A real-space effective c-axis lattice constant theory of superconductivity

0.54 MB·English

by  X. Q. Huang

A real-space effective c-axis lattice constant theory of superconductivity
X. Q. Huang1,2∗
1Department of Telecommunications Engineering ICE, PLAUST, Nanjing 210016, China
2Department of Physics and National Laboratory of Solid State Microstructure, Nanjing University, Nanjing 210093, China
(Dated: January 27, 2010)
Based on the recent developed real-space picture of superconductivity, we study the stability of
thesuperconductingvortexlatticesinlayeredsuperconductors. Itisshown thattheeffectivec-axis
lattice constant play a significant role in promoting the superconducting transition temperature in
these materials. An unified expression Tcmax = 10c∗ −28 can be applied to estimate the highest
0 possible Tcmax for a given layered superconductor with an effective c-axis lattice constant c∗. For
1 the newly discovered iron-based superconductors, our results suggest that their Tc cannot exceed
0 60 K, 50 K and 40 K for the 1111, 21311 and 122 series, respectively. In the case of copper-based
2 oxidesuperconductors,it seems thatthehighestTc canreach about 161 Kwithout applyingof the
external pressure. In our theoretical framework, we could interpret the experimental results of the
n
completely different superconducting transition temperatures obtained in two very similar cuprate
a
J superconductors(La2−xBaxCuO4 of40KandSr2−xBaxCuO3+δ of98K).Inaddition,thephysical
reason why the superconductivity does not occur in noble metals (like gold, silver and copper) is
7
discussed. Finally, we argue that the metallic hydrogen cannot exhibit superconductivity at room
2
temperature, it even cannot bea superconductor at any low temperature.

]
l PACSnumbers: 74.20.-z,74.25.Qt,74.90.+n
e
-
r
t I. INTRODUCTION is very difficult to reach60 K by means of elements sub-
s
. stitution or by applying hydrostatic pressure, and this
t
a prediction is still correct up to now. As pointed out by
Finding the room temperature superconductors has
m Chu recently, “The discovery of iron-based pnictide su-
been an elusive dream of scientists in the field of con-
perconductors may have reinvigorated the field of high-
-
d densed matter physics. In 1986, the discovery of high temperature superconductivity, but the cuprate super-
n temperature cuprate superconductivity by Bednorz and conductors are still in the game”.7
o Müler1 shocked the superconductivity community and
In addition, physicists considered that a high pressure
c rekindled the dream of room-temperature superconduc-
[ can make the hydrogen molecules into atoms and make
tivity. Sincethen,greateffortshavebeendevotedtofind- it a metallic conductor. Moreover, it was predicted ac-
1 ingoutnewmaterialswithahighercriticaltemperature.2 cording to the BCS theory8 in 1968 by Ashcroft9 that
v Many cuprate superconductors have since been discov-
solid metallic hydrogen may be superconducting at high
7 ered and the highest temperature superconductor was
temperature, perhaps even roomtemperature. However,
6
HgBa Ca Cu O with T = 138 K. Its T can be in-
0 2 2 3 8+δ c c to get metallic hydrogen is still a distant dream, not to
creased as high as 164 K under a high pressure of about
5 mention the realization of the superconductivity. Based
. 30 GPa.3 Later, several materials have been claimed to on our theory,10,11 we would like to point out that even
1
be room-temperature superconductors which have been
0 if there exists metallic hydrogen, it is also impossible to
quickly provedto be false. However,how to enhance the
0 exhibit the superconductivity at any low temperature.
superconducting transition temperature is always one of
1 In this paper, the discussionof enhancement of super-
: central concerns in the field of superconductivity. conductivity is presented in the framework of the real-
v
i In2008,Japaneseresearchersdiscoveredthesupercon- spacesuperconductingvortexlattices. We showthatthe
X ductivity in the iron-oxypnictide family of compounds.4 effectivec-axislatticeconstantplayakeyroleinpromot-
r LiketheCu-basedsuperconductors,1–3thenewsupercon- ing the superconducting transition temperature in the
a
ductors have layered atomic structures and can conduct layered superconductors.
electricity without resistance at relatively high temper-
atures than the conventional low-temperature supercon-
ductors. Similarly,thenewdiscoveryhastriggeredinten- II. A COMPARISON BETWEEN TWO
sive research worldwide and the maximum critical tem- SUPERCONDUCTING THEORIES
ESTABLISHED IN MOMENTUM SPACE AND
peraturehasbeenraisedfromtheinitialreportofT =26
c
KinLaO1−xFxFeAs4toabout55KinSmO1−xFxFeAs5. REAL SPACE
Facing the rapid increase of T , some researchers seem
c
too optimistic about the new discovery and they even A. BCS theory of superconductivity
claimed that these compounds could be useful for de-
veloping room-temperature superconductivity. In fact, In the framework of the BCS theory,8 the supercon-
morethanoneyearago,weprovedtheoreticallythatthe ducting current is carried by the so-called Cooper pairs
maximum T of the new iron arsenide superconductors of electronswhich are held together by lattice vibrations
c
2
- k
lustrated in Fig. 1(b). More importantly, the Coulomb
(cid:1) interaction is the elementary electrical force that causes
(cid:0)(cid:3)
(cid:2)
(a) two negative electrons to repel each other, hence, two
important questions arise: (1) How can the real-space
-
k (cid:0)(cid:3) (cid:1) repulsions among electrons (or pair-pair repulsions) and
(cid:4)(cid:5)(cid:6)(cid:7)(cid:6)(cid:7)(cid:8)(cid:7)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:9)(cid:8)(cid:6)(cid:7) (cid:2) the attractions among electrons and ions be eliminated
~ to support the formation of the Cooper pairs? (2) Nor-
(cid:12) (cid:22)(cid:23)(cid:24)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:23)(cid:26)(cid:25)(cid:30)(cid:31)(cid:23) !"# !"$%&
mally, there are two kinds of interactions in the super-
(b) ~ conducting materials, one is the very strong short-range
’()*+,-)./0)1)23./*45 (cid:1)67 electron-electroninteractions[see Fig. 1(b)], the otheris
the rather weak long-range electron-phonon interactions
- k k [seeFig. 1(a)]. Now,thequestioniswhycanthestronger
short-range electron-electron interactions be completely
ignored in the BCS theory?
B. Wigner crystal and the real-space
superconducting theory
x »
(cid:14)(cid:15)(cid:6)(cid:6)(cid:4)(cid:10)(cid:11)(cid:16)(cid:4)(cid:12)(cid:8)(cid:11)(cid:17)(cid:8)(cid:18)(cid:10) (cid:19)(cid:20)(cid:20)(cid:7)(cid:21)
Asweknowtherearemanytheoriesaboutthecauseof
Figure 1: (a) Two electrons are bound into a Cooper pair in
thesuperconductivityincupratehightemperaturesuper-
themomentumspace,thepairedelectronshaveoppositespin
conductors. Unfortunately,mostofthesetheoriescontra-
and opposite momentum and gives rise to the superconduc-
dict each other and they may be on the wrong track as
tivity. (b)Intherealspace,thepairedelectrons(onemoving
emphasized by Anderson6. In our opinion, the natural
left, the other moving right.) can be separated by a coher-
encelength(theCooper-pairsize)ofupto100nm. Andthere strong repulsion between two electrons is impossible to
mayexist107 electrons(orelectronpairs)betweenthepaired be totally overpowered by a lattice vibration (known as
electrons. a phonon) and all superconductors should share exactly
the same physical reason. The present situation of one-
material one-mechanism of superconductivity should be
(anexchangeofphonon)inthematerial. Theusualanal- changed. Inotherwords,anewandunifiedtheorywhich
ysis in the BCS theory relies on a momentum-space pic- can explain all superconducting phenomena has proved
ture,theCooperpairsareformedinmomentumspace(k- more important and urgent today.
space) and the paired electrons have opposite spin and With increasingly better samples and advances in ex-
opposite momentum (k↑,−k↓), as shown in Fig. 1(a). perimentaltechniques(forexampletheScanningTunnel-
Of course, if the two electrons are to remainin the same ing Microscopy12), a vast amount of data from these ex-
paired state forever, then they must undergo a continu- perimental studies reveal that the superconducting elec-
ous exchange of virtual phonon in the BCS picture. trons are more likely to congregate in some real space
AlthoughthephysicscommunityacceptsthattheBCS quasi-one-dimensionalriversofchargeseparatedbyinsu-
theory can successfully describe the behaviors of most latingdomains.12–15Obviously,acorrectandreliablethe-
metallic superconductors, it is commonly believed that ory of superconductivity has to take into account these
the phonon based BCS theory is invalid for the descrip- new results. Although many researchers have been try-
tion of the behavior of the high-temperature supercon- ing to replace the conventional superconducting picture
ductors. As pointed out by Anderson,6 the need for a (dynamicscreening)witharealspacepicture,buthowto
bosonicglue(phonon)incupratesuperconductorsisfolk- constructa proper model relatedto the formationof the
lore rather than the result of scientific logic. It is well one-dimensional charge rivers will still be a major chal-
knownthatthe superconductingelectrons(asdefined by lenge for those devotingthemselves to crackthe mystery
BCS theory) are a momentum space order phase, while of high-temperature superconductivity.
at the same time a disorder phase in real-space. From In recent years, we have tried to propose a real space
both the physics and mathematics point of view, the superconducting mechanism which may provide new in-
fundamental differences between two descriptions (one sights into the nature of the superconductivity.10,11 In
orderandonedisorder)ofsuperconductingelectronsim- our scenarios, all the superconducting electrons can be
ply that the BCS theory may not be scientifically self- considered as the ‘inertial electrons’ moving along some
consistent. From the perspective of the real-space corre- quasi-one-dimensional real-space ballistic channels, as
lations,twoboundelectrons(aCooperpair)canseparate shown in Fig. 2(a). In the previous paper,11 it was
by a coherence length of ξ up to 100 nm, which is also provedtheoreticallythatastaticone-dimensionalcharge
calledthecoherencelength. Itshouldbenotedthatthere stripe can be naturally formed inside the superconduct-
are about 107 randomly distributed electrons (or pairs) ing plane and the Coulomb repulsion between electrons
betweenthemintheconventionalsuperconductors,asil- can be suppressed completely, as indicated in Fig. 2(a).
3
89:;<=>?@ABC@D?ECB>?AB=FGH=IJC>KI?LCM><LCIGABD<GK?BJJIA<BD>K=KCN
(cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148)
(cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138)
{|d}|~(cid:127)(cid:128)(cid:129)d(cid:130)(cid:131)}(cid:132) OPPQRSQTUS v v
s 1 2 +
g
+ (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) +
(cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:138) (cid:141)
hw 8j: kMl=NmnmGo(cid:133)(cid:134) V^X Y_‘(cid:133)a-b(cid:134)Z[V](cid:135)(cid:133)Z(cid:134)(cid:135)[(cid:136)c(cid:135)d(cid:133)-(cid:134)(cid:135)X y (cid:139)(cid:140)(cid:148) (cid:149) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148)(cid:141)(cid:143) (cid:148) (cid:144) (cid:148)(cid:146)(cid:147) (cid:142)
(cid:148) x (cid:148) (cid:148) (cid:148) (cid:148) (cid:148) (cid:148)d (cid:148)
(cid:145)
Figure 3: When two electrons moving along the same direc-
(cid:137)uv xqyz (cid:137)uv xqyz ytiocnoo(ryd-ainxaist)ew),itthhethxe-saaxmiseCvoeulolocmityb(roerpsutlasitoicn, bweitthwetehnetshameme
i
h g VeXYZ[f(cid:214) \] pqrs VWXYZ[(cid:214) \] pqrs cbaanckbgreoufunldly. overcome (fB = F1 −F2) by the positive ion
perconducting vortex lattices. It is easy to find that the
competitive interactions between stripes can be greatly
reduced by increasing the stripe-stripe spacing and this,
in turn, enhances the stability of the superconducting
state and increases the corresponding superconducting
tuv xqyz tuv xqyz transition temperature. For the layered superconduct-
pqrs pqrs ing materials, there are usually two ways to control the
stripe-stripe spacing inside the superconductors. The
Figure 2: (a) The short-range repulsive forces among elec-
trons can be canceled out by arranging them into one- first is the carrier concentration and the second is the
dimensional real-space charge river (charge stripe). A stable c-axis lattice constant of the samples. Normally, a su-
one-dimensionalchargestripeisconfinedbythedomain-walls perconducting sample with a low carrier concentration
ofthepositiveions. Itisprovedthattheelectron-electronre- and a largec-axis lattice constantmay have a higher su-
pulsionsinsidethestripecan beeliminated completely asin- perconductingtransitiontemperature. Ofcourse,thetoo
dicatedinthefigure. (b)-(e)Thelong-rangeelectron-electron low carrier concentration is not conducive to the forma-
repulsionsamongdifferentstripescanalsobecanceledoutby
tionofsuperconducting vortexlattices due tothe lackof
arranging the stripes into some periodic Wigner crystals. To
effective competition among electrons.
achieve the highest superconducting transition temperature,
To form a stable superconducting vortex lattice, both
the superconducting vortex lattices should be in the follow-
electron-electronrepulsiveinteractionswithinthe charge
ingfourstablestructures,(b)and(c)thevortexlatticeswith
tetragonal symmetry, while (d) and (e) having the trigonal stripesandbetweenthe stripes shouldbe inhibited thor-
symmetry,where h is a positive integer. oughly. The former situation was studied and resolved
in one of our previous papers,11 here we will pay our at-
tention to the later situation. It will be shown that the
Accordingto the principle ofminimum energy,forthree- nature of the electron-electron repulsive interaction can
dimensional bulk superconductors, the one-dimensional even be changed into an attractive type because of the
chargestripescanfurtherself-organizeintosomethermo- positive ion background.
dynamically stable vortex lattices (the Wigner crystal) Ourtheoryisbasedontheexperimentalfactsthatthe
with trigonal or tetragonal symmetry, as shown in Figs. superconductingelectronsmaybeassociatedwithalong-
2(c)-(f). It is not difficult to prove that the electron- rangespatialcoherencechargeorderedstate. Inorderto
electron repulsions among different stripes can also be studythestabilityofthe superconductingvortexlattices
completely canceled out due to the symmetries in the of Fig. 2, we can consider the simplest case where the
Wigner crystals. stripe-stripeinteractioncanbesimplifiedastheelectron-
electron interaction, as shown in Fig. 3. We will show
that by adjusting the parameters δ the electron interac-
tioncannotonlybe repulsivebut canbe alsoattractive.
III. THE STABILITY OF THE
Moreover,there is a special value of δ that can lead to a
SUPERCONDUCTING VORTEX LATTICES
completelysuppressionofthe Coulombrepulsion,thatis
On the basis of our theory as described in Fig. 2, the f =F −F (1)
B 1 2
superconducting critical temperature of the layered su-
perconductors is closely related to the stability of the In the following, we will discuss how electron-electron
vortexlattices. Obviously,thestripe-stripeinteractionis repulsion between the two adjacent charge stripes be
the most importantfactor relevantto the stability of su- overcome? It is worth noting is that this is merely the
4
‰¯ +‰ ‚+ +” +¿ ¸ (a) (n=1)
˘†‡·(cid:181)¶ ›fi ›– ›` ›ˆ a Æ(cid:226)(cid:217)(cid:219)(cid:212)ª l=10
y + + «˙‹ +›fl • +›˜›(cid:192) + (cid:223)(cid:214)(cid:221)(cid:224) repulsive l=1
…¯ d la d›(cid:176) ›´ a (cid:222)¨ attractive
x » … ' „x d(cid:190) (cid:213)(cid:217)
(cid:150)(cid:151)(cid:152)(cid:152)(cid:153)(cid:152)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:159)(cid:152)(cid:153)(cid:152)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:160)¡¢(cid:154)£(cid:158)⁄ ¥ƒ§¤'“¢¤' (cid:212)(cid:221)
(cid:216)
(cid:215)(cid:211)
Figure 4: A diagram illustrates how the Coulomb repul- (cid:211)
sion between two electrons be eliminated by considering the (cid:213)(cid:217)
electron-ion interactions. Note that the electron A and B (cid:211) (cid:204)¸
(cid:216) ¨(cid:201)¨ ¨(cid:201)˚ ¸(cid:201)¨
(which is not shown in this figure) are located inside two ad- (cid:213)(cid:221)¨(cid:201)¸ (b)
jacent stripes, respectively. (cid:212)(cid:218) l=10
(cid:217)
(cid:211)
(cid:220)
(cid:211)
well-known Coulomb screening effect of the positive ion (cid:218)(cid:219)(cid:217) l=1
(cid:217)
background. For simplicity, apart from the Coulomb (cid:213)¨(cid:201)¨
(cid:216) l
repulsion between two electrons, we consider only the (cid:215)
nneeaarreesstt-nneeiigghhbboorrss((boontlhyininxxa-dnidreyc-tdioirne)cteiolencst)roann-dionnexint-- (cid:211)(cid:212)(cid:213)(cid:212)(cid:214) (cid:230) l (cid:229) l
(cid:210)
teractions in this study, as shown in Fig. 4. Here we (cid:209) (cid:228)
analyze mainly the forces applied on the electron A of (cid:204)¨(cid:201)¸
¨(cid:201)˝˛ ¨(cid:201)˚¨ ¨(cid:201)˚ˇ d /a ¨(cid:201)˚˝ ¨(cid:201)˚— ¨(cid:201)˚˛
Fig. 4 and the discussion is similar to another electron
B (not shown in this figure).
According to Fig. 3 and Fig. 4, the Coulomb forces Figure5: AnalyticaltotalconfinementforceFtotalonelectron
appliedtotheelectronAcanbedividedintothreeparts. Aversusδ/afordifferentlwiththespecialconditionQ=+e
The first part is electron-electron (A and B) repulsion (or n=1). Fig. (b) is the enlarged figure of the rectangular
which is given by area of Fig. (a).
e2 The resultantforce ofthis partonthe electron A canbe
f = , (l =1,2,3,···) (2)
B 4πε [(l+2)a+2δ]2 expressed as:
0
F =F(12)+F(34)+F(56)+F(78). (7)
The second part is contributed by the eight ions (they x x x x x
aremarkedby1,2,3,4,5,6,7and8inFig. 4,andeach
The third part is contributed by the other eight ions
ion carries a positive charge ne) around the elect on A, aroundthe elect on B, the correspondingresultant force
they are ′
F can be written as:
x
F(12) = f +f
x 1 2
= −n e2 δ , (3) F′ =−n e2 4 δ+(i+2)a . (8)
4πε0[(a/2)2+δ2]3/2 x 4πε0 Xi=1 (a/2)2+[δ+(i+2)a]2 3/2
h i
F(34) = f +f Nowwecanobtainageneralformulaofthe totalforce
x 3 4
e2 a−δ Ftotal applied to the electron A
= n , (4)
4πε0[(a/2)2+(a−δ)2]3/2 F =f +F +F′. (9)
total B x x
Figure 5 shows the relationship between total confine-
Fx(56) = f5+f6 ment force Ftotal and δ/a for different l with the special
e2 a+δ condition Q=+e (or n=1). From the Fig. 5(a), it can
= −n , (5)
4πε0[(a/2)2+(a+δ)2]3/2 beseenclearlythattherearetwomainregionswherethe
right region indicates the repulsive interaction between
electron A and B, while the left region indicates the at-
tractiveinteractionbetweentwoelectrons. Furthermore,
F(78) = f +f
x 7 8 for a given l, as better shown in the enlarged Fig. 5(b)
e2 2a−δ that there exists a special position λ (the deviation of
= n . (6) l
4πε0[(a/2)2+(2a−δ)2]3/2 the electron from its initial equilibrium position a/2) at
5
º(cid:237)ºØ
l (l )
1
△E ∼ △ξ. (11)
ß (cid:238)(cid:239)(cid:240) a e ξ2
l= œ
º(cid:237)ºŁ
From Eq. (11), it is not difficult to find that the ∇E
x e
œ canbeeffectivelyreducedbyincreasingξorbydecreasing
△ξ. In fact, the physical essence of E can root to the
c
º(cid:237)º(cid:231) (cid:238)(cid:239)ø energy of △Ee. Physically, we argue that the parameter
λ of Fig. 6 is closely related to the △ξ of Eq. (12), for
convenience,we assume that they are linearly related by
(cid:238)(cid:239)ł
º(cid:237)º(cid:236) λ∼△ξ. Thenaccordingtotheaboverelation(λ=1/ξ2)
(cid:238)(cid:239)(cid:247)
and Eq. (12), the energy E can be simply expressed as:
(cid:238)(cid:239)(cid:246) c
(cid:238)(cid:239)ı (cid:238)(cid:239)(cid:240)æ
(cid:238)(cid:239)(cid:244) (cid:238)(cid:239)(cid:243) 1 1 1
(cid:238)(cid:239)(cid:242) E ∼ △ξ ∼ λ∼ . (12)
º(cid:237)ºº c ξ2 ξ2 ξ4
(cid:231) Ł x Ø /a (x Œ/ºa) Œ(cid:236) Œ(cid:231)
This equation tells us that in order to obtain a more
ß stable superconducting vortex lattice with a higher tem-
perature, the stripe-stripe spacing should be sufficiently
Figure 6: The relationship between λl (the deviation of the
electron from its initial equilibrium position a/2) and ξl/a, increased. Amoredetaileddiscussionwillbegiveninthe
where ξl isthespacing of theelectronA and B. Therelation next section.
canbequantitativelydescribedasthefollowingequationλ=
a2/ξ2, as indicated as the blueline in thefigure.
IV. THE EFFECTIVE C-AXIS LATTICE
CONSTANT AND THE SUPERCONDUCTING
which the force F is equal to zero, indicating a com- CRITICAL TEMPERATURE
total
pletesuppressionoftheCoulombinteractionbetweentwo
electrons. From Fig. 4 and Fig. 5, we can see that as l Basedontheabovediscussions,wewillfurtherexplore
increases, the corresponding electron-electron (or stripe- how to enhance the superconducting transition temper-
stripe)spacingξ andthe value(λ )ofzeroforceposition ature of the layered superconductors. In our scenario,
l
will decrease respectively. a higher T means a smaller energy E , consequently, a
c c
Moreover, according to the numerical results, we can largerξ accordingto Eq. (12). It was pointed out in the
obtain the λ as a function ξ /a (where ξ is the spacing previous section, for the doped layered superconducting
l l l
of the electronA andB) for electronA with l from 1 to materials, there are two ways (the carrier concentration
10, as shown in Fig. 6. The relationship between them andthec-axislatticeconstant)toadjustthestripe-stripe
canbequantitativelydescribedasthefollowingequation:
λ=a2/ξ2, illustrated as the blue line in Fig. 6.
In our theoretical framework, for a stable supercon-
Table I: The relationship between the effective c-axis lattice
ducting vortex lattice, the superconducting electrons constant c∗ and the highest superconducting transition tem-
should develop their self-organized at the correspond- peraturesTcmaxwhichareachievedinthelayeredcuprateand
ing equilibrium positions (λ = 0) due to competition iron pnictidesuperconductors so far.
l
between the stripes. This result implies that the for-
mation of an ordered superconducting vortex lattice is Materials c(A˚) c∗(A˚) Tcmax(K)
always accompanied by the existence of electromagnetic ∗LBaa21−−xxBKaxxFCeu2OA4s2 1133..221320 66..660165∗∗ 3480
energy (Ec) inside the superconducting phase because ∗(Sr4V2O6)Fe2As2 15.673 7.837∗ 46
of the interaction among the vortex lines. This energy ∗SmO1−xFxFeAs 8.447 8.447 55
may directly influence the stability of the superconduct- La2Ca1−xSrxCu2O6 19.420 9.710∗ 60
ingvortexlatticeandthecorrespondingsuperconducting DyBaSrCu3O7 11.560 11.560 90
transitiontemperatureofthesuperconductor. Inthefol- Tl2Ba2CuO6+δ 23.239 11.620∗ 92
lowing, we will discuss qualitatively which factors might YBa2Cu3O7−δ 11.676 11.676 93
influence the energy Ec. We know that the interaction Sr2CuO3+δ 12.507 12.507 95
energy between two electrons spaced ξ apartcan be pre- Sr2−xBaxCuO3+δ 12.780 12.780 98
sented by TlCaBa2Cu2O7 12.754 12.754 103
∗
Tl2CaBa2Cu2O8 29.318 14.659 119
1 (Tl0.5Pb0.5)Sr2Ca2Cu3O9 15.230 15.230 120
Ee ∝ . (10) Hg2Ba2Ca2Cu3O8 15.850 15.850 133
ξ
HgBa2Ca2Cu3O8+δ 16.100 16.100 136
∗ ∗
Tl2Ca2Ba2Cu2O10 35.900 17.950 128
Supposethe distancebetweentwoelectronsdecreasesby ∗
Hg2Ba2Ca3Cu4O10 19.008 19.008 126
△ξ, then the interaction energy E will increase by
e
6
c ToobtaintheanalyticalrelationshipbetweentheTmax
T =10 - 28 c
(cid:254)(cid:253)(cid:255) (cid:20)(cid:21)(cid:22) (cid:24) andc∗,theexperimentalresultsofTableIarerepresented
(cid:23) + in Fig. 7. Surprisingly, except for the last two samples
(cid:254)(cid:252)(cid:255) (cid:25)(cid:6)(cid:25)(cid:4) + (Tl2Ca2Ba2Cu2O10 and Hg2Ba2Ca3Cu4O10), the figure
showsclearlythatalloftheexperimentaldataalmostfall
(cid:25)(cid:5)Z(cid:4)
on a same straight line which can be described with the
(cid:254)(cid:0)(cid:255)
equation:
K) (cid:254)(cid:3)(cid:255) 687 9: BCD
( 4345 ?>=A= Tcmax =10c∗−28. (13)
TEFGc (cid:254)(cid:255)(cid:255) /02131;<>=?@= It should be emphasized that the Eq. (13) proposed
here has its scope of applicationsince our superconduct-
(cid:253)(cid:255)
ing mechanism requires an effective competition among
∗
(cid:6)(cid:2)(cid:4) electrons and stripes, too intensive (for a small c ) or
(cid:252)(cid:255)
∗
(cid:5)(cid:2)(cid:4) %& too weak (for a large c ) competition is not conducive
’
(cid:0)(cid:255) (cid:1)(cid:2)(cid:4) (cid:26)(cid:27) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)()*+*+,-. tisoetshteimfaotremdatfiroonmoEf qs.upe(1rc3o)ntdhuacttinwghevnorct∗exislaltetsiscet.haInt
(cid:3)(cid:255) (cid:28)(cid:29)(cid:30)(cid:31)(cid:30) !"#(cid:14)$(cid:15)(cid:16)(cid:17)(cid:12)(cid:18)(cid:19)(cid:12) 2h.i8bÅit,tthheescuoprerrecsopnodnudcintigvitlayy.erOedf cmouatresrei,altshecraenmnuosttebxe-
" ∗
a maximum value of c which also led to the failure
max
(cid:255) of the establishment of the superconducting vortex lat-
(cid:252) (cid:253) (cid:254)(cid:255) (cid:254)(cid:3) (cid:254)(cid:0) (cid:254)(cid:252) (cid:254)(cid:253) (cid:3)(cid:255)
tice inside the materials. We note from the figure that,
HIJJKKJLMNOJLPQRNSTQMMNLJLUVSMQVMWXY
as unexpected, the superconducting transition tempera-
Figure 7: The relationship between the experimental data turesofTl2Ca2Ba2Cu2O10 andHg2Ba2Ca3Cu4O10 devi-
ofthehighestsuperconductingtransitiontemperaturesTcmax ate seriously from the linear equation of Eq. (13). We
and the newly defined parameter of effective c-axis lattice thinkthattheremaybetworeasonsforthesedeviations.
∗
constant c for the layered cuprate and iron pnictide super- First, since the carrier concentrations of these two su-
conductors. We argue that the possible maximum Tcmax for perconductors are unadjustable, it is likely that the vor-
SmO1−xFxFeAs of 55 K (1111 series), (Sr4V2O6)Fe2As2 of tex lattices are not in best consistent with those of the
46 K (21311 series) and Ba1−xKxFe2As2 of 38 K (122 series) crystal structures. Second, their effective c-axis lattice
cannot exceed 60 K, 50 K and 40 K, as indicated by three
constants have been very close to the maximum value of
cyan lines in thefigure. c∗ . Personally,we believe that the maximum Tmax of
max c
the cuprate superconductors may be raised to the theo-
reticalvalue (about 161K,as markedinFig. 7) without
spacing ξ inside the superconductors. Our theory sug- applying of the external pressure. Then, can we predict
gests that the changing of the carrier concentration can thelimitsoftheTmaxaccordingtoourtheory? Fromthe
c
only modestly increaseTc, while the increasingofthe ef- viewpoint of lattice stability and competition among su-
fectivec-axislatticeconstantcanleadtoasignificanten- perconductingelectrons,amaximumvalueofc∗ values
max
hancement of Tc of the superconducting materials. This rangefrom19Å(thec∗ ofHg2Ba2Ca3Cu4O10 )to19+3Å
argument has been well confirmed by numerous exper- (where 3Å is about the thickness of two atomic layers).
imental investigations in the cuprate and iron pnictide Hence by Eq. (13), we conclude that the limits of the
superconductors, as shown in Table I and Fig. 7. Tmax(∼ 192K) is very difficult to break through 200 K
c
Table I shows the experimental data of the c-axis lat- in three-dimensional bulk superconductors.
tice constant c and the highest superconducting tran- As is well known, soon after the discovery of the iron-
sition temperatures Tmax for the layered cuprate and based superconductors, some physicists have believed
c
newly discovered iron pnictide superconductors. Also in room temperature superconductivity may be possible in
Table I, we introduce the effective c-axis lattice constant these new compounds. We think that these predictions
∗ ∗
c whichcanbe dividedintotwodifferentclasses: c =c are apparently lack of solid scientific basis. As can be
∗
and c =c/2 depending on the number of the supercon- found from Table I and Fig. 7, the experimental data of
ducting planes inside a unit cell of the superconductors. three iron-basedsuperconductors: SmO1−xFxFeAs of 55
The former class indicates that the superconducting lay- K (1111 series),5 Ba1−xKxFe2As2 of 38 K (122 series)16
ers are separated by a distance of the c-axis lattice con- and(Sr V O )Fe As of46K(21311series)17 According
4 2 6 2 2
stant, while the superconducting layer-layer separation to the theory suggested in this paper, we argue that the
along the c-axis is reduced by half for the later class. possible maximum Tmax for 1111, 122 and 21311 series
c
Importantly, although the crystal structure and physical cannot exceed 60 K, 40 K and 50 K, as indicated in Fig.
property vary considerably among the superconductors, 7 (three cyan lines).
it is easy to find that the Tmax increases monotonically Furthermore, we will show that our theory can pro-
c
withtheincreasingoftheeffectivec-axislatticeconstant. vide a qualitative explanation of why the superconduc-
7
tivity does not occur in noble metals (for example, Ag, (a) (b)
}~
Au, and Cu) and a also new insights into the problemof
the superconductivity in metallic hydrogen. It is a com-
mon knowledge that gold, silver and copper are the best |
conductorsofelectricalcurrent. However,aquestionhas
long plagued the physics community: why an ideal con- {zxyz
ductor that is not a superconductor? With the help of
the Fig. 2 and Eq. (13), we try to unravel this mys- [[
[(cid:149)(cid:150)(cid:151)
tery in a very simple way. Under normal circumstances,
good conductors always have an extremely high level of
carrierconcentration. One canassume thatthese metals
vwxyz
have the superconducting properties, in our view, it is
necessary that the superconducting electrons should be [(cid:149)(cid:150)(cid:151)(cid:152)(cid:153)
arrangedasthe specificvortexlatticesofFig. 2withthe \]^_‘abcdefg
stripe-stripe spacingsξ ∼2Å(≪2.8Å). In this case,the hijklmjnnopqqrimstu
crowded vortex lattices are unstable owing to the strong
( ) ( )
electromagnetic interactions between vortex lines, as a (cid:127)(cid:128) (cid:132)(cid:128)(cid:133)(cid:134)(cid:135) (cid:137)(cid:139)(cid:140)(cid:141)(cid:142) (cid:143)(cid:144) (cid:132)(cid:128)(cid:133)(cid:134)(cid:135) d (cid:137)(cid:139)(cid:147)(cid:148)(cid:142)
result,thechargecarriersaremorelikelytobeformedin (cid:129)(cid:130)(cid:131) (cid:131) (cid:136) (cid:138) (cid:129)(cid:130)(cid:131) (cid:131) (cid:145)(cid:146) (cid:138)
a random and lower-energy stable non-superconducting Figure8: Twoanalogoussuperconductors(a)La2−xBaxCuO4
phase. Thesediscussionscanalsobeappliedtothe over- of40K,and(b)Sr2−xBaxCuO3+δ of98K.Inourtheoretical
doped high-T superconductors where the stripe-stripe framework of the effective c-axis lattice constant, they have
c
spacinginsidethesuperconductingCuOplanesisfarless completely different relation between the c-axis lattice con-
∗ ∗
stant c and the effective c-axis lattice constant c : c = c/2
than 2.8Å.
for La2−xBaxCuO4 with two superconducting planes inside
Physicists predicted that hydrogen in solid-state can one unit cell, while c∗ = c for Sr2−xBaxCuO3+δ with only
be a high-temperature superconductor, and maybe even onesuperconductingplaneinsideoneunitcell. Wethinkitis
a room-temperature superconductor. Indeed, this pre- thedifferenceoftheeffectivec-axislatticeconstantthatleads
diction sounds very interesting and attractive, but we totheverydifferencebetweentwosamplesinsuperconducting
would like to point out that it cannot be realized as de- transition temperature.
sired. Fromtheperspectiveofatomicstructure,thereare
the same number of carriers (free electrons) in metallic
hydrogen, lithium (BCC, a = b = c = 3.51Å), sodium which look very similar to that of La2−xBaxCuO4 but
(BCC, a = b = c = 4.29Å) and potassium (BCC, with partially occupied apical oxygen sites, as shown in
a=b=c=5.33Å)withthesamenumberofatoms. The Fig. 8(b). Why can two almost the same superconduct-
theoretical predicted lattice constants of metallic hydro- ing materials have completely different superconducting
gen (BCC, a=b=c=2.89Å) are smaller than those of transition temperatures? Although several factors have
theotherthreematerials,indicatingthatmetallichydro- beenconsideredinwhatenhancetheTc,18itisintuitively
gen has the highest carrier concentration among them. obvious that the actual reason has not been elucidated
As lithium, sodium and potassium do not exhibit super- yet. In our theoretical framework of the effective c-axis
conductivity,nowwecanboldlypredictthatthemetallic lattice constant, it is most likely that the introducing of
hydrogen with the highest carrier concentration cannot partially occupied apical oxygen sites could cause one of
beasuperconductoratanylowtemperature,nottomen- theCuOlayersnolongertofunctionasthesuperconduct-
tion the room temperature superconductivity. ingplane. Therefore,thetwoanalogoussuperconductors
ofFig. 8havecompletelydifferentrelationbetweencand
Finally, it is worth to note that the suggested real-
∗ ∗ ∗
space effective c-axis lattice constant theory of super- c : c =c/2forLa2−xBaxCuO4ofFig. 8(a),whilec =c
conductivity has been excellently confirmed by recent for Sr2−xBaxCuO3+δ of Fig. 8(b). From the experi-
measurements.18 Fig. 8(a) shows a unit-cell of cuprate mental results of Sr2−xBaxCuO3+δ (c∗ = c = 12.78Å)18
and Eq. (13), one can get immediately Tmax ≈ 99.8 K
La2−xBaxCuO4of 40 K with the c-axis lattice constant c
which is in good agreementwith the experimental result
c = 13.23Å. It is easy to find from the figure that
(Tmax ≈98 K).18
superconducting layer (CuO plane) spacing (the effec- c
tive c-axis lattice constant) of the superconductor is
∗
c = c/2 = 6.615Å. If there are experimental methods
thatcancauseoneoftheCuOplanetolosetheabilityof V. CONCLUDING REMARKS
conductingsuperconductingcurrent,thentheeffectivec-
∗
axislatticeconstantcanbepromotedtoc =c=13.23Å, We have proposed the real-space effective c-axis lat-
consequently,the correspondingmaximumTmax maybe tice constant theory of superconductivity based on the
c
dramatically enhanced to 104 K calculated according to self-organized picture of the superconducting electrons
Eq. (13) . This prediction has just been experimen- and studied the stability of the superconducting vortex
tallyverifiedbyGaoetal.18incuprateSr2−xBaxCuO3+δ lattices in layered superconductors. It has been shown
8
clearlythattheeffectivelatticeconstantplayasignificant (La2−xBaxCuO4 of 40K andSr2−xBaxCuO3+δ of98 K)
role in promoting the superconducting transition tem- would have such a different superconducting transition
perature inlayeredsuperconductors,suchas the copper- temperatures,thelatterisabout2.5times oftheformer.
based and newly discovered iron-based compounds. Ac- The physical reasonwhy the superconductivity does not
cordingto a largenumber of experimentaldata,we have occur in gold, silver and copper has also been provided
obtainedanimportantequation: Tmax =10c∗−28which basedonthesuggestednewmechanism. Finally,wehave
c
can be used to estimate the highest possible Tmax for a argued that the metallic hydrogencannot exhibit super-
c
∗
givenlayeredsuperconductor,where c is the effective c- conductivity at any low temperature. We think that the
axis lattice constant. This result suggests that the max- real-space effective c-axis lattice constant theory of su-
imum possible Tmax of the iron-based superconductors perconductivity may finally shed light on the mysteries
c
cannot exceed 60 K, 50 K and 40 K for the 1111, 21311 of superconductivity. With these results, the scientists
and 122 series, respectively. It should be noted that this will be able to proceed to the materials design for the
predictionhas stood the test oftime for abouttwo years new superconductors that may have a higher supercon-
by many experiments. Furthermore, we have tried to ducting transition temperature.
explain why two very similar cuprate superconductors
∗ Electronic address: [email protected] 11 X.Q. Huang, arXiv:0910.5511 (2009).
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