousandinhomo-

1 geneoussuperconductivityinthecaseofaquasi-two-dimensional superconductorwithan extended
2 saddle point in the energy dispersion under a parallel magnetic field. At low temperature, a huge
metastability region appears, limited above by a steep superheating critical field (H ) and below
sh
] by a strongly reentrant supercooling field (H ). We show that the Pauli limit (H ) for the upper
l sc p
e critical magnetic field is strongly enhanced due to the presence of the Van Hove singularity in the
- density of states. The formation of a non-uniform superconducting state is predicted to be very
r
t unlikely.
s
.
t PACSnumbers: 74.60.Ec
a
m
- The theoretical phase diagram of two-dimensional withξkσ =ξk σh,ξk =ǫk µandh=µBH andwhere
d − −
isotropic superconductors under parallel magnetic fields V, µ, H, S and µ are respectively the attractive in-
n B
showsdistinctivefeaturessuchasatricriticalpointwhere teraction constant, the chemical potential, the magnetic
o
a high temperature second order phase transition gives field, the area of the system and the Bohr magneton.
c
[ wayto alowtemperature firstordertransition[1,2]and Theksumsinthe interactiontermfollowthe usualBCS

consequently to a region of metastability. Furthermore, restrictions.
1
in such systems, superconductivity may persist at mag- S-wave symmetry is assumed throughout the paper,
v
netic fields higher than the first order critical field if the for simplicity. The effects of d-wave symmetry in the
9
7 formation of an inhomogeneous superconducting phase, case of a superconductor with constant density of states
3 i.e.,theso-calledFulde-Ferrel(FF)phase,[2,3,4]occurs. have been addressed by several authors. [9] It leads to
1 Superconductingcopper-oxideshavestrongquasi-two- visible modificationsinthe phase diagramfor inhomoge-
0 dimensional character, but the energy dispersion of the neous superconductivity with, in particular, the appear-
3
CuO planes is clearly anisotropic and in particular, it ance of a low temperature kink in the phase boundary
0 2
contains extended saddle points as documented by an- between the FF phase and the normal phase. However,
/
t gleresolvedphotoemissionexperiments.[5]Extendedsad- the phase diagram for homogeneous superconductivity
a
m dle points have also been found in the case of the two- remains very similar to that of an s-wave superconduc-
dimensional Hubbard model by Assaad and Imada [6] tor.
-
d using the Quantum Monte Carlo technique. Such sad- Minimizing the free energy[2, 10] or, equivalently, fol-
n dle points lead to strong Van Hove singularities (VHS) lowing a BCS mean field approach in the case of an
co in the density of states and recently, it has been sug- homogeneous gap function, ∆ = −VS−1 kha−k↓ak↑i,
gested [7] that such singularities could provide an expla- one obtains the two-dimensional gap ePquation 1 =
:
v nation for the strange out-of-plane upper critical field of VS−1 [1 f(ξ∆) f(ξ∆)](2ξ∆)−1 with ξ∆ =ξ∆
Xi the High-Tc superconductors.[8] In this paper, we study σh, ξ∆P=p −ξ2 +∆p↑2 a−nd wph↓ere fp(x↑) is the Fekrmσ i disktr−i-
k k
r the phase diagram of a two-dimensional superconductor bution funpction. Taking the limit ∆ 0, one obtains
a with an extended saddle point in the energy dispersion →
under parallel magnetic field.
Inastrictlytwo-dimensionalsuperconductorunder an 1 ωD N(ξ) ξ h ξ+h
= dξ tanh − +tanh (2)
in-plane magnetic field, orbital frustration due to mag- V Z 2ξ (cid:20) 2t 2t (cid:21)
0
netic field is not present and the upper critical field is
where ω is the usual frequency cutoff and t = k T.
determinedbyZeemanpairbreaking. Therefore,wecon- D B
Fromthisequation,oneextractsthetemperaturedepen-
sider the Hamiltonian
dence of the critical field (H ) that induces the second
sc
H = ξ a† a (1) order phase transition from the homogeneous supercon-
kσ kσ kσ
ducting state to the normal state. Below a certain tem-
Xkσ
perature (the tricritical point temperature), the phase
V
−S δk1−k2,k4−k3a†k1+q/2↑a†−k2+q/2↓ transition becomes of first order and the field given by
k1kX2k3k4 thepreviousequationbecomesasupercoolingfieldabove
a a whichthenormalstateisalocalminimumofthefreeen-
× −k3+q/2↓ k4+q/2↑
2
ergy. Note that this critical field has no dependence on
the Fermi surface shape, only density of states depen- a = 1/2: a = 0:
dence. sc sc
3
1st order 1st order
In a two-dimensional system, a Van Hove singularity
sh FF, sh
in the density of states results usually from the presence FF
of a saddle point in the energy dispersion ǫ(k). In the
case of a simple saddle point, ǫ(q) q2 q2, one has
a logarithmic singularity in the densi∼ty oxf−statyes. In the =0) 2
cwaistehohfigahnerexptoewnedresdasnadddthleisploeaindts, tǫo(qa)p∼ow|qexr|-nla−w |dqiyv|emr-, H(Tsc
gence in the density of states N(ǫ) (ǫ ǫ )−α with H/c
vh
α=1 1 1. Besidestwo-dimensio∼nalsy−stems,power-
−n−m
law divergences are present in one dimensional systems. 1
In a one-dimensional system, a q2 dispersion leads to a
inversesquarerootdivergenceinthedensityofstates. In
the following, we will assume that the VHS is pinned at
theFermilevelasobservedinphotoemissionexperiments
in the copper-oxides[5] andalsoindicatedby theoretical
0
studies of correlated two-dimensional models.[11] 0.0 0.2 0.4 0.6 0.8 1.0
Assuming N(ǫ)=No ǫ ǫvh −α, in the weak coupling Tc/Tco
| − |
limit, Eq. 2 can be rewritten as
α 1 h FIG.1: Thephasediagramofaparamagneticallylimitedtwo-
= G (3) dimensional superconductor with constant density of states
g (2t)α (cid:18)2t(cid:19)
(α = 0) and with a power-law divergence in the density of
states (α = 1/2) pinned at the Fermi level. The reentrant
with g =VN /2 and
o behavior of the supercooling field H observed in BCS su-
sc
perconductorsisstronglyenhancedifaVHSispresentinthe
∞ ξ−α 1 1 DOS.
G(x)= dξ + .
Z α (cid:20)cosh2(ξ+x) cosh2(ξ x)(cid:21)
0
−
(4)
The upper limit of the integralin Eq. 2 was extended to
field is not reachedat zero temperature, but at an inter-
infinity due to the fast decay ofthe integrandand there-
mediatetemperature. Thisreentrantbehaviorforthesu-
fore, in the weak coupling limit, the supercooling field
percoolingfieldisknowninBCSsuperconductors,[1]and
has no dependence on the frequency cutoff. The zero
ithasbeenrecentlyobservedinthinaluminumfilms.[13]
fieldcriticaltemperatureobtainedfromEq.3isgivenby
In the case of a VHS superconductor, this maximum is
2t [g/(α α2)]1/α which implies the well known en-
c0 ∼ − enhancedrelativelytothezerotemperaturesupercooling
hancement of critical temperature which motivated the
field and the reentrance becomes more pronounced.
Van Hove scenario of high T cuprate superconductors.
c
The zero temperature supercooling field (hsc,0) is given The strong reentrantbehavior of hsc can be explained
byh =(2g/α)1/α.[12]Therefore,h (1 α)1/αt in the following way. At zero temperature, the Fermi
sc,0 sc,0 c0
∼ − surface splitting in the normal ground state due to the
and one has a much largerenhancement of t than that
c0
Zeeman coupling creates a no-pairing or blocking region
of h in the limit of very strong VHS, α 1. The
sc,0
→ aroundthezerofieldFermisurfacesince,inordertocon-
reason behind the different enhancements is reflected in
tributetotheformationofaCooperpairofmomentumq,
the fact that for fixed finite temperature and zero mag-
twostateswithoppositespinsandofmomentak+q and
netic field, the pairingsusceptibility divergesinthe limit
k must be either both empty or both occupied at zero
α 1, but it does not diverge for fixed finite magnetic
−
→ temperature. In the case of homogeneous superconduc-
field and zero temperature (note that tanh(x) is linear
tivity, Cooper pairs have zero total momentum, q = 0.
for small x).
At finite temperature, thermal excitations provide low
In Fig. 1, the temperature dependence of the reduced
energy pairing possibilities within this no-pairing region
upper critical field (or supercooling field) obtained nu-
and therefore, the pairing susceptibility growswith tem-
merically from Eq. 2 is displayed for both a supercon-
perature. In the presence of a VHS at the Fermi level,
ductor with a VHS and a superconductor with constant
the number of these thermal excitations is much larger
density of states (BCS superconductor). The VHS ex-
due to the strong splitting of the Fermi surface in the
ponent α = 1/2 is obtained in the case of a quadratic
saddle point regions and therefore, this effect is more
one-dimensionalenergy dispersionor for example, in the
pronounced.
case of an extended saddle point with quartic dispersion
as found in the two-dimensional Hubbard model.[6] One Given a certain value of the superconducting gap, the
observesinFig.1thatthemaximumsecondordercritical free energy difference between the superconducting and
3
the normal state can be determined from [1] ItiswellknownthattheFulde-Ferrelstatebecomesen-
hanced in the presence of Fermi surface nesting.[16] One
∆ d(1/V)
F (T,H) F (T,H)= ∆′2d∆′ (5) mightimaginethataVHSpinnedattheFermilevelcould
s − n Z0 d∆′ provide a similar nesting effect in the Fulde-Ferrel state.
The transitionto the normalstate with variationoffield However,oneshouldbeawareofadifference: whileFermi
or temperature occurs when the zero gap local extreme surface nesting does not enhance homogenous supercon-
of the free energy becomes the absolute minimum. If ductivity, in the case of VHS nesting it is considerably
the finite gap local minimum of the superconducting enhanced. It is therefore possible that the Fulde-Ferrel
phase convergesto the normalstate localextreme as the region of the phase diagram is narrower or that it even
transition is approached, the transition is of second or- does not exist. We will show that at zero temperature
der, otherwise it is of first order and there is a region and finite magnetic field, saddle points lead in fact to
of metastability limited below by the supercooling field very poor Fermi surface nesting and indeed the Fulde-
h and above by the superheating field h . The su- Ferrel region of the phase diagram is absent in the case
sc sh
perheating field is the highest field for which there is ofanextendedsaddlepoint. Furthermore,this phasere-
still a finite gap solution of Eq. 2.[1] It can be shown mains absent even if one improves the nesting property
that below this field, there is a finite gap local mini- by considering an isotropic Fermi surface with vanishing
mum in the free energy. The first order critical field Fermi velocity.
h and the superheating field determined numerically Letusconsiderthelatterfirst. Inorderforanisotropic
1
from Eqs. 2 and 5 are shown in Fig. 1. At zero tem- energydispersiontohaveanVHSinthedensityofstates,
perature, the superheating field is just the zero temper- itmustbeoftheformξk =ǫk ǫvh =a sign(q)q b where
− · | |
ature gap function, h = ∆ = [A(α)g]1/α t , q = k k . Again, we assume k = k . The density
with A(α) = 2πcosec[s(h1,0− α)π]0P−α(0) where Pν∼(x)c0is of stat−es fvohr the above model is NF(ǫ) ∼vha−1/bb−1ξ1b−1
the Legendre function of the first kind. Consequently, and therefore the inhomogeneous gap equation can be
h (1 α)1/αh and there is a huge metastability rewritten as
sc,0 sh,0
∼ −
region. In Fig. 1, it is also apparent that the reduced
temperature of the tricritical point is slightly larger in ωD π dθtanh ξ2+t +tanh ξ2−t
1=V dξN(ξ) (cid:16) (cid:17) (cid:16) (cid:17) (8)
the case of the VHS superconductor. Z Z π ξ++ξ−
0 0
Atzerotemperature,thefreeenergydifferenceisgiven
by with ξ± =[ξ 1/bsgn(ξ) 1/2qcosθ]b h where we have
neglectedaq|2|/ktermsin±ceq k an±dwhere[x]bshould
N 4 F
∆F(0,H)= o h2−α αA(α)∆2−α . (6) beunderstoodassgn(x)xb. ≪Inordertokeeptheexpres-
4 2α(cid:20)1 α − (cid:21) | |
− − sions simpler, we have considered a = 1. Note that in
The zero temperature critical field associated with the the one-dimensionalcase,the angleintegrationwouldbe
first order transition is usually denominated by Pauli absent.
limit or Chandrasekhar-Clogston limit [14]. This field In the weakcoupling limit, the zerotemperature criti-
is easily extracted from the previous result, hp = α(1 calfield,forq˜=q/(2h1/b) 1,canbe obtainedfromthe
α)A(α)/4}2−1α∆0 and therefore, this limit to sup{ercon−- equation hc,0(q˜)=[2gG(q˜)≤]α1 with
ductivity is as strongly enhanced as the zero temper-
ature superconducting gap due to the presence of the π dθ ∞ 2ξ−1+1b
G(q˜)= dξ
VHS. For α = 0, one recovers the well known result, Z0 π Z(1−q˜cosθ)b (ξ1b +q˜cosθ)b+(ξ1b q˜cosθ)b
hp = ∆0/√2.[1] In the limit α 1, hp ∆0/2. Note − (9)
→ →
that recentexperiments indicate that the in-plane upper Note that now the blocking region is no longer ξ = h
critical field in the copper-oxides exceeds considerably but ξ = h(1 q˜cosθ)b. G(x) has a maximum for
the BCS Pauli limit.[15] x=1andthere−forethemaximumH (q)isreachedwhen
c
The previous results are independent of the shape of q˜=q/(2h1/b)=1. Thisfieldissignificantlyenhancedby
theFermisurface. Now,wewillconsiderfiniteqsolutions the VHS since it is proportional to V α1, with α=1 1.
of the gap equation, that is, we will search for a Fulde- − b
However, there is only a weak enhancement in compar-
Ferrel phase in the phase diagram. Following the BCS
ison with the q = 0 supercooling field as observed in
meanfieldapproachinthecaseofaninhomogeneousgap
Fig. 1 in the case of α = 1/2. The FF phase bound-
function, ∆ = VS−1 a a , and taking the
q − kh −k−q↓ k↑i ary lies clearly below the first order critical field h1 and
limit ∆q →0,one obtainPs the followingtwo-dimensional therefore, this phase will not be observed.
gap equation,[2]
The same conclusion is reached if now one considers
1= VS Xp 1−fξ(pξp++q/q2/↑2↑+)−ξpf−(qξ/p2−↓q/2↓). (7) aam(oe|qnmxe|ernng−tyu|mdqiys|rpmees)rtsrwiiochtneerdweitqtoh=assakmd−dallkelvrphe,ogiiwnoitntsh,arξtohkue=nvdaǫltkuhe−essoǫavfdhtdh=lee
For a given temperature, the FF critical field is deter- points kvh and kvh by a cutoff. In this case, the zero
−
mined by searching for the highest field solution of this gappairingsusceptibility(the integrandofEq.8)hasits
equation for any value of q. maximum value for q = 0 even for a simple quadratic
4
1.14 ever, one should be aware that the Fulde-Ferrel state is
VHS q=(q,0) 1.0 extremelysensitivetonestingpropertiesoftheFermisur-
1.12 VHS q=(q,q)/21/2 faceandthatwehaveonlyconsiderthesaddlepointcon-
BCS tribution to the formation of this state and ignored the
1.10 0.8 rest of the Fermi surface. In the case of homogeneous
) )
0 0 superconductivity, such procedure is justified since the
S( 1.08 S(
C H saddle point contribution clearly dominates. In the case
B 0.6 V
cq)/ 1.06 cq)/ of Fulde-Ferrel superconductivity, if some portion of the
( ( Fermi surface is strongly nested, it may lead to a higher
S S
BC 0.4 VH critical field than that of homogeneous superconductiv-
1.04
c c ity.
1.02 0.2 Concerning the experimental relevance of the results
presented in this paper to the particular case of the
1.00
cuprates, an obvious statement is that an huge metasta-
0.0
0.0 0.5 1.0 1.5 2.0 bility region in the phase diagram is reflected by a low
q/2h1/b temperaturehystereticbehaviorwhichcanbeprobed,for
instance, by resistive critical field measurements [17] or
tunnelling measurementsofdensity ofstates (asrecently
FIG.2: Thepairingsusceptibilityasafunctionoftherenor- in thin Al films [13]). Unfortunately, in-plane critical
malized pairmomentumq/(2h1/b)foraBCSsuperconductor fields of the high-T superconductors are presently out-
c
and a superconductorwith an extended VHSat a small, but sidetheexperimentalmagneticfieldrange. Onecanpar-
finitetemperature, obtained numerically from Eq.8. tially circumvent this difficulty by considering strongly
overdoped or underdoped cuprates with lower critical
fields. A study of hysteresis in these materials would
saddle point. InFig. 2, plots ofthe zero gappairing sus-
provide a test of the validity of the extended Van Hove
ceptibility at zero temperature and for a fixed magnetic
scenario for High T cuprate superconductivity.
c
field are displayed for a BCS superconductor and for an
extended saddle point with m=n=4. The direction of In conclusion, we have shown that a two-dimensional
momentum q for the latter is chosen to be along the x superconductorwithanextendedsaddle pointinthe en-
axis or along the diagonal of the Brillouin zone. For the ergydispersionpinnedattheFermilevelhasatlowtem-
saddle point, the maximum of the pairing susceptibility peraturealargemetastabilityregioninthetemperature-
is for q = 0. In the case of a simple quadratic saddle parallelmagnetic field phase diagramand that the Pauli
point, the maximum remains at q =0. If one “weakens” limit,H ,fortheuppercriticalmagneticfieldisstrongly
p
thesaddlepointbychoosingexponentssmallerthantwo, enhanced in this extended Van Hove scenario. Fulde-
themaximumshiftstofiniteq,andrapidlybecomesfixed Ferrel superconductivity is absent from the phase dia-
at q˜= 1. One could therefore conclude that the Fulde- gram unless there are Fermi surface sections (away from
Ferrel state is absent in systems with energy dispersions the saddle point region) with very good nesting proper-
containing saddle points close to the Fermi level. How- ties.
[1] K. Maki and T. Tsuneto, Prog. Theor. Phys. 31, 945 tivity (John Wiley & Sons, New York, 1965), Chapter
(1964). 5.
[2] H.Shimahara, Phys.Rev.B 50, 12760 (1994). [11] J. Gonz´alez, F. Guinea, and M. A. H. Vozmediano, Eu-
[3] P.Fulde and A.Ferrel, Phys.Rev. 135, A550 (1964) rophys. Letts. 34, 711 (1996); V. Y. Irkhin et al, Phys.
[4] A. I. Larkin and Yu N. Ovchinnikov, JEPT 20, 762 Rev. Lett.89, 076401 (2002).
(1965). [12] Note that the fact apparently hsc,0 diverges as α → 0
[5] K. Gofron, J. C. Campuzano, A. A. Abrikosov, M. Lin- justreflectsthefactthatinthislimittheupperlimitfor
droos,A.Bansil,H.Ding,D.Koelling,andB.Dabrowski, energy in the integrals can not be extended to infinity.
Phys.Rev.Lett. 73, 3302 (1994). The correct expression in this limit is hsc,0 = (ω−α +
[6] F. F. Assaad and M. Imada, Eur. Phys. Journal B 10, α)−1/α.
2g
595(1999);M.Imada,A.Fujimori,andY.Tokura,Rev. [13] V.Yu.Butko,P.W.AdamsandE.I.Meletis,Phys.Rev.
Mod. Phys. 40, 1039 (1998). Lett. 83, 3725 (1999).
[7] R.G. Dias, J. Phys. Cond. Matt. 12, 9053 (2000). [14] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962); A.
[8] A.P Mackenzie et al, Phys. Rev.Lett. 71, 1238 (1993). M. Clogston, Phys. Rev.Lett. 9, 266 (1962).
[9] K.YangandS.L.Sondhi,Phys.Rev.B57,8566(1998); [15] A. S. Dzurak et al, Phys.Rev.B 57, R14084 (1998).
K.Maki and H.Won, Czech. J. Phys. 46, 1035 (1996). [16] N. Dupuis,Phys. Rev.B 51, 9074 (1995).
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5
B 51, 1378 (1995); P. W. Adams, P. Herron, and E. I.
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