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Disordered quantum Hall ferromagnets and cooperative transport anisotropy
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Physica E 22 (2004) 82 – 85 www.elsevier.com/locate/physe
Disordered quantum Hall ferromagnets and cooperative transport anisotropy J.T. Chalkera , D.G. Polyakovb;∗;1 , F. Eversb , A.D. Mirlinb; c;2 , P. W0ol1ec; b a Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK f ur Nanotechnologie, Forschungszentrum Karlsruhe, Karlsruhe 76021, Germany c Insitut f ur Theorie der Kondensierten Materie, Universit at Karlsruhe, Karlsruhe 76128, Germany b Insitut
Abstract We discuss the behaviour of a quantum Hall system when two Landau levels with opposite spin and combined 3lling factor near unity are brought into energetic coincidence using an in-plane component of magnetic 3eld. We focus on the interpretation of recent experiments under these conditions (Phys. Rev. Lett. 86 (2001) 866; Phys. Rev. B 64 (2001) 121305), in which a large resistance anisotropy develops at low temperatures. Modelling the systems involved as Ising quantum Hall ferromagnets, we suggest that this transport anisotropy re1ects domain formation induced by a random 3eld arising from isotropic sample surface roughness. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.43.Cd; 73.21.−b; 75.10.−b; 75.10.Nr Keywords: Quantum Hall e@ect; Electron–electron interactions; Quantum Hall ferromagnetism; Ising model; Domain walls
Two very striking experimental observations of large electronic transport anisotropy for quantum Hall systems in tilted magnetic 3elds have been reported recently [1,2]. In both cases, anisotropy appears at integer values of the 3lling factor with an in-plane magnetic 3eld component B tuned to bring two Landau levels of opposite spin into energetic coincidence. While the orientation of B itself de3nes an axis within the sample, the fact that large anisotropy appears in resistivity only below a characteristic temperature T ∼ 1 K suggests it has a cooperative origin. ∗ Corresponding author. Tel.: +49-7247-826412; fax: +497247-826368. E-mail address: [email protected] (D.G. Polyakov). 1 Also at A.F.Io@e Physico-Technical Institute, 194021 St.Petersburg, Russia. 2 Also at Petersburg Nuclear Physics Institute, 188350 St.Petersburg, Russia.
In view of the phenomenological similarities, it is natural to compare these Landau levels coincidence experiments and the earlier discovery of resistance anisotropy near half-3lling of high Landau levels [3], attributed to the formation of a uniaxial charge density wave with a period set by the cyclotron radius Rc [4]. Most importantly, the nature of the electron states near the chemical potential and their average occupation is quite di@erent in each case: two separate orbital Landau levels with opposite spin and a combined 1 in the coincidence experiments, as against a single, spin-polarised and roughly half-3lled Landau level in the other case. A uniaxial spin- or charge-density wave at the point of coincidence could straightforwardly explain the transport anisotropy [1,2]. However, recent Hartree–Fock calculations for bilayer systems [5] with parameters relevant in the present context give a fully polarised homogeneous ground state
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.221
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
at coincidence, as do calculations for one-band models, in which Hartree–Fock solutions probe both spinand charge-density modulations [6] or the Hamiltonian with realistic interaction potentials is diagonalised exactly for a small number of electrons [7]. We thus rule out stripe phases. The observations [1,2] therefore present a puzzle, which we argue can be understood in terms of domain walls created by disorder, with a characteristic size much larger than Rc . Exchange interactions lead, at a combined 3lling = 1 for the crossing levels, to a 3rst-order transition between two ground states in which one or other level is completely 3lled. Within a Hartree–Fock treatment, the excitation gap remains non-zero through this transition [8]. Observations of a quantised plateau in Hall resistivity xy and deep minimum in diagonal resistivity xx , both persisting through the transition [9], as well as the variation of the activation gap with B [10], provide support for such a picture. The coincidence transition is one example of a broad class of cooperative phenomena in quantum Hall systems, involving ferromagnetism of either spin or pseudospin variables [11,12]. We thus represent the two Landau levels involved using two states of a pseudospin, with ferromagnetic interaction and Ising anisotropy [13]. The total 3eld Btot measured from its value at coincidence acts on pseudospins as a Zeeman 3eld. Our account involves three distinct ingredients. (i) We suggest that domains are induced by a random Zeeman 3eld acting on the pseudospins, which arises from the interplay between isotropic sample surface roughness and the in-plane 3eld B . (ii) We show that a random 3eld generated by this mechanism is intrinsically endowed with anisotropic correlations, and that the correlation anisotropy is large enough to explain the observed anisotropy in resistivity. (iii) We argue that transport in a multi-domain sample occurs along domains walls, via the processes discussed recently in Refs. [14–16]. The onset temperature for transport anisotropy arising by this mechanism is the Curie temperature of the Ising quantum Hall ferromagnet, and we note that the reported [1,2] onset temperature of about 1 K is similar to the value for the Curie temperature expected [13] and observed elsewhere [17]. Correlations are characterised by the expectation value of pseudospin, S(r) = c † c , where c↑† (r) and c↓† (r) are coherent state creation operators for the
83
two Landau levels involved and is the vector of Pauli matrices. The order parameter has magnitude |S(r)| = S, where S = 1 at a combined 3lling = 1 and is smaller otherwise. Consider an energy functional E{S} for the system. For variations of S(r) which are smooth on the scale of Rc , E reads E = d 2 r(−DSz2 + J |∇S|2 + J |9n S|2 − hSz ): (1) Here D ¿ 0 represents Ising anisotropy, J is the spin sti@ness, the derivative 9n ≡ nˆ · ∇ acts in the direction ˆ , and J represents spatial anisotropy in the of n B spin sti@ness (for simplicity, we omit anisotropy in spin-space from the sti@ness). The e@ective Zeeman 3eld acting on pseudospins is h. For a homogeneous system, the ground state of Eq. (1) is uniform with Sz = sgn(h)S and S⊥ = 0. Domains may arise either in metastable states or because they are induced by quenched disorder. However, metastability and hysteresis [17] are not reported to be an important aspect of observations in Refs. [1,2]. We therefore turn to domains induced by disorder. Potentially the most important source of disorder in Eq. (1) is randomness in h, which leads us to the random 3eld Ising model [18]. One should distinguish the weak and strong disorder regimes: taking h to 1uctuate about mean value zero with amplitude and correlation length l, and supposing l is greater than the domain wall width w = J=D,√the boundary between the two regimes lies at l ∼ JD. At weak disorder, domain size is exponentially larger than l√and domain morphology depends on the di@erence JD in the energy density along domain walls running parallel or perpendicular to B . At strong disorder, the domain pattern is simply that of sgn(h). Let us now turn to a microscopic origin of the random Ising 3eld. One possibility is that variations in carrier density n produce changes in the value of h. Randomness of this kind is expected to be spatially isotropic, but may give rise to transport anisotropy via dependence of the domain wall energy on spatial orientation. A second possibility is that sample surface roughness changes the local value of , and hence h. To compare the likely importance of these two, we appeal to experiment, noting (e.g., from Fig. 2 of Ref. [1]) that while the coincidence transition is sharp in (width ∼ 0:5◦ ), it has a much larger width (20%) in Btot , which is indicative of its width in n.
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
The existence of sample surface roughness with an amplitude of a few nanometers and l ∼ 1 m is reported in Ref. [1] and amplitudes of up to 10 nm are well established in a variety of other contexts [19], giving gradients of at least a few tenths √ of a degree. We estimate that the condition l ∼ JD is met by h ∼ e2 cot = ˝!c , and conclude that the random 3eld originating from surface roughness constitutes intermediate or strong disorder. Moreover, it can account for transport anisotropy, as we now show. Let z(r) denote height of the sample surface above a reference plane, and let c be the critical angle of the coincidence transition. For small-angle roughness h(r) = ( − c ) + 9n z(r);
(2)
where is a proportionality constant. Crucially, by this mechanism surface roughness with a correlator C(r)= z(r )z(r + r ) − z(r )z(r + r ), which is isotropic, generates a random 3eld with spatially anisotropic correlations, since K(r) = h(0)h(r) − h2 reads K(r) = − 2 92n C(r):
2
x10 Dxx(t), Dyy (t)
84
2.5
1
x10 1.5
0.5
0
0
10
1
10
10 0
10
10 1 2
10 2
10
10 3 3
10 4 4
10
t/t o Fig. 1. Simulation data used to determine the degree of anisotropy. Mean square displacements per unit time in directions perpendicular (Dxx (t), full line) and parallel (Dyy (t), dashed line) to B , averaged over all trajectories. Inset: averages over open trajectories only, x2 (t)=t 8=7 and y2 (t)=t 8=7 , demonstrating scaling with the classical percolation exponent value.
E
(3)
To establish the characteristic degree of this anisotropy, we have carried out numerical simulations. We quantify the anisotropy by following the classical dynamics of a particle that moves along contours of h(r) for a Gaussian shape of C(r), using the method of Ref. [20]. Taking nˆ along the y-axis, the quantities Dxx (t) ≡ x2 (t)=t and Dyy (t) ≡ y2 (t)=t should approach the eigenvalues, Dxx and Dyy , of the di@usion tensor, for times t which are large compared to the correlation time, t0 . Evidence that Dxx (t) and Dyy (t) indeed tend to a 3nite limit, with Dxx ∼ 8Dyy , is presented in Fig. 1. The orientation of this anisotropy, with the larger di@usion constant in the direction perpendicular to B , is as observed in Refs. [1,2], and its magnitude is about the same as that determined at low temperature using a Hall bar sample [1]. The foregoing discussion is based on the idea that transport occurs along boundaries between domains. In order to substantiate this, we next examine transport properties of domain walls between oppositely magnetised phases of the Ising quantum Hall ferromagnet. Recalling that the domain wall forms the boundary between a region on one side with 3lling factors for the coincident Landau levels of ↑ 1 and ↓ 0, and a region on the other with interchanged 3lling factors, ↑ 0 and ↓ 1, the simplest structure one
y (a) E
y (b) Fig. 2. Schematic summary of domain wall structure, showing pseudospin and excitation energies E, as a function of position y across the wall, within Hartree–Fock theory: (a) for an Ising wall stabilised by short-range scattering; (b) for a Bloch wall.
might imagine is that shown in Fig. 2(a). In this picture, the wall supports two counter-propagating modes with opposite spin polarisation, which arise as edge
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
states of the occupied Landau levels in the domains on either side. Such an Ising domain wall, in which S⊥ (r) = 0 everywhere and S(r) = 0 at the wall centre, may be stabilised by short-range scattering, which allows solutions with |S(r)| ¡ 1 [21], in contrast to Eq. (1). For a sample without short-range scattering, however, Hartree–Fock theory yields [16] the Bloch wall structure shown in Fig. 2(b). Here, S⊥ (r) = 0 within the wall. In consequence, within Hartree–Fock theory there is mixing and an avoided crossing of edge states arising from occupied Landau levels on either side of the wall. At this level of approximation, for a Bloch wall the chemical potential lies within a quasiparticle gap. To account for transport under these conditions, it is necessary to consider collective excitations. The combined consequences of continuous symmetry for the Hartree–Fock solution under rotations of S⊥ (r) about the Ising axis, and the connection between spin or pseudospin and charge that is standard for quantum Hall ferromagnets [11] have been examined in a related context in Ref. [15]. Introducing pseudospin rotation angle ’, as a function of position coordinate x along the wall and imaginary time $, the action
S= 2
dx
d$
9’ 9x
2
1 + 2 u
9’ 9$
2 (4)
is obtained for domain wall excitations [15], where in our context ∼ Jw ∼ e2 = and u ∼ e2 = ˝. A charge density (2&)−1 9’=9x is associated with these modes. This is the action for a spinless Luttinger liquid. A vital property for our argument is that left and right-moving excitations propagate independently, provided rotation symmetry about the pseudospin easy axis is exact. In short, Fig. 2(a) remains a useful picture even without short-range disorder to stabilise an Ising wall, provided only that there is no spin–orbit scattering. In this picture, transport in a multidomain sample occurs at domain boundaries, via two independent, counter-propagating sets of modes. Neglecting quantum interference e@ects, we arrive at the problem for which numerical results are given above. In conclusion, we have argued that the observations of anisotropic transport reported in Refs. [1,2]
85
can plausibly be attributed to formation of anisotropically shaped domains, induced as a result of sample surface roughness. Our numerical work demonstrates that this mechanism generates an anisotropy comparable to that found experimentally [1]. In addition, the onset temperature for strongly anisotropic transport is comparable to the critical temperature expected [13] and observed [17] in other Ising quantum Hall ferromagnets. For future work it would be interesting to investigate transport in systems with deliberately induced surface features. We thank R. Haug and U. Zeitler for extensive discussions, and E. H. Rezayi for correspondence. The work was supported by the Humboldt Foundation (J.T.C.), by the Schwerpunktprogramm “Quanten-Hall-Systeme” of DFG, and by RFBR. References [1] U. Zeitler, et al., Phys. Rev. Lett. 86 (2001) 866. [2] W. Pan, et al., Phys. Rev. B 64 (2001) 121305. [3] M.P. Lilly, et al., Phys. Rev. Lett. 82 (1999) 394; R.R. Du, et al., Solid State Commun. 109 (1999) 389. [4] M.M. Fogler, A.A. Koulakov, B.I. Shklovskii, Phys. Rev. B 54 (1996) 1853; R. Moessner, J.T. Chalker, Phys. Rev. B 54 (1996) 5006. [5] E. Demler, et al., Solid State Commun. 123 (2002) 243. [6] F. Evers, D.G. Polyakov, unpublished. [7] E.H. Rezayi, et al., Phys. Rev. B 67 (2003) 201305. [8] G.F. Giuliani, J.J. Quinn, Phys. Rev. B 31 (1985) 6228. [9] S. Koch, et al., Phys. Rev. B 47 (1993) 4048. [10] A.J. Daneshvar, et al., Phys. Rev. Lett. 79 (1997) 4449. [11] S.L. Sondhi, et al., Phys. Rev. B 47 (1993) 16419. [12] S.M. Girvin, in: A. Comtet, T. Jolicouer, S. Ouvry (Eds.), Topological Aspects of Low Dimensional Systems, de Editions Physique, Paris, 1999. [13] T. Jungwirth, et al., Phys. Rev. Lett. 81 (1998) 2328; T. Jungwirth, A.H. MacDonald, Phys. Rev. B 63 (2001) 035305. [14] V.I. Falko, S.V. Iordanskii, Phys. Rev. Lett. 82 (1999) 402. [15] A. Mitra, S.M. Girvin, cond-mat/0110078. [16] L. Brey, C. Tejedor, Phys. Rev. B 66 (2002) 041308. [17] V. Piazza, et al., Nature 402 (1999) 638. [18] A.P. Young (Eds.), Spin Glasses and Random Fields, World Scienti3c, Singapore, 1997. [19] C. Orme, et al., Appl. Phys. Lett. 64 (1994) 860; R.L. Willett, et al., Phys. Rev. Lett. 87 (2001) 126803; K.B. Cooper, et al., Solid State Commun. 119 (2001) 89. [20] F. Evers, W. Brenig, Z. Phys. B 94 (1994) 115. [21] S. Yarlagadda, Phys. Rev. B 44 (1991) 13 101.
Physica E 22 (2004) 82 – 85 www.elsevier.com/locate/physe
Disordered quantum Hall ferromagnets and cooperative transport anisotropy J.T. Chalkera , D.G. Polyakovb;∗;1 , F. Eversb , A.D. Mirlinb; c;2 , P. W0ol1ec; b a Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK f ur Nanotechnologie, Forschungszentrum Karlsruhe, Karlsruhe 76021, Germany c Insitut f ur Theorie der Kondensierten Materie, Universit at Karlsruhe, Karlsruhe 76128, Germany b Insitut
Abstract We discuss the behaviour of a quantum Hall system when two Landau levels with opposite spin and combined 3lling factor near unity are brought into energetic coincidence using an in-plane component of magnetic 3eld. We focus on the interpretation of recent experiments under these conditions (Phys. Rev. Lett. 86 (2001) 866; Phys. Rev. B 64 (2001) 121305), in which a large resistance anisotropy develops at low temperatures. Modelling the systems involved as Ising quantum Hall ferromagnets, we suggest that this transport anisotropy re1ects domain formation induced by a random 3eld arising from isotropic sample surface roughness. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.43.Cd; 73.21.−b; 75.10.−b; 75.10.Nr Keywords: Quantum Hall e@ect; Electron–electron interactions; Quantum Hall ferromagnetism; Ising model; Domain walls
Two very striking experimental observations of large electronic transport anisotropy for quantum Hall systems in tilted magnetic 3elds have been reported recently [1,2]. In both cases, anisotropy appears at integer values of the 3lling factor with an in-plane magnetic 3eld component B tuned to bring two Landau levels of opposite spin into energetic coincidence. While the orientation of B itself de3nes an axis within the sample, the fact that large anisotropy appears in resistivity only below a characteristic temperature T ∼ 1 K suggests it has a cooperative origin. ∗ Corresponding author. Tel.: +49-7247-826412; fax: +497247-826368. E-mail address: [email protected] (D.G. Polyakov). 1 Also at A.F.Io@e Physico-Technical Institute, 194021 St.Petersburg, Russia. 2 Also at Petersburg Nuclear Physics Institute, 188350 St.Petersburg, Russia.
In view of the phenomenological similarities, it is natural to compare these Landau levels coincidence experiments and the earlier discovery of resistance anisotropy near half-3lling of high Landau levels [3], attributed to the formation of a uniaxial charge density wave with a period set by the cyclotron radius Rc [4]. Most importantly, the nature of the electron states near the chemical potential and their average occupation is quite di@erent in each case: two separate orbital Landau levels with opposite spin and a combined 1 in the coincidence experiments, as against a single, spin-polarised and roughly half-3lled Landau level in the other case. A uniaxial spin- or charge-density wave at the point of coincidence could straightforwardly explain the transport anisotropy [1,2]. However, recent Hartree–Fock calculations for bilayer systems [5] with parameters relevant in the present context give a fully polarised homogeneous ground state
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.221
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
at coincidence, as do calculations for one-band models, in which Hartree–Fock solutions probe both spinand charge-density modulations [6] or the Hamiltonian with realistic interaction potentials is diagonalised exactly for a small number of electrons [7]. We thus rule out stripe phases. The observations [1,2] therefore present a puzzle, which we argue can be understood in terms of domain walls created by disorder, with a characteristic size much larger than Rc . Exchange interactions lead, at a combined 3lling = 1 for the crossing levels, to a 3rst-order transition between two ground states in which one or other level is completely 3lled. Within a Hartree–Fock treatment, the excitation gap remains non-zero through this transition [8]. Observations of a quantised plateau in Hall resistivity xy and deep minimum in diagonal resistivity xx , both persisting through the transition [9], as well as the variation of the activation gap with B [10], provide support for such a picture. The coincidence transition is one example of a broad class of cooperative phenomena in quantum Hall systems, involving ferromagnetism of either spin or pseudospin variables [11,12]. We thus represent the two Landau levels involved using two states of a pseudospin, with ferromagnetic interaction and Ising anisotropy [13]. The total 3eld Btot measured from its value at coincidence acts on pseudospins as a Zeeman 3eld. Our account involves three distinct ingredients. (i) We suggest that domains are induced by a random Zeeman 3eld acting on the pseudospins, which arises from the interplay between isotropic sample surface roughness and the in-plane 3eld B . (ii) We show that a random 3eld generated by this mechanism is intrinsically endowed with anisotropic correlations, and that the correlation anisotropy is large enough to explain the observed anisotropy in resistivity. (iii) We argue that transport in a multi-domain sample occurs along domains walls, via the processes discussed recently in Refs. [14–16]. The onset temperature for transport anisotropy arising by this mechanism is the Curie temperature of the Ising quantum Hall ferromagnet, and we note that the reported [1,2] onset temperature of about 1 K is similar to the value for the Curie temperature expected [13] and observed elsewhere [17]. Correlations are characterised by the expectation value of pseudospin, S(r) = c † c , where c↑† (r) and c↓† (r) are coherent state creation operators for the
83
two Landau levels involved and is the vector of Pauli matrices. The order parameter has magnitude |S(r)| = S, where S = 1 at a combined 3lling = 1 and is smaller otherwise. Consider an energy functional E{S} for the system. For variations of S(r) which are smooth on the scale of Rc , E reads E = d 2 r(−DSz2 + J |∇S|2 + J |9n S|2 − hSz ): (1) Here D ¿ 0 represents Ising anisotropy, J is the spin sti@ness, the derivative 9n ≡ nˆ · ∇ acts in the direction ˆ , and J represents spatial anisotropy in the of n B spin sti@ness (for simplicity, we omit anisotropy in spin-space from the sti@ness). The e@ective Zeeman 3eld acting on pseudospins is h. For a homogeneous system, the ground state of Eq. (1) is uniform with Sz = sgn(h)S and S⊥ = 0. Domains may arise either in metastable states or because they are induced by quenched disorder. However, metastability and hysteresis [17] are not reported to be an important aspect of observations in Refs. [1,2]. We therefore turn to domains induced by disorder. Potentially the most important source of disorder in Eq. (1) is randomness in h, which leads us to the random 3eld Ising model [18]. One should distinguish the weak and strong disorder regimes: taking h to 1uctuate about mean value zero with amplitude and correlation length l, and supposing l is greater than the domain wall width w = J=D,√the boundary between the two regimes lies at l ∼ JD. At weak disorder, domain size is exponentially larger than l√and domain morphology depends on the di@erence JD in the energy density along domain walls running parallel or perpendicular to B . At strong disorder, the domain pattern is simply that of sgn(h). Let us now turn to a microscopic origin of the random Ising 3eld. One possibility is that variations in carrier density n produce changes in the value of h. Randomness of this kind is expected to be spatially isotropic, but may give rise to transport anisotropy via dependence of the domain wall energy on spatial orientation. A second possibility is that sample surface roughness changes the local value of , and hence h. To compare the likely importance of these two, we appeal to experiment, noting (e.g., from Fig. 2 of Ref. [1]) that while the coincidence transition is sharp in (width ∼ 0:5◦ ), it has a much larger width (20%) in Btot , which is indicative of its width in n.
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
The existence of sample surface roughness with an amplitude of a few nanometers and l ∼ 1 m is reported in Ref. [1] and amplitudes of up to 10 nm are well established in a variety of other contexts [19], giving gradients of at least a few tenths √ of a degree. We estimate that the condition l ∼ JD is met by h ∼ e2 cot = ˝!c , and conclude that the random 3eld originating from surface roughness constitutes intermediate or strong disorder. Moreover, it can account for transport anisotropy, as we now show. Let z(r) denote height of the sample surface above a reference plane, and let c be the critical angle of the coincidence transition. For small-angle roughness h(r) = ( − c ) + 9n z(r);
(2)
where is a proportionality constant. Crucially, by this mechanism surface roughness with a correlator C(r)= z(r )z(r + r ) − z(r )z(r + r ), which is isotropic, generates a random 3eld with spatially anisotropic correlations, since K(r) = h(0)h(r) − h2 reads K(r) = − 2 92n C(r):
2
x10 Dxx(t), Dyy (t)
84
2.5
1
x10 1.5
0.5
0
0
10
1
10
10 0
10
10 1 2
10 2
10
10 3 3
10 4 4
10
t/t o Fig. 1. Simulation data used to determine the degree of anisotropy. Mean square displacements per unit time in directions perpendicular (Dxx (t), full line) and parallel (Dyy (t), dashed line) to B , averaged over all trajectories. Inset: averages over open trajectories only, x2 (t)=t 8=7 and y2 (t)=t 8=7 , demonstrating scaling with the classical percolation exponent value.
E
(3)
To establish the characteristic degree of this anisotropy, we have carried out numerical simulations. We quantify the anisotropy by following the classical dynamics of a particle that moves along contours of h(r) for a Gaussian shape of C(r), using the method of Ref. [20]. Taking nˆ along the y-axis, the quantities Dxx (t) ≡ x2 (t)=t and Dyy (t) ≡ y2 (t)=t should approach the eigenvalues, Dxx and Dyy , of the di@usion tensor, for times t which are large compared to the correlation time, t0 . Evidence that Dxx (t) and Dyy (t) indeed tend to a 3nite limit, with Dxx ∼ 8Dyy , is presented in Fig. 1. The orientation of this anisotropy, with the larger di@usion constant in the direction perpendicular to B , is as observed in Refs. [1,2], and its magnitude is about the same as that determined at low temperature using a Hall bar sample [1]. The foregoing discussion is based on the idea that transport occurs along boundaries between domains. In order to substantiate this, we next examine transport properties of domain walls between oppositely magnetised phases of the Ising quantum Hall ferromagnet. Recalling that the domain wall forms the boundary between a region on one side with 3lling factors for the coincident Landau levels of ↑ 1 and ↓ 0, and a region on the other with interchanged 3lling factors, ↑ 0 and ↓ 1, the simplest structure one
y (a) E
y (b) Fig. 2. Schematic summary of domain wall structure, showing pseudospin and excitation energies E, as a function of position y across the wall, within Hartree–Fock theory: (a) for an Ising wall stabilised by short-range scattering; (b) for a Bloch wall.
might imagine is that shown in Fig. 2(a). In this picture, the wall supports two counter-propagating modes with opposite spin polarisation, which arise as edge
J.T. Chalker et al. / Physica E 22 (2004) 82 – 85
states of the occupied Landau levels in the domains on either side. Such an Ising domain wall, in which S⊥ (r) = 0 everywhere and S(r) = 0 at the wall centre, may be stabilised by short-range scattering, which allows solutions with |S(r)| ¡ 1 [21], in contrast to Eq. (1). For a sample without short-range scattering, however, Hartree–Fock theory yields [16] the Bloch wall structure shown in Fig. 2(b). Here, S⊥ (r) = 0 within the wall. In consequence, within Hartree–Fock theory there is mixing and an avoided crossing of edge states arising from occupied Landau levels on either side of the wall. At this level of approximation, for a Bloch wall the chemical potential lies within a quasiparticle gap. To account for transport under these conditions, it is necessary to consider collective excitations. The combined consequences of continuous symmetry for the Hartree–Fock solution under rotations of S⊥ (r) about the Ising axis, and the connection between spin or pseudospin and charge that is standard for quantum Hall ferromagnets [11] have been examined in a related context in Ref. [15]. Introducing pseudospin rotation angle ’, as a function of position coordinate x along the wall and imaginary time $, the action
S= 2
dx
d$
9’ 9x
2
1 + 2 u
9’ 9$
2 (4)
is obtained for domain wall excitations [15], where in our context ∼ Jw ∼ e2 = and u ∼ e2 = ˝. A charge density (2&)−1 9’=9x is associated with these modes. This is the action for a spinless Luttinger liquid. A vital property for our argument is that left and right-moving excitations propagate independently, provided rotation symmetry about the pseudospin easy axis is exact. In short, Fig. 2(a) remains a useful picture even without short-range disorder to stabilise an Ising wall, provided only that there is no spin–orbit scattering. In this picture, transport in a multidomain sample occurs at domain boundaries, via two independent, counter-propagating sets of modes. Neglecting quantum interference e@ects, we arrive at the problem for which numerical results are given above. In conclusion, we have argued that the observations of anisotropic transport reported in Refs. [1,2]
85
can plausibly be attributed to formation of anisotropically shaped domains, induced as a result of sample surface roughness. Our numerical work demonstrates that this mechanism generates an anisotropy comparable to that found experimentally [1]. In addition, the onset temperature for strongly anisotropic transport is comparable to the critical temperature expected [13] and observed [17] in other Ising quantum Hall ferromagnets. For future work it would be interesting to investigate transport in systems with deliberately induced surface features. We thank R. Haug and U. Zeitler for extensive discussions, and E. H. Rezayi for correspondence. The work was supported by the Humboldt Foundation (J.T.C.), by the Schwerpunktprogramm “Quanten-Hall-Systeme” of DFG, and by RFBR. References [1] U. Zeitler, et al., Phys. Rev. Lett. 86 (2001) 866. [2] W. Pan, et al., Phys. Rev. B 64 (2001) 121305. [3] M.P. Lilly, et al., Phys. Rev. Lett. 82 (1999) 394; R.R. Du, et al., Solid State Commun. 109 (1999) 389. [4] M.M. Fogler, A.A. Koulakov, B.I. Shklovskii, Phys. Rev. B 54 (1996) 1853; R. Moessner, J.T. Chalker, Phys. Rev. B 54 (1996) 5006. [5] E. Demler, et al., Solid State Commun. 123 (2002) 243. [6] F. Evers, D.G. Polyakov, unpublished. [7] E.H. Rezayi, et al., Phys. Rev. B 67 (2003) 201305. [8] G.F. Giuliani, J.J. Quinn, Phys. Rev. B 31 (1985) 6228. [9] S. Koch, et al., Phys. Rev. B 47 (1993) 4048. [10] A.J. Daneshvar, et al., Phys. Rev. Lett. 79 (1997) 4449. [11] S.L. Sondhi, et al., Phys. Rev. B 47 (1993) 16419. [12] S.M. Girvin, in: A. Comtet, T. Jolicouer, S. Ouvry (Eds.), Topological Aspects of Low Dimensional Systems, de Editions Physique, Paris, 1999. [13] T. Jungwirth, et al., Phys. Rev. Lett. 81 (1998) 2328; T. Jungwirth, A.H. MacDonald, Phys. Rev. B 63 (2001) 035305. [14] V.I. Falko, S.V. Iordanskii, Phys. Rev. Lett. 82 (1999) 402. [15] A. Mitra, S.M. Girvin, cond-mat/0110078. [16] L. Brey, C. Tejedor, Phys. Rev. B 66 (2002) 041308. [17] V. Piazza, et al., Nature 402 (1999) 638. [18] A.P. Young (Eds.), Spin Glasses and Random Fields, World Scienti3c, Singapore, 1997. [19] C. Orme, et al., Appl. Phys. Lett. 64 (1994) 860; R.L. Willett, et al., Phys. Rev. Lett. 87 (2001) 126803; K.B. Cooper, et al., Solid State Commun. 119 (2001) 89. [20] F. Evers, W. Brenig, Z. Phys. B 94 (1994) 115. [21] S. Yarlagadda, Phys. Rev. B 44 (1991) 13 101.