- Email: [email protected]

Ferromagnetism and high Tc superconductivity Benoy Chakraverty Lepes, C.N.R.S, BP 166, 38042 Grenoble Cedex, France

Abstract The analogy between magnetism and superconductivity is important for our understanding of both the phenomena in an uniﬁed way. The age-old dilemma between Stoner versus non-Stoner ferromagnetism also exists in high Tc superconductors. Between the Bloch-wall stiffness and superconducting phase stiffness the analogy is rigorous and complete. Holes in d-band usher in ferromagnetism; the high Tc superconductivity in the copper oxide materials is also heralded by the ubiquitous d-holesFis that fortuitious? r 2002 Elsevier Science B.V. All rights reserved. Keywords: Ferromagnetism; Superconductivity

We shall show in this communication that the two most well-known phase transitions in nature, ferromagnetism and superconductivity have a great deal in common. Of course both are second-order transitions and as such develops in its ground state broken internal symmetry in conﬂict with the mother Hamiltonian which has the unbroken symmetry. In case of ferromagnetic phase transition, the symmetry broken is the spin rotation (O3) symmetryFthe system chooses an unique spin orientation corresponding to the magnetisation direction while the Hamiltonian (Heisenberg for example) has full rotational symmetry. The broken symmetry state leads to spin waves or goldstone modes. In case of superconductivity, if we assume singlet electron pairs in the ground state the spin-rotation symmetry is never broken; the broken symmetry is more subtleFit corresponds to the superconducting order parameter which is two-dimensional (has an amplitude and a phase) blocking its phase along some arbitrary but unique phase direction in the two-dimensional azimuthal plane and thus breaking the azimuthal symmetry of rotation along the Z-axis, the so-called Uð1Þ symmetry. The resultant goldstone mode is the phase excitation mode which is pushed to plasma frequency in a charged metallic system; otherwise a metal will change its colour when it becomes superconducting! We must add that while in the ferromagnetic ground state the E-mail address: [email protected] (B. Chakraverty).

time reversal symmetry is gone for the electrons (up-spin and down-spin electrons do not have the same energy) this symmetry is a must for the singlet BCS cooper pairs. This is why one thinks of ferromagnetism and superconductivity as natural enemies (note in case of triplet superconductors the two can coexist). The role of holes in ferromagnetism is known from time immemorial (the Slater–Pauling curve). A full d-band is not ferromagnetic; to make it so we have to line up a majority of upspins but to do so we need to empty electronic states into which the down-spins can be promoted to and hence the need for holes. Strangely enough a half-ﬁlled d-band (one electron per atom or site) is not necessarily ferromagnetic. While it is so for Fe, in many of the transition metal oxides, the half-ﬁlling makes the system insulating (the Mott insulator) and concommitantly antiferromagnetic. Once again a slight deviation from half-ﬁlling renders the system ferromagnetic (Lanthanum oxides when calcium doped, i.e. doped with holes) [1]. Some of the most exciting developments in this ﬁeld have been precisely thisFthat addition of holes even to a semiconductor with magnetic atoms (by hole injection) which gives rise to strong ferromagnetism [2]. What about superconductivity? All classical superconductors are metallic to start with and adding holes or electrons makes no difference as far as superconductivity goes. But the high Tc superconductors are exceptional. The parent compound is insulating and being exactly half-ﬁlled d-band is insulating and antiferromagnetic. At a critical hole concentration the antiferromagnetism

0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 1 7 6 - 3

B. Chakraverty / Journal of Magnetism and Magnetic Materials 242–245 (2002) 9–12

10

disappears and superconductivity appears at T ¼ 0: Batlogg’s recent experiments [3] on hole injection in organic molecular crystals like anthracine show that once again at a critical hole concentration the system becomes superconducting! In both ferromagnetism and high Tc superconductors holes play a crucial role. Why? The simplest Hamiltonian to illustrate what is happening is X X tij cni cj 7jUj nim nik : ð1Þ H¼ ij

i

This Hamiltonian is often studied, known as the Hubbard Hamiltonian, when the sign before jUj is positive. The positive sign implies that an up-spin electron on the site i repels a down-spin electron on the same site, nim is an electron operator giving the probability of the site i; being occupied by an up-spin electron (a given site can at best be occupied by 2 electrons at a cost U). The ﬁrst term gives us the kinetic energy of electrons hopping from site j to i; c0 s are the electron creation and annihilation operators which give us the amplitude of hopping. When we have a negative sign in front of jUj it tells us that two up-spin and downspin electrons would prefer the same site as they would gain an energy jUj: This pairing of electrons would lead to a singlet state and eventually to a BCS superconductivity. Thus, the same Hamiltonian can be used to describe ferromagnetism and superconducting order parameter. The BCS pairing order parameter can be written as the thermodynamic average /Cbcs S ¼ /cim cik S:

ð2Þ

The superconducting energy gap is given by Dbcs ¼ jUj/Cbcs S:

ð3Þ

We may note that the fermi system of electrons is unstable to the formation of superconducting pairs for inﬁnitesimal U i.e. for Ua0: The question that we shall ask is whether superconducting Tc ¼ jDbcs j as the BCS theory asserts. Putting that question awhile for later, let us deﬁne the ferromagnetic order parameter as the magnetic moment /MS ¼ 12/nim nik S;

ð4Þ

where the quantisation axis for spin is assumed to be in the Z-direction along which the magnetic moment is supposed to be aligned breaking the spin rotationsymmetry. This is the so-called Stoner model for itinerant electrons and gives the energy gap for spin excitation as Dstoner ¼ UjMj:

ð5Þ

We may note that in contrast with the superconducting case, now we need a critical value of jUc j so that jMja0: The so-called Stoner criterion is UNðEF Þg0 for ferromagnetism to appear, where NðEF Þ is the density

of states at the fermi level. The inevitable question to ask is if the ferromagnetic Curie temperature Tc CDstoner ? The following table is eloquent. Material

Moment/ atom

Tc (K)

Exchange splitting, Um (K)

Fe Co Ni Gd EuS

2.22 1.72 0.61 7.1 6.5

1043 1388 627 292 25

21,000 12,000 4,500 100,000

The problem with the Stoner model is that it is a mean ﬁeld theory that implicitly assumes an inﬁnite transverse spin stiffness which obliges the spin direction to remain locked while the transition to the metallic state is expected to occur by progressive loss of amplitude of the magnetic moment. The counterpoint to this scenario was pointed out and demonstrated by several authors [4]. Here, one assumes a finite transverse stiffness such that the local spin direction can rotate away from the global magnetisation, while the amplitude of the magnetic moment does not necessaily vary. At T ¼ Tc the loss of ferromagnetism occurs due to total disorientation of the moments. The cost in energy is not the Stoner exchange energy but a Bloch-wall or spin stiffness energy. We can write qE jrM 2 j ¼ rm ; O M2

ð6Þ

where rm is the Bloch-wall stiffness energy. The thermodynamic average of the directional ﬂuctuation is /jrM 2 jS ¼

kTM 2 : rm

ð7Þ

Now we can use the Lindemann criterion of melting to deﬁne the Tc as /jrM 2 jS ¼ M 2 ;

kTc ¼ rm :

ð8Þ

For a simple metal with hole pockets rm can be calculated from ﬁrst principles [5] and is given by 8 9 Z /nk0 m þ nk0 k Sr2k0 ek0 < = : ð9Þ rm ¼ d 3 k 0 2 2 : /nk0 m nk0 k Sðrk0 ek0 Þ ; DEexc We note that for a full R band, the ﬁrst term on the righthand side is zero i.e. k0 r2k0 ek0 ¼ 0; hence one needs holes to give a ﬁnite spin stiffness. One can also write spin stiffness in the following alternative empirical forms: rm ¼ c20 wm ¼ JS2 a2d ¼ KA d2 :

ð10Þ

B. Chakraverty / Journal of Magnetism and Magnetic Materials 242–245 (2002) 9–12

Here, we have c0 as the spin wave velocity, wm is the uniform magnetic susceptibility, J is the exchange constant, with a being a lattice distance of dimension d; (the product JS2 can be obtained from spin wave dispersion $ ¼ 4JS2 q2 Þ; KA is the anisotropy constant while d is a measurable domain wall-thickness. Tc can be written in the scaled universal form [6] rm xd2 n 2 ¼ : kTc 4p d 2

ð11Þ

Here, x is a magnetic correlation length ðxE12ðJ 00 ðq ¼ pﬃﬃﬃ 0ÞÞ=Jðq ¼ 0ÞEa 6Þ: We show in the next table the results. ( rm (meV=A)

( 1 Þ x1 ðA

4prm x=kTc

FeF14 CoF17.5 NiF7

1.5 1.45 1.5

1.3 1.3 1.04

We see that rm mimics Tc perfectly. High Tc superconductors: We now show how the situation is analogous in the high Tc problem. In fact, there would not be any problem if these new superconductors followed even remotely the BCS behaviour which says that kTc ¼ 1:74 Dbcs ; the superconducting gap at the fermi level. If the gap goes up the Tc must go up. The following table shows the experimental Tc and measured gap values for a variety of cuprates; this is actually the general trend.

YBa2 Cu3 O6þx

Tc ðKÞ

D0 ðmeVÞ

x ¼ 0:92 x ¼ 0:85 x ¼ 0:69 x ¼ 0:57 x ¼ 0:53

91 89 59 47 25

13 14 25.3 27.1 30.6

Bi2 Sr2 Ca1x Dyx Cu2 O8þd d ¼ 0:15 d ¼ 0:12 d ¼ 0:1 d ¼ 0:075

92.75 79.5 66.2 40

19 21 22.5 25

The experimental trend goes quite the opposite way, highest superconducting temperature for optimum doping is associated with the smallest superconducting gap! This shows the limitation of the BCS theory which is a mean ﬁeld theory and implicitly assumes that as we approach Tc the gap amplitude diminishes progressively through single particle excitation of electron–hole pairs until the system becomes a fermi metal at the transition temperature. This vision neglects the essential fact that the superconducting order parameter /Cbcs S and hence

11

the superconducting gap is a complex quantity, has an amplitude and phase (it is two-dimensional just as ferromagnetic order parameter is three-dimensional in the spin space) and as a result no superconductivity can appear if the whole physical space does not have the same phase for the order parameter. The order parameter is written as /Cbcs S ¼ jCjexp iy and the BCS theory assumes an inﬁnite phase rigidity so that only excitation that can destroy superconductivity is destruction of amplitude (we pay Dbcs Þ: On the other hand, if we assume a ﬁnite phase rigidity, phase excitation can cost less energy than gap energy and will lead to loss of all phase coherence at T ¼ Tc and hence loss of superconductivity [7]. It emerges that there are two characteristic energy scales to the problem, one relates to the gap Dbcs and the other is related to a phase coherency energy scale Dc given, respectively, by _vf _2 ; Dbcs C ; Dc C xc 2mdc2

ð12Þ

where vf is the fermi velocity and xc is the Cooper-pair coherence length, while m is an electron mass and dc is an average distance between the Cooper pairs. The competition between these two energy scales will determine whether phase or amplitude ﬂuctuation will bring out destruction of superconductivity at T ¼ Tc : If we have a low-density electron system like the high-temperature superconductors, dc bxc ; then Dc !Dbcs and phase ﬂuctuation will predominate . On the other hand, in the usual metallic system of highdensity electrons xc bdc ; Dbcs !Dc ; phase excitation can be ignored and loss of superconductivity will be entirely determined by loss of amplitude and this is the classical BCS behaviour [8]. A more fundamental approach is to calculate the superconducting phase stiffness for a typical Hamiltonian we described above (negative U BCS Hamiltonian) and calculate the resultant Tc : In two dimensions p kTc ¼ rs ðTc Þ; 2 " 1 X qek 2 qf ðEk Þ 1 q2 ek rs ¼ þ : qkx 2N k @Ek 2 qkx2

# ek ef bEk : tanh 1 Ek 2

ð13Þ

Here, r is the superconducting stiffness, Ek ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃs e2k þ D2bcs : We see that the expression is identical to that given earlier, Eq. (9) for magnetic stiffness. While rs is calculated for superconductors with negative U; rm is to be evaluated for magnetic system with

12

B. Chakraverty / Journal of Magnetism and Magnetic Materials 242–245 (2002) 9–12

positive U [9]. The analogy between ferromagnetism and high Tc superconductors is summarised in the ﬁnal table.

X –Y ferromag.

High Tc supercond.

Order parameter, Mxy ¼ /Sþ S Correlation, fnGðr; tÞ ¼ /Sþ ðoÞSðrÞS

c ¼ jC0 j/exp iyS

Susceptibility, w ¼

qMr qH>

Exchange splitting, Dstoner Spin stiffness, rm Tc Erm 5Dstoner

jC0 j2 /exp iðyo yr S R

Gðr; tÞdr

BCS gap, Dbcs Phase stiffness, rs Tc Ers 5Dbcs

References [1] J.M.D. Coey, M. Viret, S. Von Molnar, Adv. Phys. 48 (1999) 167. [2] H. Ohno, et al., Nature 408 (2000) 944 in these proceedings. [3] B. Batlogg, et al., Nature 406 (2000) 704. [4] V. Korenmann, et al., Phys. Rev. B 16 (1977) 4032; J. Hubbard, Phys. Rev. B 20 (1978) 4584; Ph. Nozieres, Lecture Notes College de France, Paris, 1988. [5] C. Herring, in: G. Rado, H. Suhl (Eds.), Magnetism, Vol. 4, Academic Press, NY, 1966.343pp [6] E. Brezin, J. Zinnjustin, Phys. Rev. Lett. 36 (1976) 6921. [7] B.K. Chakraverty, et al., Physica C 235–240 (1994) 2323. [8] V.J. Emery, S.A. Kivelson, Nature 374 (1995) 434; B.K. Chakraverty, T.V. Ramakrishnan, Physica C 282–287 (1997) 290. [9] P.J.H. Denteneer, G. An, J.M.J. van Leeuen, Phys. Rev. B 47 (1993) 6259.