The distinction between type I and type II superconductivity is re-examined
in the context of the SO(5) model recently put forth by Zhang. Whereas in
conventional superconductivity only one parameter (the Ginzburg-Landau
parameter $\kappa$) characterizes the model, in the SO(5) model there are two
essential parameters. These can be chosen to be $\kappa$ and another parameter,
$\beta$, related to the doping. There is a more complicated relation between
$\kappa$ and the behaviour of a superconductor in a magnetic field. In
particular, one can find type I superconductivity even when $\kappa$ is large,
for appropriate values of $\beta$.Comment: 7 pages, 5 figures. Talk given at MRST Conference on High Energy
  Physics (2002), Perimeter Institute, Waterloo, Ontari

MacKenzie, R.

English

arXiv

R. MacKenzie

Crossref

Magnetic behaviour of SO(5) superconductors

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UdeM-GPP-TH-02-101
Magnetic behaviour of SO(5) superconductors
R. MacKenzie
Laboratoire-René-J.-A.-Lévesque, Université de Montréal
C.P. 6128, Succ. Centre-ville, Montréal, Qc H3C 3J7, Canada
Abstract. The distinction between type I and type II superconductivity is re-examined in the
context of the SO(5) model recently put forth by Zhang. Whereas in conventional superconductivity
only one parameter (the Ginzburg-Landau parameterκ) characterizes the model, in the SO(5) model
there are two essential parameters. These can be chosen to beκ and another parameter,β related to
the doping. There is a more complicated relation betweenκ a d the behaviour of a superconductor
in a magnetic field. In particular, one can find type I superconductivity even whenκ is large, for
appropriate values ofβ .
INTRODUCTION
In this talk, recent work on magnetic properties of the SO(5)model of high-temperature
superconductivity (HTSC) will be presented. After reviewing the case of conventional
superconductivity and some relevant facts of HTSC, the SO(5) model will be introduced.
The behaviour of a system described by this model when placedin a magnetic field will
be analysed. The main conclusion is that in strongly underdope superconductors, the
critical value of the Ginzburg-Landau (GL) parameterκ can be much larger than the con-
ventional value. Thus, a large value ofκ can be associated with a type I superconductor.
The application of these ideas to HTSC will be discussed briefly. This work forms the
bulk of Refs. [1, 2].
PRELIMINARIES
Conventional superconductivity
In this section we briefly review some features of conventional superconductivity
(SC). This material is well-known, and can be found in greater d tail in almost any
introductory SC textbook, such as Tinkham [3].
The phase transition in a conventional superconductor can be described at low ener-
gies by an effective theory, known as a Ginzburg-Landau (GL)theory, written in terms
of the SC order parameterφ (a complex field representing the Cooper pair amplitude)
and the electromagnetic field. This GL theory can be expressed in t rms of a Helmholtz
free energy, which takes the following form:
F =
∫
dx
{
fn−
a21
2
|φ |2+ b
4
|φ |4+ 1
2m∗
∣
∣
∣
∣
(
−ih̄∇− e
∗A
c
)
φ
∣
∣
∣
∣
2
+
h2
8π
}
.
Here, fn is a constant,h = ∇ × A is the microscopic magnetic field anda1, b are
parameters. The minimum of the potential is|φ |2 = a21/b≡ v2.
There are two characteristic length scales in this model: thcoherence lengthξ =
(h̄2/m∗a21)
1/2 and the magnetic field penetration depthλ = (m∗c2/4πe∗2v2)1/2. These
are, roughly, the Compton wavelengths of the scalar and electromagnetic fields, respec-
tively. By scaling out all dimensionful quantities, one finds that the behaviour of a SC
described by the above free energy is determined by one dimens onless parameter, the
GL parameterκ = λ/ξ . Some typical values for this parameter appear in Table 1.
TABLE 1. Typical parameter values
for various categories of superconduc-
tors.
λ (Å) ξ (Å) κ
Simple metal 300 1000 .3
Alloy 2000 50 40
High-Tc 2000 20 100
There are two very different classes of (conventional) SCs,depending on the value of
κ . If κ < 1/
√
2, the material is said to be type I, while ifκ > 1/
√
2, it is said to be type
II. The behaviour when a magnetic field is applied to a superconductor differs greatly
for these two classes.
There are several ways to see this. One way is to consider a configuration where a
magnetic field equal to the so-called thermodynamic critical field H0c is applied to the
superconductor.1 The sign of the energy of a surface separating SC and normal regions is
the telling quantity. If it is positive, the system would prefer to minimize the amount of
surface for a given magnetic flux; this is achieved if the flux penetrates in a macroscopic
region. In contrast, if the surface energy is negative, the flux will form a lattice of flux
tubes of the minimum allowable flux. One can calculate numerically the surface energy
of a boundary between SC and normal regions. Even quantitatively, one can argue that
for smallκ the surface energy is positive, while for largeκ it is negative (see Figure 1).
An alternative way to determine the distinction between type I and type II supercon-
ductors is to consider the energetics of vortices of varyingwinding number. Since the
magnetic flux of a vortex is proportional to its winding number, the energy per unit wind-
ing number indicates whether it is energetically favourable for a given amount of flux to
penetrate in many unit-winding-number vortices or in one large vortex. The former will
1 If a weak magnetic field is applied to a superconductor, the field is expelled; if a strong field is applied,
SC is destroyed and the field penetrates the material. The critical field is the transitional value, i.e., that
where the (Gibbs) free energy of the normal phase in the magnetic field is equal to that of the SC phase in
the field’s absence.
 (a) Small κ
h
f
 (b) Large κ
h
f
FIGURE 1. Field profiles at SC/normal surface for (a) Type I and (b) TypeII superconductors.f is a
rescaledφ . The slow variation off whenκ ≪ 1 gives rise to a positive contribution to the surface energy;
the slow variation ofh has the opposite effect in the opposite limit. (Figure from "Magnetic Properties of
SO(5) Superconductivity" by M. Juneau, R. MacKenzie and M.-A. Vachon, in Annals of Physics, Volume
298, 421, copyright 2002, Elsevier Science (USA), reproduce by permission of the publisher.)
be the case if the energy per unit winding number is of positive slope, while the latter
will be true if it is of negative slope (see [1]).
The magnetization curves of type I and type II superconductors also highlight their
different behaviour in a magnetic field (see Figure 2). As themagnetic field is increased
in a type I superconductor, the field is completely expelled until H0c is reached, at which
point SC is destroyed macroscopically. In contrast, in a type II superconductor, at a
lower field H0c1 the field starts to penetrate the superconductor in vortices; when the
upper critical fieldH0c2 is reached, SC is finally destroyed. These critical fields vary as a
function ofκ ; one findsH0c2 =
√
2κH0c . The transition point between type I and type II
SC occurs when these two critical fields are equal, which occurs atκ = κc = 1/
√
2.
High-temperature superconductivity
In this section, a couple of relevant facts of HTSC are present d. The key observation
which leads to the SO(5) model of HTSC is that these materialsexhibit two very
different phases at low temperature, depending on the degree of doping. At sufficiently
high doping, one sees SC, while at lower values of the doping (including the undoped
case) the materials are antiferromagnetic (AF), as shown inFigure 3.
HcHc1 Hc2
type I
type II
B
H
FIGURE 2. B vs.H for type I and type II superconductors.
µ c
T
AF
SC
µ
FIGURE 3. Approximate phase diagram for high-temperature superconductors and for SO(5) super-
conductivity.
An important property of HTSCs is that, as mentioned above (se Table 1), they are
“highly type II”, i.e., κ is very much larger than its critical value. (Values in the vicinity
of 100 are typical; the lowest value we have seen reported is 17.)
However, as we shall see presently, in the SO(5) model,κ a one does not determine
the magnetic behaviour (i.e., the type) of a superconductor. Indeed, it is possible that, in
spite of having a very largeκ , HTSCs (if described by the SO(5) model) might exhibit
type I behaviour under certain conditions.
SO(5) SUPERCONDUCTIVITY
Motivation; Ginzburg-Landau model
As mentioned above, the presence of both AF and SC in HTSCs sugge ts the possi-
bility of a sort of unification of these phenomena, both of which nvolve spontaneous
symmetry breaking. This possibility was put forth by Zhang [4], who wrote down a
model in terms of a five-component real order parameter. The five components are the
real and imaginary components of the SC order parameterφ and the three components
of the AF order parameterη. A GL theory which has an approximate SO(5) rotational
symmetry can then be written down; the free energy is
F =
∫
dx
{
h2
8π
+
h̄2
2m∗
∣
∣
∣
∣
(
∇+
ie∗A
h̄c
)
φ
∣
∣
∣
∣
2
+
h̄2
2m∗
(∇η)2+V(φ ,η)
}
,
where the potential is
V(φ ,η) =−a1
2
2
φ2− a2
2
2
η2+
b
4
(φ2+η2)2.
There are now three relevant length scales:λ , ξ andξ ′ (the characteristic length of the
η field). Rescaling now reduces the number of essential parameters to two, which can be
taken to beκ andβ ≡ (a22/a12). The latter is related to the doping;β = 1 at the AF-SC
boundary, andβ < 1 in the SC phase.
The potential forβ < 1 is shown in Figure 4. There are two important features of the
potential. First, it is minimized at a nonzero value ofφ andη = 0, so the ground state is
indeed SC. Second, ifφ is somehow forced to be zero, thenη will be nonzero.
In fact, there are two situations whenφ is indeed forced to be zero: firstly, in the core
of a vortex [4, 5, 6, 1], and secondly, if the superconductor is placed in a sufficiently
strong magnetic field.
Magnetic properties
This “induced antiferromagnetism” can have a dramatic effect on the critical fields,
and on the type of superconductor described by the model. Theeffect on the thermody-
namic critical fieldHc arises because it is found by comparing the Gibbs free energies of
SC and AF (rather than normal) states. The other critical fieldsHc1,2 are affected because
vortex energetics are affected by the AF core.
Both Hc andHc2 can be calculated analytically [2]:
Hc = H
0
c
√
1−β 2;
Hc2 = H
0
c2(1−β ) =
√
2κH0c (1−β ).
As in the conventional case, equality of these critical fields indicates the boundary of
type I/II behaviour. We can thus obtain a curve in theβ -κ plane which represents the
0
0.5
1
φ
0
0.25
0.5
0.75
1
n
0.25
0.5
0.75
1
FIGURE 4. SO(5) potential as a function ofφ andη .
boundary separating type I and type II behaviour (Figure 5).Analytically, this curve is
given by
κc(β ) =
1√
2
√
1+β
1−β .
This expression is confirmed (numerically) by analysis of surface energy and vortex
energetics.
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
ka
pp
a
beta
type I
type II
FIGURE 5. Curve delineating type I vs type II superconductivity in theκ-β plane. (Figure from
"Magnetic Properties of SO(5) Superconductivity" by M. Juneau, R. MacKenzie and M.-A. Vachon, in
Annals of Physics, Volume 298, 421, copyright 2002, Elsevier Science (USA), reproduced by permission
of the publisher.)
Application to high-temperature superconductivity
In HTSC, as mentioned above,κ is typically of order 100; thus, HTSCs are considered
highly type II. However, as the previous section demonstrates, in the SO(5) modelβ also
plays a role in the nature of the superconductor. In particular, for anyκ > 1/
√
2, if β is
sufficiently large, the material is type I. We can invert the above expression forκc(β ) to
obtain the following expression forβc(κ), valid if κ ≫ 1 βc(κ)≃ 1−κ−2. For example,
if κ = 100, the material is a type I superconductor ifβ < 0.9999 while it is type II if
β >0.9999. (Sinceβ <1 in the SC state, there is only a minute range ofβ corresponding
to type II.) If κ = 17 (the smallest value reported for a HTSC),β > 0.996 for the material
to be type II – still a small window, but much greater than thatfor κ = 100.
The parameterβ is related to measurable quantities:β = 1− 8m∗h̄−2ξ (µ2 − µ2c ),
whereµ, µc are the chemical potential and the critical chemical potential (that at the
SC-AF boundary) andξ is the charge susceptibility.
Thus, we have the possibility of a fairly dramatic test of theSO(5) model; however,
the experimental situation in the underdoped region appears rather delicate. For example,
the appearance of inhomogeneities (stripe formation, phase separation) could mask the
appearance of type I behaviour. Nonetheless, since the degre to which a superconductor
is type II is reduced as the doping is reduced, one could hope to s e signs of a reduction
in the rigidity of the vortex lattice.
ACKNOWLEDGMENTS
This work was supported by the Natural Science and Engineerig Research Council of
Canada.
REFERENCES
1. Juneau, M., MacKenzie, R., Vachon, M.-A., and Cline, J.,Phys. Rev. B (Rapid Communications), 65,
140512 (2002).
2. Juneau, M., MacKenzie, R., and Vachon, M.-A.,nn. Phys., 298, 421 (2002).
3. Tinkham, M.,Introduction to superconductivity, 2nd edn., McGraw-Hill, 1996, ISBN 007064876.
4. Zhang, S.-C.,Science, 275, 1089 (1997).
5. Arovas, D., Berlinsky, A., Kallin, C., and Zhang, S.-C.,Phys. Rev. Lett., 79, 2871 (1997).
6. Alama, S., Berlinsky, A., Bronsard, L., and Giorgi, T.,Phys. Rev. B, 60, 6901 (1999).


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