Table Of ContentMagnetic skyrmion lattices in heavy fermion superconductor UPt
3
A. Knigavko1 and B. Rosenstein2
1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C.
2Electrophysics Department, National Chiao Tung University, Hsinchu, Taiwan 30043, R.O.C.
9 TopologicalanalysisofnearlySO(3)spinsymmetricGinzburg–Landautheory,proposedforUPt3by
Machida et al, shows that there exists a new class of solutions carrying two units of magnetic flux:
9
the magnetic skyrmion. These solutions do not have singular core like Abrikosov vortices and at
9
1 low magnetic fields they become lighter for strongly type II superconductors. Magnetic skyrmions
repel each other as 1/r at distances much larger then the magnetic penetration depth λ, forming
n
a relatively robust triangular lattice. The magnetic induction near Hc1 is found to increase as
Ja (H−Hc1)2. This behavioragrees well with experiments.
2
PACS numbers: 74.20.De, 74.25.Ha, 74.60.Ec, 74.70.Tx
] Heavy fermion superconductors have surprising prop- clear. For a short range repulsion, if one fluxon pene-
n
o ertiesonboththemicroscopicandthemacroscopiclevel. tratedthesample,manymorecanpenetratealmostwith
c Charge carrier’s pairing mechanism is unconventional. no additional cost of energy. This leads to the infinite
- The vortex state is also rather different from that of s- derivative of magnetization. On the other hand for a
r
p wave superconductors: there exist unusual asymmetric longrangeinteractionmakingaplaceforeachadditional
u
vortices,andphasetransitionsbetweennumerousvortex fluxon becomes energy consuming. Derivative of magne-
s
. latticestakeplace[1]. ForthebeststudiedmaterialUPt3 tization thus becomes finite.
t
a severalphenomenologicaltheorieshavebeenputforward It is generallyassumedthat althoughvorticesin UPt3
m [2–4] which utilize a multicomponent order parameter. differ from the usual Abrikosov vortices in many details
- Inparticular,greatefforthas beenmade to qualitatively [8] two important characteristics are preserved. First,
d
and quantitatively map the intricate H T phase dia- their size λ is well defined: magnetic field and interac-
n −
gram. Most attention has been devoted to the region of tions between vortices vanish exponentially beyond this
o
c magnetic fields near Hc2. length. Second, their energy is proportional to logκ.
[ Magnetizationcurvesof UPt3 nearHc1 arealsorather However,inthisnoteweshowonthebasisoftopological
unusual (see Fig.1). Theoretically, if the magnetization analysis of a model by Machida et al. [9] that there ex-
2
v is due to penetration of vortices into a superconducting istsanadditionalclassoffluxonswhichwecallmagnetic
5 sample then one expects 4πM to drop with an infinite skyrmions. They carrytwounits ofmagnetic fluxN =2
3 derivativeatHc1(dotted−line). Ontheotherhandexper- and do not have a singular core, similar to ATC texture
4 imentally 4πM continuestoincreasesmoothly(squares in superfluid 3He [10]. We show that their line energy
12 and triang−les represent rescaleddata taken from refs. [5] ε 2ε0, ε0 (Φ0/4πλ)2 is independent κ and is smaller
≈ ≡
8 and [6] respectively). Such a behavior was attributed to than that of Abrikosov vortices for strongly type II su-
9 strong flux pinning or surface effects [5]. However both perconductors like UPt3 (κ 50). Magnetic skyrmion
t/ experimental curves in Fig.1, as well as the other ones lattice becomes the ground s∼tate at low magnetic fields
ma found in literature, are close to each other if plotted in (H−Hc1)/Hc1 ≪1. Wefurtherfindthattherepulsionof
units ofHc1. There mightbe amorefundamentalexpla- magnetic skyrmions is, in fact, long range: V(r) 1/r.
- nationofthe universalsmoothmagnetizationcurvenear This allowsus to produce a nice fit to the magneti∼zation
d
n Hc1. If one assumes that fluxons are of unconventional curve(solidlineinFig.1)whichisuniversal(independent
o type for which interaction is long range then precisely of κ).
c this type of magnetization curve is obtained. The order parameter in the weak spin–orbit coupling
:
v Magnetization near Hc1 due to fluxons carrying N model of UPt3 is a three dimensional complex vector:
Xi units of flux Φ0 ≡ hc/2e, with line energy ε and mu- ψi(~r) [2]. The Ginzburg–Landau free energy reads:
tual interaction V(r), is found by minimizing the Gibbs
ar energy of a very sparse triangular lattice: F =Fsym+∆F, (2)
B BH
G(B)= [ε+3V(a)] , (1) F = αψ ψ∗+ β1(ψ ψ∗)2+ β2 ψ ψ 2 (3)
NΦ0 − 4π sym − i i 2 i i 2 | i i|
wexhpe[reλar]=, t(hΦe0/mBa√gn3e)t12icisinldatutcitcieonsphacaisngth.eWchoennveVnt(iro)n∼al +K1∆(cid:0)|DFx=ψi−|2γ+|ψ|xD|2yψ−i|λ2|(cid:1)ψ+z|2K+2|D∆2χzψ|ψi|i2B+i|281πBj2, ((45))
rbaenhga−ev,ioVr(Br) ∼ 1[l/orgn(,Hth−enHocn1e)]−fi2nd[s7]B, whi(leHif iHt cis1)nlo+n1g. +Pi=x,yhk1i (cid:0)|Dxψi|2+|Dyψi|2(cid:1)+k2i|Dzψi|2i, (6)
∼ ∼ −
The physical reason for this different behavior is very where ∂ i(2e/¯hc)A and B = ( A~) . We
j j j j j
D ≡ − ∇×
1
separatedeq.(2)into a symmetric partF whichis in- ∆~l ~l(~l ∆~l)+2j (~l ∂ ~l)=0. (9)
sym k k
− · ×
variant under the spin rotation group SO(3) acting
spin
on the index i of the order parameter, and into terms Eq.(8) shows that the superconducting velocity is given
breaking the SO(3)spin symmetry (anisotropy, coupling by np∇~mp = −∇~ϑ, where the angle ϑ specifies the ori-
toantiferromagneticspinfluctuationsandspin-orbitcou- entation of vector ~n or m~ in the plane perpendicular to
pling). Although ∆F is crucial in explaining the double ~l (see insert in Fig.2). Thus, ϑ is the superconducting
superconducting phase transition in UPt3 at zero exter- phase.
nal magnetic field and the shape of Hc2(T) curve on the Now we proceed to classify the boundary conditions.
H T phasediagram,itcanbeconsideredasasmallper- Magnetic field vanishes at infinity, while topology of the
tur−bationinthe lowtemperature superconducting phase orientationofthe triad~n,m~,~l atdifferentdistantpoints
(phase B) well below its critical temperature T T− isdescribedbythefirsthomotopygroupofvacuumman-
≪ c ≃
.45 K and at low magnetic fields H Hc1. Indeed, ifold: π1(SO(3))=Z2 [10]. It yields a classification of
estimation of coefficients of ∆F at T≃= T−/2 yields: solutions into two topologically distinct classes (”odd”
c
γ/α .2,λ/α .05 and ∆χH2 / α2 10−6, and and ”even”). This classification is too weak, however,
≃ ≃ (cid:16) 2 c1(cid:17) (cid:16)2β1(cid:17) ≃ for our purposes because it does not guarantee nontriv-
also k K [2]. Therefore, in a certain range of mag-
≪ ial flux penetrating the plane. We will see that con-
netic fields and temperatures there is an approximate
figurations having both ”parities” are of interest. In
O(3) symmetry and we first turn to minimize F .
sym the presence of the magnetic flux possible configurations
In the vacuum of phase B the order parameter is
are further constrained by the flux quantization condi-
ψ~ = ψ0(~n+im~)/√2,ψ02 ≡ α/β1, ~n ⊥ m~, ~n2 = m~2 = 1. tion. The vacuum manifold is naturally factored into
Stability requires: α > 0, β1 > 0 and β2 > −β1. SO(3) SO(2) S2 where the S2 is set of directions of
The symmetry breaking pattern of phase B is as fol- ~l and→the SO(⊗2) is the superconducting phase ϑ. For
lows. Both the spin rotations SO(3) and the U(1)
spin a given number of flux quanta N, the phase ϑ makes N
gauge symmetries are spontaneously broken, but a diag-
windingsatinfinity,seeFig. 2. Thefirsthomotopygroup
onal subgroup U(1) survives. The subgroup consists of
of this part is therefore fixed: π1(SO(2))=Z. If, in addi-
combined transformations: rotations by angle ϑ around
the axis ~l ~n m~ accompanied by gauge transforma- tion, ~l is constant, there is no way to avoid singularity
≡ × in the phase ϑ where ψ~ =0. However, general require-
tions eiϑ. Each vacuum state is specified by orientation | |
of a triad of orthonormal vectors ~n, m~, ~l. The vacuum ment that a solution has finite energy is much weaker.
It tells us that the direction of ~l should be fixed only
manifold is therefore isomorphic to SO(3). Topological
at infinity [11]. The relevant homotopy group is non-
defects might be of two kinds: regular and ”singular”.
To find regular topological line defects, it is enough to trivial: π2(S2)=Z. The second homotopy group appears
becausefixing~latinfinity(say,up)effectively”compact-
consider the London approximation [10], i.e. to assume
thattheorderparametergraduallychangesinspacefrom ifies” two dimensional physical space into S2. Unit vec-
one vacuum to another. The triad ~n, m~, ~l then be- tor~l winds towards the center of the texture. The new
topological number is Q = (1/8π) ε ~l ∂~l ∂ ~l d2r.
comes a field. The Abrikosovvortex is a singular defect: ij i j
it has a core where the modulus of the order param- Therefore, all configurations fall iRnto cla(cid:16)sses×char(cid:17)acter-
eter vanishes and energy diverges logarithmically. Ac- ized by the two integers N and Q. For regular so-
cordingly, a cutoff parameter – the correlation length – lutions, however, these two numbers are not indepen-
should be introduced and one obtains logκ dependence dent. Uponintegratingthesupercurrentequation,eq.(8)
for the vortex line energy. This fact alone means that along a remote contour and making use of the identity
if there exists a regular solution it is bound to become εpqslp(∂ilq)(∂jls)=(∂inp)(∂jmp) (∂imp)(∂jnp),weob-
−
energeticallyfavorableforlargeenoughκ. Belowwecon- tain: Q=N/2. We call these regular solutions magnetic
sider a situation when the external magnetic field is ori- skyrmions.
ented along z axis and all configurationsare translation- The lowestenergysolutionwithin the Londonapprox-
ally invariant in this direction. We use dimensionless imation corresponds to N/2 = Q = 1 (or N/2 = Q =
−
units: r λr˜, A (Φ0/2πλ)A˜, B Φ0/2πλ2 B˜ and +1). We analyzeacylindricallysymmetricsituationand
F = (ε0/≡2πλ2)F˜,≡where λ (Φ0/2≡π)(cid:0) β1/4παK(cid:1) 1 (the choose the triad~n, m~,~l in the form:
≡
tilde will be dropped hereafter). Thepfree energy takes
~l=~e cosΘ(ρ)+~e sinΘ(ρ),
form z ρ
~n=(~e sinΘ(ρ) ~e cosΘ(ρ))sinϕ+~e cosϕ, (10)
z ρ ϕ
2 −
F =1/2 ∂ ~l +(~n∂ m~ A )2+B2 (7) m~ =(~e sinΘ(ρ) ~e cosΘ(ρ)) cosϕ ~e sinϕ,
L (cid:16) k (cid:17) k − k k z − ρ − ϕ
and the field equations are whereρandϕarepolarcoordinatesandΘ=~e ~l. Bound-
z
ary conditions are: Θ(0)=π andΘ( )=0. The vector
c
np∇~mp−A~ =∇~ ×(cid:16)∇~ ×A~(cid:17)=~j, (8) potential is given by A~ = A(ρ)~eϕ. T∞he general form of
2
′
such a configuration is shown in Fig.2. The unit vec- ′′ Θ sinΘ 2+cosΘ
Θ + + +2A =0, (12)
tor ~l (solid arrows) flips its direction from up to down ρ ρ (cid:18) ρ (cid:19)
as it moves from infinity toward the origin. The phase
which follow from the cylindrically symmetric Ansatz of
ϑ (arrow inside small circles in Fig.2) winds twice while
eq.(10). Calculations for R from R = 5 till R = 600
completing an ”infinitely remote” circle. If in eq. (7)
were done by means of a finite element method. The
only the first term were present we would deal with a
energy per unit cell in a wide range of R is satisfactory
standard SO(3) invariant nonlinear σ-model [12]. Being
described (deviation at R = 10 is 1%) by the function
scaleinvariant,itpossessesinfinitelymanypureskyrmion
ε 2+5.62/R. Note that in the limit R we re-
solutions Θ (ρ;δ) = 2arctan(δ/ρ), which have the same cell ≃ →∞
s coverourpreviousvariationalestimate: ε ε =2.
cell ms
energy equal to 2 (in units of ε0) for any size δ of a Thedominantcontributiontomagneticskyrm→ionenergy
skyrmion. However, in the present case the structure of
at large R comes from the first term ε , similar to the
the order parameter is more complex and the above de- s
analytical variational state described above. The contri-
generacy is lifted by the second and third terms of eq.
butiontoε frommagneticfield,ε ,issmallforlarge
(7). cell mag
R but becomes significant in denser lattices. The most
Below we make use of the functions Θ (ρ;δ) to explic-
s interestingfeatureofthesolutionisthatthesupercurrent
itly construct the variational configurations. We show
contribution ε to the energy of magnetic skyrmion is
that as size of these configurations increases the en- cur
negligibly small for all considered values of R. This is
ergy is reduced to a value arbitrarily close to the ab-
to be compared with the usual Abrikosov vortex where
solute minimum of ε = 2. Substituting eq.(10) into
ms athighκ the totalenergyis dominatedby magneticand
eq.(7) and integrating over the x y plane we obtain
− supercurrent contributions which are of the same order
the energy of the magnetic skyrmion in the form: ε =
ms of magnitude. Most of the flux goes through the region
εs+εcur+εmag,where εs ≡ ρdρ Θ′2/2+sin2Θ/2ρ2 , where the vector~l is oriented upwards. In other words,
ε ρdρ[(1+cosΘ)R/ρ+(cid:0)A]2 and ε (cid:1)
cur mag the magnetic field is concentrated close to the center of
ρdρ(A≡/ρ+RA′)2. The first term εs is the same as i≡n a magnetic skyrmion.
nRonlinear σ-model without magnetic field. It is bound Line energy of Abrikosov vortices ε for the present
v
from below by 2, the energy of a pure skyrmion. The model was calculated numerically (beyond London ap-
secondtermεcur,the”supercurrent”contribution,ispos- proximation) in [13]. For κ = 20 and 50 we obtain
itive definite. One still can maintain zero value of this 2ε /ε 3.5 and 4.4 respectively. Therefore we expect
v ms
≈
term when the field Θ(ρ) is a pure skyrmion Θs(ρ;δ) thatthelowercriticalfieldofUPt3isdeterminedbymag-
of certain size δ. Assuming this one gets: A(ρ) = netic skyrmions: hc1 = εms/2N. Returning to physical
(1+cosΘ)/ρ = 2ρ/(ρ2 +δ2). The third term, the units,
− −
magnetic field contribution (which is also positive def-
inite), becomes εmag = 8/3δ2. It is clear that when Hc1 =Φ0/4πλ2. (13)
δ we obtain energy arbitrarily close to the lower
bo→und∞: ε 2+8/3δ2 2. Single magnetic skyrmion To find magnetization, we now utilize eq.(1). Interac-
ms ≤ → tions among magnetic skyrmions follows easily from the
therefore blows up.
energy of a unit cell of the hexagonal lattice: V(r) =
If many magnetic skyrmions are present, then their
2(ε 2)/6 1.87/r. The resulting averagedmagnetic
cell
interactions can stabilize the system. They repel each inducti−on, in u≃nits of Φ0/2πλ2, reads
other,aswewillseeshortly,andthereforeformalattice.
Since they are axially symmetric, the interaction is axi- B 0.25(H/Hc1 1)2. (14)
≃ −
ally symmetric and thus a triangular lattice is expected.
This agrees very well with the experimental results, see
Assumethatthelatticespacingisa. Attheboundariesof
the hexagonalunit cells the angle Θ is zero,while at the Fig.1. Forfields higher thenseveralHc1 Londonapprox-
imation is not valid anymore since magnetic skyrmions
centersitisπ. ThemagneticfieldB iscontinuousonthe
will start to overlap. In this case one expects that ordi-
boundaries. Therefore, to analyze a magnetic skyrmion
naryAbrikosovvortices,whichcarryoneunitofmagnetic
lattice we should solve eqs.(8)–(9) on the unit cell with
flux, become energetically favorable. The usual vortex
these boundary conditions demanding that two units of
picture has indeed been observedathighfields by Yaron
fluxpassthroughthecell(byadjustingthevalueofmag-
etal.[14]Curiously,ourresultissimilartoconclusionsof
netic field on the boundary). We have approximatedthe
hexagonal unit cell by a circle of radius R = 33/4a/√π Burlachkovet al. [15] who investigated stripe-like (quasi
one dimensional) spin textures in triplet superconduc-
and the same area, and performed numerical integration
tors. Having establishedthe magnetic skyrmionsolution
of the equations
of F we next estimated how it is influenced by var-
sym
A′ A 1+cosΘ ious terms of ∆F eqs.(5)–(6). It was found that these
′′
A + A =0, (11)
ρ − ρ2 − − ρ perturbations do not lead to destabilization of a mag-
netic skyrmion.
3
In conclusion, we have performed a topological classi- [8] T.A.Tokuyasu, D.W.Hess, J.A.Sauls, Phys. Rev. B41,
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spin
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try breaking terms, describes heavy fermion supercon- JETP73(1),170(1991);K.Machida,T.Fujita,T.Ohmi,
J. Phys.Soc. Jpn.62, 680 (1993).
ductorUPt3 andpossiblyotherp-wavesuperconductors. [9] K.Machida,T.Ohmi,J.Phys.Soc.Jpn.67,1122(1998).
A new class of topological solutions in weak magnetic
[10] M.M.Salomaa, G.E.Volovik, Rev. Mod. Phys. 59, 533
fieldwasidentified. Thesesolutions,magneticskyrmions,
(1987).
do not have normal core. At small magnetic fields the [11] This follows from the presence of (∂i~l)2 term in Fsym of
magnetic skyrmions are lighter then Abrikosov vortices
eq. (7), which cannot be ”gauged away” similar to the
andthereforedominatethephysics. Magneticskyrmions
corresponding term for theSO(2) part.
repel each other as 1/r at distances much larger then [12] R.Rajaraman, Solitons and instantons (North-Holland,
magnetic penetration depth forming a relatively robust 1982).
triangular lattice. Hc1 is reduced by a factor logκ as [13] A.Knigavko,B.Rosenstein,Phys.Rev.B58,9354(1998).
compared to that determined by usual Abrikosov vortex [14] U.Yaron et al.,Phys. Rev.Lett. 78, 3185 (1998).
(see eq.(13)). [15] L.I.Burlachkov, N.B.Kopnin, Sov. Phys. JETP 65(3),
Thefollowingcharacteristicfeatures,inadditiontothe 630 (1987).
[16] M.E.Zhitomirsky,K.Ueda,Phys.Rev.B53,6591(1996).
slopeofthemagnetizationcurve,canallowexperimental
identification of a magnetic skyrmions lattice.
1. Unit of flux quantization is 2Φ0.
2. Superfluid density ψ~ 2 is almost constant through-
| | -4 M/HHHHHHH
outthemixedstate. ThiscanbetestedusingSTMtech- C 1
niques.
3. Duetothefactthatthereisnonormalcore,inwhich
usually dissipation and pinning take place, one expects
that pinning effects are reduced.
It is interesting to note that our results are actually
applicable to another model of UPt3 with accidentally
degenerateAErepresentations[16]. Althoughthismodel
H / H
adopts the strong spin–orbit coupling scheme, it has a C 1
structure closely related to F of eqs.(3)–(4) at low
sym
temperatures where both order parameters become of
FIG.1. Magnetization of magnetic skyrmion lattice (solid
equal importance and can be viewed as a single three
line)andexperimentaldataforUPt3fromRef.5(squares)and
dimensional order parameter. Ref.6 (triangules). Magnetization of Abrikosovvortex lattice
The authors are grateful to B. Maple for discussion of (dotted line) is given for comparison.
results of Ref. 5, to L. Bulaevskii, T.K.Lee and J. Sauls
for discussions and to A. Balatsky for hospitality in Los
Alamos. The work is supported by NSC, Republic of
China, through contract #NSC86-2112-M009-034T.
L
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[4] K.A.Park, R.Joynt,Phys. Rev.Lett. 74, 4734 (1995). rotates twice as an infitely remote contour is circumvented.
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4
