Topological defects and Goldstone excitations in domain walls between ferromagnetic quantum Hall effect liquids

4 Pages·0.23 MB·English

by  Vladimir I. Fal'ko

Topological defects and Goldstone excitations in domain walls between ferromagnetic
quantum Hall effect liquids
Vladimir I. Fal’ko ∗†, S.V. Iordanskii ‡∗
∗ School of Physics and Chemistry, Lancaster University, LA1 4YB, UK
† Departement de Physique, Universit´e Joseph Fourier Grenoble I, Grenoble, France
9 ‡ Landau Institute for Theoretical Physics, ul. Kosygina 1, Moscow, Russia
9 (today)
9
1 Itisshownthatthelow-energyspectrumofaferromagneticquantumHalleffectliquidinasystem
withamulti-domainstructuregeneratedbyaninhomogeneousbareZeemansplittingε isformedby
n Z
excitationslocalized atthewallsbetweendomains. Forastep-likeε (r),thedomainwallspectrum
a Z
J includes a spin-wave with a linear dispersion and a small gap due to spin-orbit coupling, and a
low-energy topological defects. The latter are charged and may dominate in the transport under
7
conditions that thepercolation through thenetwork of domain walls is provided.

]
l 73.40.Hm, 64.60.Cn, 75.10.-b
l
a
h Duetotheelectron-electronexchangeinteraction,two- skyrmionic-type textures onto the latter case. For the
- dimensional (2D) electrons form a ferromagnetic liquid sake of simplicity, we consider a strained quantum well
s
e [1] in the vicinity of odd-integer filling factors. Recently, [7] in which εZ(r) =0, but only as an averageover the
h i
m an idea has been proposed [2–4] that the dissipation 2Dplane,whereaslocallyitfluctuatesbetweensmallpos-
. and thermodynamics of the liquid at the filling factor itive and negative values, ε2(r) =ε2. Mesoscopic-size
t Z Z
a ν = 1 may be dominated by a topological spin-texture regions of a negative and positive ε generate a multi-
Z
m (skyrmion) which provides a non-trivial degree of map- domainstructure of the fe(cid:10)rromag(cid:11)neticliquid. A soft dis-
- ping of a 2D plane onto a sphere representing the local order in the sign of the Zeeman energy is different from
d
directionofspin-polarization n,n=1. Asaparticle-like thetraditionallystudiedpotentialdisorder,sincethefor-
n −→
excitation,theskyrmioncarrieselectricalchargeequalto mercannotbeelectrostaticallyscreened. Inagivensam-
o
c its topologicalcharge,and has a reduced contribution of ple, the lines where εZ(r) = 0 define a contour of walls
[ the exchange to its energy [3], as compared to the dis- betweenoppositelypolarizeddomains-theirspatialcon-

sociation energy of a spin-exciton into an electron-hole figurationis determinedby a distribution ofstressesand
1
v pair[5]. Ithasbeensuggested[3,6]thattheconductivity bytheinterfaceroughnessinagivenquantumwelldevice
3 σ at exactly ν = 1 may be provided by thermal ac- [7,8], or by inhomogeneity of the nuclear spin polariza-
xx
5 tivation of skyrmion/anti-skyrmion pairs. However, the tion [9]. Below, we analyze the spin-texture of a domain
0
excessZeemanenergyofaskyrmionmaycompensatethe wall between two oppositely polarized ν = 1 ferromag-
1
gainintheexchangeenergy,duetoalargenumberofim- neticliquids(DW),itsexcitationsandapossibleeffectof
0
9 properlypolarizedelectronsinit. Tostabilizeskyrmions, topologicaldefectsintheDWonthetransportproperties
9 one should reduce the bare value of the single-electron of the 2D system.
/ Zeemansplitting, ε inthe materialaccommodatingthe Several features of a domain wall between quantum
t Z
a
2D electron gas. In GaAs/AlGaAS heterostructures and Hall effect ferromagnets can be assessed qualitatively.
m
quantumwells,onemayachievethisgoaleitherbyusing Due to the stiffness of a polarization field, it can be re-
- the hydrostatic pressure [7,8] which induces strains and garded as a smooth variation of n(r) created by con-
d −→
n progressively changes the conduction band Land´e factor ditions imposed by the crystal: from nz(x > 0) = 1 to
o from negative to positive values, or by optically polariz- n (x < 0) = 1. The long-range character of the vari-
z
c ing nuclear spins [9] of Ga and As in the vicinity of the ation of the o−rder parameter n(r) across the DW, at
−→
:
v heterostructure, which affects εZ via the hyperfine in- the length scale Lw λ = ¯hc/eB, is guaranteed by
≫
i teraction [10]. Both of these two methods have recently the smallness of the Zeeman energy as compared to the
X p
allowedonetoobtainthe2Delectrongaswithanalmost electron-electron exchange, ε . A spin-texture in-
Z 0
r ≪ℑ
a zero value of the single-electron Zeeman splitting [7–9], sidethedomainwallcanbedescribedasamappingofthe
althoughitisnotcleartowhichextendtheinducedvari- 2d plane on to the unit sphere, such that n(r) retraces
−→
ation of ε is spatially homogeneous over the plane of a path from the ’south’ to the ’north’ pole when a path
Z
the 2D gas. in the real space traverses the border between domains.
In this paper, we study the effect of an inhomogene- In the case of a flat border between ε > 0 and ε < 0
Z Z
ity of the single-electron spin splitting onto the ferro- regions,thegeodesicsonaspherenaturallyminimizethe
magnetic quantum Hall effect liquid at ν = 1 under the DW exchange energy.
extreme condition that the externally modified Zeeman Without spin-orbit coupling, the DW energy would
energy approaches zero value and extend an analysis of be degenerate with respect to the choice of a geodesic,
1
i.e., with respect to the angle ϕ between the pla- formation, and derived the 2D σ-model for the polar-
nar component of the polarization vector n = ization n = (sinνcosϕ,sinνsinϕ,cosν) using the gra-
−→ −→
(sinνcosϕ,sinνsinϕ,cosν) and the locally determined dient expansion [17]. Below, we parametrize the po-
normal direction to the line where n = cosν = 0. This larization by using Euler angles: ν, ν < π, with re-
z
| |
degeneracy results in a soft collective mode correspond- spect to the normal to the plane and ϕ with respect
ing to the rotationofthe spincoordinatesystemrelative to the direction across the domain wall. The mean
to the orbital one. Symmetry transformations respon- field n(r,t) enters into a unitary transformation, U =
−→
sible for this residual degeneracy belong to the group (1+n )1/2+i(1+n )−1/2(n σ n σ ) /√2, which
z z y x x y
SO2 of rotations of a spin-texture around the axis −→l z hlocally determines the spinor part−of the eilectron wave
- the magnetic field orientation [11,12]. The spin-orbit
function. The gradient expansion with respect to the
coupling of a form proposed in [13] lifts the degeneracy
matrices U( iλ )U†, U( i∂ )U† was based on the as-
of the DW ground state and orients the field n in the − ∇ − t
−→ sumption that the chemical potential in the system lies
middle of the DW perpendicular to the border between
in the gap, and we took into account the Landau level
domains. As a one-dimensional object, the DW has an
mixing both by interactions and the spin-orbit coupling.
actionthatresemblestheactionofaclassicalsine-Gordon
That brings us to the action in the form
model [14,15]. Its low-lying excited states have the form
of(a)spin-wavesand(b)oftopologicallystabledefects - [F +(F +F )+F +F ]dr/(2π) , (2)
t ∇ C Z so top
kinks in the angle ϕ. −F
Z
The ground state of a single straight domain wall is which is composed of the gradient terms (F + F ),
∇ C
characterized by one properly chosen geodesic line on a the Zeeman energy F , the spin-orbit coupling term
Z
stipohnesr,eϕ|−→n|1=, o1f.aSgpeiond-ewsaicv.esAcokrinrekspcoanndbteodsemscarilblerdotaas- Fwshoe,rethµe itsoptohleogsiicnagllet-eerlemctrFotnopch=em(icµal−pℑot1e/n2t)ial,draδnρd,
a ∆ϕ=2≪π rotation of the polarizationvector −→n around Ft = (i/2λ2)cosν∂tϕ. The latter term should bRe kept
the axis −→l z. Topological classes of kinks are given by tocal−culatethespin-wavespectrumoftheDW[18]. The
the homotopic group π1(SO2) = Z, or, equivalently, by polarizationfieldstiffnessandtheZeemanenergyarede-
the degree of mapping of a line representing the middle scribed by
of the DW, n = 0 onto the equator of a unit sphere.
z
As a 2D object, a kink can be viewed as a mapping of a F∇ = 1 ( ni)2/8= 1 sin2ν( ϕ)2+( ν)2 /8
ℑ ∇ ℑ ∇ ∇
planeintoasphere,R2 →S2,suchthatasetofgeodesics Xi (cid:16) (cid:17)
collected upon moving along the domain wall covers the F = ε /2λ2 sign(x)n = ε /2λ2 sign(x)cosν. (3)
Z Z z Z
entire sphere n = 1. Since the excess density of elec-
|−→| The Ze(cid:0)emanen(cid:1)ergy in Eq. (3(cid:0)) corresp(cid:1)ondsto the model
trons, δρ carried by a polarization texture is equal to
of a step-like ε (r) = ε sign(x). The exchange interac-
[2,3] Z Z
tion of electrons at the n-th Landau level with electrons
from a completely filled Landau level n=0 is given by
δρ=[(∂ cosν)(∂ ϕ) (∂ cosν)(∂ ϕ)]/4π, (1)
y x − x y = ∞V(R)e−R2/2L (R2/2)RdR, where R = r/λ.
ℑn 0 n
the electrical charge of a kink is equal to the degree of For an unscreened Coulomb interaction, V(r) = e2/rχ,
R
mapping R2 S2 provided by the defect, which coin- 1 = 0/2,and 0 = π/2e2/λχ. The’Coulombterm’
→ ℑ ℑ ℑ
cides with its topological charge classified by π1(SO2). FC in Eq. (2) is the repsult of the higher order expansion
That makes kinks akin skyrmions, though the latter are in gradients [3]. It is mentioned only to be neglected
classified using the homotopic group π2(SU2/U1) = Z, later,sinceF∇ isenoughtoprovideanultravioletcut-off
and these two solitonic excitations are related to differ- in the DW defect energy calculation.
ent boundary conditions at the infinities of a 2D plane. In the main approximation, that is, before the spin-
Stability ofakinkwithrespecttothe decayofitscharge orbitcoupling is takeninto account,the domainwallen-
density into the bulk is provided by its relatively small ergy is minimized by any texture with ϕ = const and
energy, as comparedto the bulk skyrmion, which is sup- with ν(x) obeying the saddle-point equation,
ported by the quantitative analysis. The kinks we dis- λ2∂2ν =(4ε / )sign(x)sinν. (4)
cuss in this paper arealsosimilar to skyrmionic textures x Z ℑ0
on the edge of the quantum Hall effect liquid discussed Eq. (4) is the result of the variational principle applied
in [12]. to the energy F∇+FZ. Its solution should be antisym-
The quantitative analysisofthe DW energeticsin this metric, so that in the half-plane x > 0 it satisfies the
paperisbasedontheσ-modelapproach[14]. Westarted boundary conditions cosν(0)=0 and cosν( )=1. The
∞
from the Grassman functional integral for interacting optimal ν0(x) can be found in the form of
electrons where the kinetic energy,Zeeman splitting and
cosν = 1 2cosh−2 w +ln(√2+1) sign(w),
spin-orbitcouplingtermswerepresent. Then,wesplitin- 0 − | |
teractions by means of the Hubbard-Stratonovichtrans- h (cid:16) (cid:17)i
(5)
2
where w =(2x/λ) ε / , which confirms that the do- =α ( /ε )1/4(ε /¯hω )1/2. (7)
Z 0 a 0 0 Z so c
ℑ E ℑ ℑ
main wall width, L = 2λ /ε , is large, L λ.
pw ℑ0 Z w ≫ Since the pair of defects with an opposite sign has total
The result of Eq.(5) allows us to calculate the ground
stateDWenergypermagnepticlength: E0 =√ε (1 topologicalnumberN =0,theterm top playsnorolein
w Zℑ0 − determining . Note that the activFation energy in Eq.
1/2)/π [19]. a
E
(7)issmallerthantheactivationenergyofaskyrmionin
Atthisscaleofenergies,thestructureofthewallisde-
pgenerate with respect to the choice of a geodesic on the the2Dbulk, a 0 (sinceweassumethat(εso/ωc)2 <
E ≪ℑ
unit sphere alongwhich the polarization−→n rotates,that εZ/ℑ0). This prevents the kink from decaying into the
is,withrespecttotheangleϕ. Atafinerscaleofenergies, bulk excitations.
such a degeneracy is lifted by the spin-orbit coupling. It Thespectrumofspin-wavespropagatingalongtheDW
is natural to assume that h¯ω = e2 π ε , can be found by expanding the Lagrangian in the vicin-
c ≫ ℑ0 χλ 2 ≫ Z ity of the homogeneous ground state, ν = ν (w) and
which confines the ground state electrons to the lowest 0
p ϕ = 0 - up to the second order in small variations,
Landau level. Since the spin-orbit coupling [13,20], such
ϕ=eiqy/λ−iωt/h¯ϕ(w) andν =ν (w)+eiqy/λ−iωt/h¯δν(w)
asv [p σ],isnotdiagonalintheLandaulevelbasis, 0
so −→ −→
× (thecoordinateacrossthewallisnormalizedbyitswidth
it appears only in the form of ε /¯hω , ε = h¯v /λ,
so 0 c so so
ℑ L ,w =(2x/λ) ε / ). Toselectexcitationslocalized
which approves its perturbative treatment. To be more w Z 0
ℑ
near the boundary between domains, one should study
specific, we shall limit the spin-orbit coupling energy by
p
the constraint (ε /¯hω )2 < ε / , which enables us to only those solutions for which δν( )=0. In contrast,
so c Z 0 ±∞
ℑ the boundary conditions for ϕ are free, since the fluctu-
exclude a spin-wave instability of the ground state of a
ations of ϕ have no sense in the regions where sinν =0.
homogeneousliquidin the bulk [17]. The spin-orbitcou-
Therefore, it is easier to formulate the eigenvalue prob-
pling term in Eq. (2) can be found in the form of
lemforadifferentfunction,φ(w)=sinν (w)ϕ(w),which
0
F =(ε /¯hω λ)n (r)∂ n (r) (6) has more suitable boundary conditions φ( )=0.
so so 1 c z i i
ℑ ±∞
ε sin2ν Afterapplyingthevariationalprincipletotheresultof
so 1
= ℑ cosϕ∂ +sinϕ∂ ν .
x y suchanexpansion,andin the limit ofq ε / (the
2h¯ω λ{ } − 2 Z 0
c (cid:18) (cid:19) wavelength along the wall is much long≪er than tℑhe DW
p
which has a minimum at ϕ = 0 for the ground state of width), we arrive at the eigenvalue equations,
the DW with ν(r) = ν (x) and determines the energies
0
of the low-lying excitations of this object. iωφ ε ∂2 +sign(w)cosν δν, (8)
≈ Z − w 0
First of all, we analyze the energetics of a topologi- δν ε q2 2ε ε
Z(cid:2) so Z(cid:3)
cal defect. A kink corresponds to the texture with the iω Πφ+ [∂wν0] φ. (9)
− ≈ 4 − ¯hω
planarcomponentofthepolarization(followedalongthe ℑ0 ℑ0 (cid:20) crℑ0 (cid:21)
middle of the DW) making a full circle when the coor- The operator on the right hand side of equation Eq. (8)
dinate changes from y = to y = . If the DW has a spectrum of eigenvalues with gap u 1. Eigen-
texture varies along the w−al∞l much slow∞er than across values of the operator Π in Eq. (9), 0 ∼
it - an assumption which can be verified after the solu-
tion is found, the problem may be formulated as a one- Π= ∂2 + ∂2 sinν /sinν ,
− w w 0 0
dimensional one. The free energy of the wall (regarded
asa1Dsystem)relatedto theliftedresidualSO2 degen- start from zero - for the f(cid:0)unction φ(cid:1)0 =Const×sinν0(w)
eracycanbeobtainedfromEqs. (3,6)byintegratingout - and are of the order of unity for other ’excited states’
the transverse form-factor of the DW, cosν0(x): φn [17]. Thelowesteigenvaluecorrespondstotheexcita-
tion with ϕ constant across the wall, but varying along
dy ε it. This is the Goldstone mode. Other excited states
[ϕ(y)]= ℑ0 (αλ∂ ϕ)2+ so (1 cosϕ)
E ℑ0Z 8λ"rεZ y ¯hωc − # are spin-waves in the bulk of the domain affected by the
presence of a wall. Their spectrum starts above a gap of
the order of Zeeman energy, ε . If we treat Eqs. (8,9)
where α = (2 √2/2)/3π 0.37. The energy mini- Z
− ≈ perturbatively andapproximatethe lowesteigenstateby
mum of a kiqnk is provided by the texture with
φ , taking into account the expression on its left-hand
0
cosϕ(y)=1− cosh22(u) , u= y εso εZ 1/2. seixdceitiantitohnehdaiasgtohneadliasppperrsoixoinmation,we find that the soft
sinϕ(y)= 2Θsinh(u) αλ ¯hω
( cosh2(u) (cid:18) crℑ0(cid:19)
Θ = 1 is the sign of a topological charge of the de- ¯hω =(πεZ 0/A)1/2 (εso/¯hωc) εZ/ 0+(αq)2, (10)
± ℑ ℑ
fect. The charge density distribution in a single kink q
p
2
calculated using Eqs. (1) is shown in Fig. 1. The acti- where A = dwf (w)sinν (w) /u 1 is deter-
k k 0 k ∼
vation energy of the kink/anti-kink pair (two times the mined by the normalized eigenstates, f (w) and eigen-
spin-deformation energy of one kink) is equal to values, u ofPan(cid:12)(cid:12)Requation ∂2 + co(cid:12)(cid:12)sνk u f (w) =
k − w | 0|− k k
(cid:2) (cid:3)
3
0. So, the spectrum of a soft mode has the [1] S.Girvin andA.MacDonald, in Perspectives in Quantum
gap (ε /¯hω )1/2(ε / )3/4 and a linear dispersion Hall Effect, ed. by S. Das Sarma and A.Pinczuk (Wiley,
0 so c Z 0
q√εℑ at (ε /¯hω )1/2ℑ(ε / )1/4 <q < ε / . NY 1997); and refs. therein
Zℑ0 so c Z ℑ0 Z ℑ0 [2] D.-H.Lee, C.Kane, Phys.Rev. Lett. 64, 1313 (1990)
Note that, as a one-dimensional object, DW between
p [3] S.L.Sondhi et al, Phys.Rev.B 47, 16419 (1993)
oppositely polarized QHE liquids resembles a 1D anti-
[4] H.A.Fertig et al, ibid. B50, 11018 (1994); B55, 10671
ferromagnet. This analogy can be traced throughout
(1997)
the properties of kinks and linear-dispersion spin-waves, [5] Yu.Bychkov, S.Iordanskii, G.Eliashberg, Pis’ma ZhETF
and also can be extended onto identification of regimes 33, 152 (1981) (JETP Lett. 33, 143 (1981)); C.Kallin
when classically obtained solutions are sound. Classi- B.Halperin, Phys. Rev.B 30, 5655 (1984)
cal treatmentof spin-wavesis providedby their stability [6] A.Schmeller et al,Phys. Rev.Lett. 75, 4290 (1995)
against decay into kink-antikink pairs, since ¯hω. [7] D.Maude et al,Phys. Rev.Lett. 77, 4608 (1996)
a
The latter may be compared to the knownEcon≫dition [8] D.Leadley et al, ibid. 79, 4246 (1997)
[9] I.Kukushkin,K.v.Klitzing, K.Eberl, preprint
for the border between the classical solution for the 1D
[10] M.Dobers et al, Phys. Rev.Lett. 61, 1650 (1988)
Sin-Gordon model and the Fermi-gas of kinks [14,21]:
[11] R.Cˆot´e et al, ibid.78, 4825 (1997)
β2/8π = (πα2√A)−1(ε / ) < 1. It has been also
Z ℑ0 [12] A.Karlhede et al, Phys. Rev.Lett. 77, 2061 (1996)
shown previously [22] that the topological defect energy
[13] Yu.Bychkov,E.Rashba, JETP Lett. 39, 78 (1984)
inanantiferromagnetis loweredby quantumcorrections
[14] E.Fradkin,FieldTheories ofCondensed Matter Systems
arising from scattering of spin-waves on it. One should, (Addison-Wesley PC, 1991)
therefore,analyzesimilarcorrectionstotheDW-kinken- [15] A.Tsvelik, Quantum Field Theory in Condensed Matter
ergy, a. However, our estimation [17] indicates that Physics (Cambridge UniversityPress 1995)
E
those corrections do not alterate the relation ¯hω. [16] An isolated domain surrounded by a curved wall may
a
E ≫
Aswehaveshownabove,thedomainwall(DW)hasan correspond to the same topological class as a skyrmion,
excitationspectrumwithenergieslowerthantheenergies sothatitmaybecharged,butweassumethatthelinear
lengthscale,Lofthedomainstructureislargeenoughto
of excitations in the bulk. Therefore, the low-gap spin-
studytheDWlocally,asa1Dobject.Thelatterrequires
waves in it may be responsible for the low-frequency ab-
sorptionofasystemintheregimewhenthemulti-domain L>Lk =αλ(h¯ωc/εso)1/2(ℑ0/εZ)1/4.The2relationLk >
Lw is providedby theinequality (εso/¯hωc) <εZ/ 0.
structure is formed. Moreover, the fact that topological ℑ
[17] V.Fal’ko and S.Iordanskii, in preparation; analysis of a
defect in the DW is charged and has a low activation
model of smoothly varying ε will be reported in it.
Z
energy (since they are formed on the top of a texture [18] The higher-order terms in ∂ donot affect dispersion.
t
imposed by the external conditions), may lead to the [19] The saddle point equation has also solution cosν1 =
dominatingroleofkink/anti-kinkpairsinthedissipative 1 2cosh−2 2(x/λ) εZ/ 0 ln(√2+1) sign(x),
conductivity,σ whenthenetworkofDW’sformsaper- − ℑ −
colationclusterxtxhroughtheentire2Dplane. SinceDW’s whith Ew1 =√ε(cid:16)Zℑ0(1+p 1/2)/π>Ew0. (cid:17)i
[20] F. Malcher, G. Lommer and U. Rossler, Superlatt. Mi-
separate regions of a positive and negative value of ε ,
Z crostr. 2, 273 (1986); Phpys. Rev.Lett. 60, 729 (1988)
the critical regime is realized when the sample areas un-
[21] S.Coleman, Phys.Rev.D 11, 2088 (1975)
der the ε > 0 and ε < 0 domains are equal [23], that
Z Z [22] J.P.Rodriguez, Phys.Rev.B 39, 2906 (1989)
is, when ε (r) = 0. This scenario suggests that in the
h Z i [23] B.ShklovskiiandA.Efros, ElectronicProperties ofDoped
proximityofexternalconditionsproviding εZ 0,dis- Semiconductors (Springer-Verlag, Berlin 1984)
h i→
sipative transport at ν = 1, σ (T) may be determined
xx
by percolation of thermally activated kinks through an
infinite cluster of DW’s. That may be the reason why,
in the high-pressure experiment [7,8], in the vicinity of r
the pressure value where the conduction band g-factor
in GaAs should nominally change its sign, the ν = 1
activation energy falls sharply below the value expected
for the skyrmion/anti-skyrmion pairs in a homogeneous
ferromagnetic liquid.
u
AuthorsthankA.TsvelikandB.Halperinforilluminat- 4 o w
ing discussions, D.Maude, R.Nicholas and M.Potemski - -4 -2 0 2
for discussions ofexperiments. This workwas supported
FIG.1. 2Dcharge-densitydistributionaroundakink. The
by EPSRC and RFFI (grant 98-02-16245).
scratch shows the middle of the DW, where g(r) = 0. The
coordinatesacrossandalongtheDW,wandu,arenormalized
byL and L [16], respectively.
w k
4