Cryptoferromagnetism in Superconductors with Broken Time-Reversal Symmetry

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by  N. A. Logoboy

Cryptoferromagnetism in Superconductors with Broken Time-Reversal-Symmetry
N. A. Logoboy and E. B. Sonin
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
(Dated: February 2, 2008)
Thecryptoferromagneticstate(thestatewithintrinsicdomainstructure)insuperconductingfer-
romagnets subjected to external magnetic field is studied theoretically. Ferromagnetism originates
either from electron spin or the intrinsic angular momentum of Cooper pairs (chiral p-wave super-
conductors like Sr2RuO4). The phase transitions towards the Meissner and the mixed states are
8
investigated, and the magnetic phase diagrams are obtained. Cryptoferromagnetism, as a form co-
0
existenceofsuperconductivityandferromagnetism,canbedetectedbyobservationofmagnetization
0
curvespredicted in thepresent analysis.
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n PACSnumbers: 74.25.Ha,74.90.+n,75.60.-d
a
J

0 In recent years numerous experimental evidences of metry(TRS).Weusedthisnameinthetitleofthepaper
2 superconductivity-ferromagnetismcoexistence invarious following this more cautious semantics though one can-
materials were reported [1, 2, 3, 4, 5, 6]. Two types not imagine broken TRS without at least some features
]
n of such coexistence are possible: (i) The phase transi- of ferromagnetism (see further discussion).
o tionstotheferromagneticandthesuperconducting(SC) Among possible explanations why experimentalists
c statesoccursatdifferenttemperatures,sothecoexistence cannot see stray fields from DWs is the presence of do-
-
r startsbelowthelowerfromthetwotransitions. Rutheno- main structure with a period essentially smaller than
p cuprates [1] belong to this type: the superconductivity a distance between a sample surface and a probe used
u onset occurs at the temperature much lower than the by experimentalists. There were some experimental ev-
s
. temperature ofthe magnetictransition. Normallydiffer- idences of domains in SFs both in the spin [9] and the
at ent elements of the crystal structure are responsible for orbital SF [10]. The theoretical investigations of the do-
m ferromagnetismand superconductivity, and spontaneous main structure in SFs were restricted with the case of
- magnetization(ferromagneticorderparameter)isrelated zero external magnetic field [11, 12, 13, 14]. One must
d to spin. Later we call them spin superconducting ferro- discern two possible types of equilibrium domain struc-
n magnets(spinSFs). (ii)ThemagneticandtheSCtransi- ture. The first one is well known for normal ferromag-
o
tionsoccursimultaneously. Thiscantakeplaceinuncon- nets [15]. The domain structure results from competi-
c
[ ventional superconductors with triplet Cooper pairing. tion between the energy of DWs and the magnetostatic
An example of them is strontium ruthenate Sr2RuO4 energyofstrayfieldsgeneratedbythemagneticfluxexit-
1 [3, 4, 7]. The theory connects spontaneous magnetiza- ing fromthe sample surface. The periodofthe structure
v
tion in this material not with spin but with the orbital depends on the shape and the size of the sample going
1
9 intrinsicangularmomentofthep-waveCooperpairwith to infinity when the sample size grows. One can call
0 the wave function in the momentum space proportional these domains extrinsic ferromagnetic domains. Since in
3 to px+ipy (chiral p-wave superconductivity). We shall SFs the Meissner effect expels the magnetic field, it is
. call them orbital superconducting ferromagnets (orbital impossible to benefit from decreasing the bulk magne-
1
0 SFs). tostatic energy in comparison with the DW energy, and
8 Whereas proof of superconductivity in SF materials extrinsicdomainscannotappearatequilibrium[12]. But
0 is quite straightforward, a clear-cut detection of the fer- also long ago there was known another type of domains,
v: romagnetic order parameter is much more problematic. which decrease the bulk magnetostatic energy at the ex-
i The internal magnetic field is screened out by the SC pense of destroying the Meissner state [11, 13, 14]. The
X
Meissner currents and can be present only near sample size of these domains is roughly of the order of the Lon-
ar borders and defects, in particular, domain walls (DWs). don penetration depth λ and does not depend on either
Thisstronglysuppressesthestraymagneticfieldsaround shape or size of the sample. Strictly speaking the state
the sample,which aremost convincingevidence of ferro- with this domain structure at the macroscopic scales is
magnetism. Especially worrying is situation with stron- not ferromagnetic but antiferromagnetic though with a
tium ruthenate Sr2RuO4, where Kirtley et al. [8] could rather large period. We shall call such a state cryptofer-
not detect any stray field from DWs or sample edges at romagnetic, the term introduced by Anderson and Suhl
all. Thisisachallengeforthetheoryandfortheverysce- [16] for another model of ferromagnetism and supercon-
narioofchiralp-wavepairing. Difficultieswithdirectde- ductivity, in which crystal anisotropy was neglected and
tection of ferromagnetism coexisting with superconduc- spiral structure appeared instead of domains.
tivityleadtothequestionwhetheronemayusetheterm Inpresentpublicationweextendthetheoryofintrinsic
ferromagnetism atall. Indeedintheliteratureonuncon- domainstructure(cryptoferromagneticstate)onnonzero
ventional superconductors sometimes they prefer to tell external magnetic field and analyze competition of the
aboutsuperconductivitywithbroken Time-Reversal Sym- cryptoferromagnetic state with the pure Meissner state
2
and the mixed state with vortices. This allows to ob- Here h0 = H0/4πM0 and w = σ/4πM02λ are dimension-
tainthe full phase diagramofboth spin andorbitalSFs. less parameters. If h0 = 0 and ξ↑ = ξ↓ Eq. (3) coincides
We demonstrate that the measurementofmagnetization with the free energy density of Krey [11].
curvesin various areasofthe phase diagramcan provide MinimizationofenergydensityEq.(3)withrespectto
evidence of ferromagnetic or cryptoferromagnetic order ξ and ξ yields the system of two nonlinear equations
↑ ↓
in superconductors with broken TRS. for ξ and ξ :
↑ ↓
Let us consider a stripe domain structure with 1800
ξ ξ ξ +ξ
DWs in a sample subjected to external magnetic field ↑ ↓ ↑ ↓
tanh − tanh =h0,
H0 = (0,H0,0). The DWs are parallel to the yz-plane 2 2
separatingdomainswithalternatingmagnetizationM= sinh2ξ↑+sinh2ξ↓ 2(ξ↑+ξ↓)
− =w. (4)
(0,±M0,0) along the +y or −y direction. Since the H0 (coshξ↑+coshξ↓)2
orientation is preferable the width d of domains with
↑
themagnetizationMparalleltoH0 ( -domains)exceeds The magnetic induction B = h is determined by re-
the width d↓ of the domains with M↑antiparallel to H0 duced magnetic induction b=Bh/i4πM0:
( -domains). We restrict ourselves to the simplest case
w↓hen the London penetration length λ exceeds the DW b= 1 ∂E = (1+h0)tanhξ↑−(1−h0)tanhξ↓. (5)
thickness. Then the surface energy σ and the internal −2∂h0 ξ↑+ξ↓
structure of DW is not affected by fields and currents at
scales of λ. Fig.1(a)showsthephasediagramintheplanew2−h0.
Theareaofthecryptoferromagneticstateisrestrictedby
The Gibbs potential inside domains is
twolines wherethe phase transitionbetweenthe crypto-
3 h2 2πλ2 2 h H0 ferromagnetic and the Meissner states occurs:
G =Z d x(cid:18)8π + c2 j −h·M− 4·π (cid:19), (1) 1. The line E = 0, which corresponds to the limit
ξ↑,↓ . The values of w and h0 on this line are con-
where h is the magnetic field, and the electric current j necte→d b∞y the relation w=1+h20.
is connected with the magnetic field h via the Maxwell 2. The line on which domains with magnetization op-
equation h = (4π/c)j. Variation of the Gibbs po- posite to the externalmagnetic field vanish, ξ↓ =0. The
tential yie∇lds×the magnetic field h = (0,h ,0) in the equation describing this line is
↑,↓ ↑,↓
-domains and -domains:
↑ ↓ (1 h )2 1+√h
c c
w = h (1+h ) − ln , (6)
h↑,↓ =(H0 4πM0)cosh(x/λ−ξ↑,↓), (2) c p c c − 2 1−√hc
± coshξ
↑,↓ the critical size of the -domain being
↑
where x is the distance from the DW and ξ =d /2λ
↑,↓ ↑,↓ 1+√h
are reduced domain widths. ξ =ln c. (7)
↑c
1 √h
Application of the Gibbs potential Eq. (1) to orbital c
−
SFs requires some comments. As shown in Ref. [17], for
The magnetic induction on the critical line is
orbital ferromagnetism related to the intrinsic angular
momentum of Cooper pairs the spontaneous magnetiza- −1
1+√h
c
tion cannot be defined unambiguously and therefore the bc =2 hc ln . (8)
(cid:18) 1 √h (cid:19)
Landau-Lifshitz theory of ferromagnetism [15] based on p − c
this definition is not valid. Nevertheless, interaction of Let us consider the left lower corner of this diagram
magnetizationcurrentsinside narrowDWwiththe mag- wherew 1andh0 1. The criticalparametersonthe
netic field can be reduced to the expression looking like line ξ =≪0 as functio≪ns of w are
↓
the standard Zeeman energy h M0. However, here
− ·
M0 is not a magnetic moment inside the domain but is h = 1(3w)2/3, ξ =(3w)1/3. (9)
defined so that 8πM0 would be the jump 8πM0 of the c 4 ↑c
magnetic field on the DW [see Eq. (2)]. So “magnetiza-
Aside from the critical line Eqs. (4) yield:
tion”M0 isdeterminedbythe DWstructureandcannot
be used for other phenomena connected with ferromag- ξ2 ξ2 ξ3+ξ3
neticordering,e.g.,analyzingthemagnonspectrum[17]. h0 = ↑ − ↓, w = ↑ ↓. (10)
Substituting Eq. (2) into Eq. (1), adding the sur- 4 3
face energy σ of DWs, and averaging over the domain-
For small h0 hc one can solve Eqs. (10) analytically:
structure period d = d +d we arrive to the following ≪
↑ ↓
expressionfor reducedenergydensity E =G/2πM02V (V ξ =41/3 h 1 h20 h0 . (11)
is the sample volume): ↑,↓ c(cid:18) − 44/3h2(cid:19)± 41/3√h
p c c
2w (1+h0)2tanhξ↑ (1 h0)2tanhξ↓ Letus considerthe magnetizationcurveinthe crypto-
= − − − . (3)
E ξ↑+ξ↓ ferromagneticstate. The linearmagnetic permeability is
3
a) h01.0 determined from two relations connecting µ and w with
the period ξ =ξ +ξ 2ξ
↑ ↓ ↑
≈
0.8 db cothξ ξ
µ= = , w =tanhξ . (12)
dh0 ξ − coshξ
0.6 In the limit w 0 the magnetic permeability is diver-
gent: µ (2/3→w)2/3. The whole magnetization curves
≈
b(h0), which were calculated numerically, are shown in
0.4 Fig. 2(a).
Fig.2(b) showsthe dependence of the periodξ =ξ +
↑
ξ andthedifferenceofthetwodomainwidthsδ =ξ ξ
↓ ↑ ↓
−
0.2 as functions of the reduced external magnetic field h0.
Up to now we ignored the possibility of the transi-
tion to the mixed state assuming that the first critical
1 2 3 4 magnetic field Hc1 = Φ0lnκ/4πλ2 essentially exceeds
w2 the characteristicfieldsofthe cryptoferromagneticstate.
Here κ = λ/ξ0 is the ratio of λ to the coherence length
b) h 1.0
0 F ξ0. Let us now take into account this possibility. Both
S
al the field Hc1 and the parameter w depend on the pene-
t trationdepthλ, anditisusefulto introducethe reduced
0.8 orbi first critical field hc1 = Hc1/4πM0w2 = Φ0M03lnκ/σ2,
f which does not depend onλ. The reducedfree energy of
o
0.6 n the mixed state with respect to the energy of the Meiss-
w ner state is
o
0.4 ol-d m = 1+h0 H∗ 2 = 1+h0 hc1w2 2,(13)
o E −(cid:18) − 4πM(cid:19) − −
c (cid:0) (cid:1)
0.2 cool-down of spin SF wherethefieldH∗insidethemixedstatediffersfromHc1
by another logarithm factor, but we neglect it assuming
H∗ Hc1. The phase transition to the mixed state may
≈
occureitherfromtheMeissnerstatebeingdeterminedby
0.0
0 1 2 3 4 the condition = 0, or from the cryptoferromagnetic
w2 Em
state crossing the critical line on which = . At zero
m
E E
external field h0 and small w the phase transition be-
c) h 1.0 F tweenthemixedstateandthe cryptoferromagneticstate
0 al S occurs at wm ≈ √3/(2hc1)3/4. Thus whatever large hc1
t is, in the left lower corner of the phase diagram there is
0.8 rbi alwaysthespontaneousvortexphase,i.e.,themixedstate
o
f withoutexternalmagneticfield. Thefullphasediagrams
o
n at two finite values hc1 =2 and 0.83 are shown in Figs.
0.6 w 1(b) and (c). The cryptoferromagnetic state disappears
= do from the phase diagram at hc1 <0.5.
0.4 ol- Now let us analyze the phase transformations in the
o process of cooling down below the SC critical tempera-
c
ture. This process is different for spin and orbital SFs.
cool-down of spin SF
In the case of spin SFs, when the magnetic transition
0.2
occurs at much higher temperature, one may neglect
temperature dependence of M0, σ, and hc1. Then only
w2 1/λ2 τ depends on relative temperature differ-
∝ ∝
1 2 3 4 ence τ = (T T)/T . On the phase diagrams of Figs.
c c
w2 −
1(b) and (c) the state moves along straight lines paral-
lel to the horizontal axis w2. From these figures it is
FIG.1: (coloronline)Phasediagramforvariousvaluesofthe evident that just below the critical temperature the sys-
reducedlowercriticalfieldhc1: (a)hc1 →∞;(b)hc1 =2;(c) tem enters the mixed state. At further cooling down the
hc1 =0.83. Thelighter(yellow)shadedareaisthecryptofer- system crosses to the Meissner state either directly or
romagneticstate. Thedarker(blue)shadedareaisthemixed
through the area of the cryptoferromagnetic state. For
state. Therest istheMeissner state. The horizontal and the
orbital SFs the cooling process occurs differently. In
verticalarrowsshowtheprocessesofcool-downacrosstheSC
critical temperature of spin and orbital SFs respectively.
4

DW surface energy is a product of the condensation en-
a) 1.0 1 - w = 0.2 eξ0rg(τy)Hc2ξ(τ0/)√∼τ:[Φσ0/λ(ττ3)/ξ20Φ(20τ/)λ]220ξa0n.dHtehreecλo0haenrednξc0ealreengththe
0.8 ∼ ∼
2 - w = 0.4 penetrationdepth andthe coherencelength atzerotem-
b 0.6 3 - w = 0.6 perature. Then the parameters w2 ∼h−c11 ∼(λ0/ξ0)2 do
not depend on temperature whereas the reduced mag-
3
0.4 1 2 netic field does: h0 = H0/4πM0 ∝ 1/τ. Thus in the
field-cooling process the state moves along vertical lines
0.2 on the phase diagrams in Figs. 1(b) and (c). However,
as pointed out above, the cryptoferromagnetic state can
0.0 compete with the mixed state only if hc1 is high enough.
0.0 0.1 0.2 0.3 0.4 Since hc1 (ξ0/λ0)2, this requires λ0 not large com-
∼
h0 pared to ξ0. According to [8] the ratio of λ0 = 190 nm
3.0 to ξ0 = 66 nm is not too high indeed. But this means
that the DW thickness is not so small compared to λ as
b)
2.5 3 assumed in our analysis. Therefore for orbital SFs our
2 analysis can provide only a qualitative but still credible
2.0
picture of the phase transformations.
,
1.5 1
In conclusion, we analyzed the conditions for appear-
1.0 ance of intrinsic domains in superconductors with bro-
ken time reversal symmetry, in which superconductivity
0.5 coexists either with spin ferromagnetism or with ferro-
magnetismoriginatedfromtheintrinsicangularmomen-
0.0
0.0 0.1 0.2 0.3 0.4 tum of Cooper pairs (chiral p-wave superconductors like
h Sr2RuO4). Since these domains strictly speaking corre-
0
spond not to a ferromagnetic but a globally antiferro-
FIG.2: (coloronline)(a)Magnetizationcurvesand(b)mag- magneticstate,thestatewascalledcryptoferromagnetic.
neticfielddependenciesofξ =ξ(h0)(solidline)andδ=δ(h0) We considered competition of this state with the Meiss-
(dashline)areshownatdifferentvaluesofparameterw. Ver-
ner and the mixed states and found the phase diagram.
ticallinescorrespondtocriticalfieldsabovewhichtheintrin-
This phase diagram can be checked with detailed mea-
sic domain structure collapses.
surements of magnetization curves.
ThisworkhasbeensupportedbythegrantoftheIsrael
this case the “magnetization” M0 Φ/λ2 τ, and the Academy of Sciences and Humanities.
∼ ∝
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