Download New topological excitations and melting transitions in quantum Hall systems

New topological excitations and melting transitions in quantum
Hall systems
Tzu-ging Lin,1 George Simion,2 John D. Watson,2, 3 Michael J. Manfra,2, 3, 4, 5
Gabor A. Csathy,2 Yuli Lyanda-Geller,2, 3 and Leonid P. Rokhinson2, 3, 4
1 Department
of Physics and Astronomy,
Purdue University, West Lafayette, Indiana 47907 USA
2 Department of Physics, Purdue University, West Lafayette, Indiana 47907 USA
3 Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA
4 Department of Electrical and Computer Engineering,
Purdue University, West Lafayette, Indiana 47907 USA
5 Department of Materials Engineering,
Purdue University, West Lafayette, Indiana 47907 USA
(Dated: Resubmitted to Nature Physics on November 3, 2014)
Nature provides striking and varied examples of topologically non-trivial structures
ranging from snowflakes and mushroom gills to chiral ferromagnetic domain walls.
Systems in which the ground state is topologically trivial yet its excitations possess
non-trivial topological order are much less common. The presence of such topological
excitations dramatically alters the systems response to external perturbations; a classic example is the presence of vortices in superconductors. Topological excitations
are notoriously difficult to predict since they cannot be obtained as a perturbation of
the ground state. Here we report the discovery of a new type of topological excitation
which arises in two-dimensional electron systems in a magnetic field. Charge excitations in the reentrant quantum Hall regime are topologically non-trivial textures of
electron density with charge-dependent symmetry. These excitations determine the
response of electrical conduction to external strain, lead to a thermodynamicallydriven insulator-to-metal transition, and should manifest themselves in optical and
non-linear transport phenomena.
Topology and symmetry define states of matter and their responses to external forces.
How solids melt and become fluids, or how insulators become conductors is often controlled
by excitations rather than by the ground state of the system, and non-trivial topology
of excitations can alter the response. In a magnetic field the energy spectrum of twodimensional electrons is quantized into Landau levels, and electron interactions cause a
variety of ground states as a function of magnetic field. When an integer number of singleparticle energy levels are filled (filling factor ν = Φ0 /Φ is integer, where Φ0 = h/e is a
flux quanta, Φ = B/n is magnetic flux per electron, B is magnetic field and n is electron
density), the Hall resistance becomes quantized and longitudinal resistance vanishes, the
hallmarks of the integer quantum Hall effect[1]. Electron-electron interactions at ν < 1
result in fractional quantum Hall states[2] or a Wigner crystal[3]. States with non-Abelian
excitations are suggested in the 2nd Landau level (2 < ν < 4) [4]. At ν > 4 charge density
wave states arise: the unidirectional stripe phase is formed near half-integer filling, [5–7],
while between integer and half-integer ν formation of “bubble phases” lead to the reentrant
integer quantum Hall effect (RIQHE) [8, 9] with Hall resistance quantized to the nearest
integer. Charge excitations in the RIQHE regime are the focus of this paper.
2
The ground state in the RIQHE regime is thought to be a crystal of electron bubbles
with an integer number of electrons on each site [10, 11]. Electron hopping between bubbles
is forbidden by Coulomb blockade, hence an insulating state results[8, 9, 12–15]. Charge
defects in a bubble crystal (an extra electron or lack of an electron on a bubble) provide
a mechanism for conduction, but the structure of these defects has remained hidden. In
this work we present experimental and theoretical evidence that the formation of charged
defects involves a shape transformation of the underlying bubbles; the presence of charged
excitations causes the crystalline bubbles to elongate, change symmetry, and form topological textures. While such behavior is known to occur in bilayer electron systems [16], this
phenomena is unique and unanticipated in single-layer 2D systems. The textures have vortex or hedgehog symmetry, depending on the charge of the defect. At low defect densities,
controlled by temperature and magnetic field, textures do not overlap and extended topological charge defects form an insulating crystal, similar to [3, 17]. At high defect density
topological defects start to overlap and their interactions are described by the XY-model.
Experimentally, such high densities are achieved at temperatures well above the BerezinskiKosterlitz-Thouless thermodynamic phase transition [18, 19], and the crystal of topological
defects melts resulting in a sharp insulator-metal transition[20]. This new phase transition
resembles asymptotic freedom of quarks requiring them to be ”squeezed” in order to be
freed[21]. The presence of topological defects determines the symmetry of the electron gas
response to an applied electric field, most remarkably in the presence of uniaxial strain. The
experimentally observed strain dependence of resistivity cannot originate from any singleparticle effects, but can be explained by the formation of elongated topological defects driven
by strong interactions.
Magnetoresistance in a high mobility 2D electron gas is shown in Fig. 1 for 4 < ν < 5.
A stripe phase with large resistance anisotropy between [110] and [11̄0] crystallographic
directions is observed near ν = 4.5. RIQHE phases are formed near ν = 4.3 and 4.7 and are
marked as B1 and B2 on the plot. The width of RIQHE phases is temperature dependent,
with the insulating state bounded by a sharp resistance peak separating insulating and
conducting phases in the (B − T ) plane. The most striking feature shown in Fig. 1 is
strain dependence of resistivity R(ε). Uniaxial strain ε = εxx = −εyy leads to a weak
(< 0.2%) anisotropy of the exchange interaction and of the electron effective mass [22].
Indeed, for the stripe phase we see small enhancement of resistivity in x||[11̄0] and reduction
in y||[11̄0] direction, consistent with strain-dependence in p-type gases where strain-induced
anisotropy is stronger [22]. Unexpectedly, however, dR/dε changes sign across the RIQHE
states as illustrated in the insets in Fig. 1. Strain-dependent anisotropy of conductivity,
σii = σii0 + β(εii − εjj )], i, j = x, y, i 6= j, is defined by an intrinsic parameter β that reflects
the symmetry of the underlying state (note that σxx ∝ ρxx at high B). To be consistent
with the experiment, β has to change sign across the RIQHE state as a function of magnetic
field. This sign change is incompatible with the symmetry of the ground state of RIQHE,
or any other known ground state of an interacting 2D electron system..
In order to gain insight into the physics of topological excitations, we analyze the effect
of strain on resistivity from the vantage point afforded by symmetry. Conduction is due
to the motion of charge defects: For crystals with two electrons per bubble defects with
three or one electron can move through
the lattice without an energy penalty. Electric
R
current in real space is given by j = r(eEr)f (r)dr, where E is the electric field, r is the
electron coordinate, f (r) describes the charge distribution, reflecting the presence of strain
and other factors, and conductivity σii = e hri2 f (r)i. For symmetry analysis we consider
3
r to be continuous; similar results can be obtained for hopping over discrete lattice sites.
The symmetry invariant T1 = (εxx − εyy )(r2x − r2y ) describes electron hopping in an ideal
lattice deformed by strain, as shown schematically in Fig. 2(a). If the only factor due to
strain were anisotropy of f (r) ∝ T1 , which is similar to the strain-induced anisotropy of
the effective mass, then the sign of corresponding β would be the same across all RIQHE
regions in contrast to the experimental data. We will show below that the lowest energy
charge excitations involve elongation of electron bubbles in the vicinity of charge defects and
their re-arrangement into non-trivial textures, see Fig. 2b. Distributions of charge in two
(v)
possible textures, vortices and 2D hedgehogs, are described by functions T2 = (a × r)2z and
(h)
T2 = (a · r)2 respectively, where vector a is the elongation of a bubble. In the presence of
elongations, strain affects the system via the symmetry invariant T3 = (εxx − εyy )(a2x − a2y ).
(h)
(v)
Calculating hri2 f (r)i by averaging over a for f (h) (r) ∝ T2 T3 and f (v) (r) ∝ T2 T3 , we arrive
at the opposite sign of β for vortices and hedgehogs. Microscopic calculations confirm the
sign difference. We also show that vortex topological excitations dominate on one side of
RIQHE, while hedgehogs dominate on the other side, giving the symmetry change across the
RIQHE needed for the change of the sign of strain response. We note that from a symmetry
standpoint εxx − εyy behaves like Bx2 − By2 in the presence of tilted magnetic field, where
Bx and By are the in-plane components of B. Therefore it is possible that experiments
in tilted field will also reveal opposite effects on resistivity due to vortices and hedgehogs.
Similarly, linear polarization of light e leads to a symmetry invariant e2x − e2y , suggesting
that topological excitations may be also investigated in optical experiments.
In order to identify key effective interactions in the system and elucidate an appearance
of topological textures around 1ē and 3ē defects, we derive and analytically solve the model
for defects in a bubble crystal. We also perform full numerical simulations which confirm
and refine analytical results. The charge density wave phases at partial filling of Landau
levels in the Quantum Hall systems, including the bubble crystal phase, are conventionally
approached via the Hartree-Fock (HF) method [10, 11, 23, 24] , and we use its “interacting
guiding centers” version[15] in order to describe defects in the bubble crystal microscopically.
The P
HF Hamiltonian
P is the sum of effective interactions between guiding centers: HHF =
1/2( i6=j U (Rij ) + i U (0)) where i and j labels the lattice nodes, Ri is the coordinate of
the bubble i, Rij = Ri − Rj . The interaction between bubbles i and j is given by
Z
U (Rij ) =
d2 q ∗
ρ (q)[VH (q) − VF (q)]ρj (q) exp (iq · Rij ) ,
(2π)2 i
(1)
where ρi is the site i bubble density projected on the uppermost Landau Level. The Hartree
2
and exchange potentials are, respectively [23], VH (q) = 2π
e−q /2 [Ln (q 2 /2)]2 , and VF (q) =
q
R d 2 q0
−i(q×q0 )·ẑ
, where Ln is the nth Laguerre polynomial. We concentrate on
2π (2π)
2 VH (q)e
a two-electron bubble crystal and its excitations. An ideal bubble crystal at T = 0 has
round bubbles in the ground state. Calculated energies of 1ē and 3ē defect in a crystal of
round bubbles with account of displacements of lattice bubbles due to defects, similar to
displacement of electrons in the Wigner crystal due to vacancies [25], are plotted in Fig. 3a
with dashed lines. Apart from high symmetry of these defects being inconsistent with the
strain dependence of resistivity, their activation energies are in the range of several Kelvin
and cannot explain high conductivity at 100 mK.
The activation energy of charged defects is reduced if two-electron bubbles around 1ē and
3ē defects elongate. We will see that elongations are smaller than the magnetic length λ.
4
This allows us to formulate an analytical model based on the small parameter λ/w, where
w ∼ 8λ is the lattice constant of the two-electron bubble crystal. The wavefunction of an
elongated two-electron bubble with guiding centers separated by 2a is:
Ψξ,a (r1 , r2 ) = α (ψa (r1 , r2 ) + ξψ−a (r1 , r2 ))
(2)
where α is the normalization coefficient, ξ is a variational parameter,
ψa (r1 , r2 ) = [u−2 (r1 + a)u−1 (r2 − a)ei(r1 −r2 )×a/2 − u−1 (r1 − a)u−2 (r2 + a)ei(r2 −r1 )×a/2 ],
u−2 and u−1 are single-electron wavefunctions on the 3rd Landau level (3LL) in the symmetric gauge with angular momentum projections m = −2 and m = −1 [26]. This is a
variational wavefunction, similar in spirit to the one proposed for round bubbles [11]. The
wavefunction (2) reflects both the magnitude and orientation of the bubble elongation.
Analytical expressions for electron density of an elongated bubble projected on the 3LL
and interaction energy between elongated bubbles can be found in the Supplementary Material. The total energy includes a contribution from interactions between elongated bubbles
and the charge defect. The wavefunction of a 1ē bubble u−2 and has round shape. For a 3ē
bubble the wavefunction is a Slater determinant of u−2 , u−1 , and u0 . Its exact shape can be
determined by energy minimization, but the physics is primarily determined by elongations
of the crystalline two-electron bubbles, so we neglect the detailed structure of the 3ē bubble
and model it as having all three guiding centers on top of each other.
We express interactions of a single charge defect at a site k with 2ē bubbles via elongations
ai and effective dipole moments µi (ai ), which represents rotated and rescaled ai . Retain−3
ing terms up to Rik
we get asymptotic expansion in λ/w for the elongation-dependent
contribution to energy:
" "
#
#
X
µ
·
R̂
v
ik
i
i
∓
H± =
+
cos (2ϕi ) + Uf± (i)
2
3
R
2R
ik
ik
i6=k
+
1 X µi · µj − 3(µi · R̂ij )(µj · R̂ij )
,
3
2 i6=j
Rij
(3)
where (+) and (−) signs correspond to 1ē and 3ē defects, R̂ij = Rij /Rij and ϕi is the angle
between ai and Ri . The first three terms of Eq. (3) come from the interaction of i-th bubble
with the charge defect, the fourth term is a dipole-dipole interaction between elongated
bubbles at sites i and j. Expressions for µi , vi , and Uf± (i), are given in Supplementary
Materials.
The ground state of a deformed bubble crystal in the vicinity of a charge defect is obtained
by minimizing the energy (3). For 1ē defects the energy minimum is at ϕi = 0 and π,
corresponding to vortices and anti-vortices, and for 3ē defect the energy minimum is at ϕi =
π/2 and 3π/2, corresponding to 2D hedgehogs and anti-hedgehogs. Numerical minimization
of the full Hartree-Fock Hamiltonian, in which two-electron bubbles are projected onto state
(2) and defects are introduced in the system, confirms an appearance of hedgehogs and
vortices in the presence of 1ē and 3ē defects, respectively. Activation energies of defects
in the presence of textures computed by analytical minimization of (3) or in full numerical
simulation are shown by solid and dotted lines in Fig. 3a, respectively. Energies of topological
defects are an order of magnitude smaller than excitation energies for charge defects in the
5
absence of textures, and agree with energy scale observed in experiments. Some difference
between energies obtained using analytical model and full numerical simulation is discussed
in the Supplementary Materials.
Density profiles of vortex and hedgehog defects are plotted in Fig. 2b. Accoording to the
last term in (3), the interaction between elongated lattice bubbles is essentially described by
the XY-model, in which the textures are topological excitations. Energy minimization using
analytical model provides the magnitudes of elongations ai (of the order of the magnetic
length λ) given in the Supplementary Materials. An important result of minimization is
that single texture is extended over a finite distance L ∼ 10.5w from the charge defect. As
distance Rik from the center of the charge defect increases, bubble elongation ai decreases.
At Rik ≥ L round bubbles (ai = 0) become energetically favorable. This analytically
obtained bubble size agrees with the results of numerical simulations, where a crossover
from logarithmic dependence of energy per defect on defect separation to almost separationindependent interaction energy is observed at 2L ≈ 21 for ν = 4.22, see Fig. 3d. In the
numerical simulation, the magnitude of elongations of 2ē bubbles on the boundary of defects
steeply increases at Rik < L.
At low defects density, ρd < 1/(4L2 ), charged defects interact via residual Coulomb
interaction and form a superlattice superimposed on the bubble crystal, similar to the Wigner
[3] or Abrikosov[17] lattice. At a given temperature the equilibrium defect density is achieved
via charges coming from and leaving to the contacts. Other channels of equilibration are
very slow at low T ; a calculated barrier for annihilation of hedgehog defects ∼ 10K, of vortex
defects ∼ 15K, and for the recombination of vortex and hedgehog ∼ 0.5K.
When T increases or ν is shifted away from the RIQHE center, the density of defects
increases, see Fig. 3b. When ρd > 1/(4L2 ) the energy (3) includes an XY-model interaction
between elongated bubbles belonging to different defects. Estimating this energy from our
analytical model, we take the lower bound on the “exchange constant” J, given by µi ·µj term
in (3). Assuming J is defined by the magnitude of elongations a ∼ λ in the overlap region,
we get J ∼ e2 a2 /κL3 . This value is in good agreement with the slope of the logarithmic part
of the numerically obtained curve, see Fig 3d. The free energy of topological excitations is
given by[27]
E = (πJ − 2T ) ln(L/w),
(4)
where L is the size of the system, and the core of topological excitations is w. Such a
system must exhibit the Berezinski-Kosterlitz-Thouless (BKT) transition [18, 19, 28] at
TBKT = πJ/2. However, the transition in the RIQHE differs from a BKT transition due to
the finite size of topological defects. For ρd < 1/(4L2 ) there is no overlap between defects,
energy is not a logarithmic function of inter-defect distance and, thus, BKT transition
is not possible. The temperature TL , at which ρd = 1/(4L2 ), is much higher than the
estimated TBKT ≈ 5 mK. Thus, TBKT itself is not observed experimentally. The unbinding
of topological defects at TL is what precludes the divergence of logarithmic interactions
in (4). It constitutes melting of the defect crystal. We plot both TL (B) calculated from
the analytic model and from the numerical simulations over the experimentally measured
Rxx (T, B) in Fig. 3c. TL (B) describes the observed phase diagram of the insulator-toconductor transition rather well. The role of charged impurities in the heterostructure and
how they affect position of the phase boundary is discussed in the Supplementary Material.
We note that alternative explanations of the observed R(T ) , such as activated transport,
with R ∝ exp(−Ta /T ), or Efros-Shklovsky variable-range hopping
(VRH)in the presence
of the Coulomb gap in a disordered conductor, with R ∝ exp −(TES /T )1/2 , contradict the
6
experimetal data. The best activation energy fit is Ta = 3.1 K, which is inconsistent with
high mobility measured at 0.03 K. The best fit to the VRH dependence corresponds to the
localization length Lloc = kB TES κ/e2 ∼ 70 nm ∼ w/2, contradicting to existence of the
RIQHE state and precluding formation of a bubble crystal.
Estimating that melting of an ideal bubble crystal due to dislocations [25, 29] occurs at
Tm ≈ 400 mK > TL , we conclude that topological defects in a bubble crystal persist above
TL , contributing to the observed strain dependence of resistivity. Strain εxx = −εyy = ε is
included using deformed coordinates x → x(1 + ε), y → y(1 − ε), in which the lattice is
identical to the unstrained one, but the interaction becomes anisotropic. The linear in strain
Hamiltonian is Hε = −εŨf (a) cos[2(φi − θik )], where (φi − θik ) and θik are polar angles of
ai and Rik , and Ũf (a) is defined in the supplement. Note that εa2 cos[2(φi − θik )] is the T2
invariant of our symmetry considerations. Minimizing the total energy we find that strain
leads to small rotations of elongated bubbles
δφi = εγi sin[2(φi − θik )] = ±εγi sin(2θik ),
(5)
where the (+) sign corresponds to vortex (φi = 0, π) and (-) sign to hedgehog (φi =
π/2, 3π/2) textures, γi = β1i Uf . Effect of strain on defect textures is sketched in Fig. 2.
A strain dependent conductivity arises from re-adjustment of textures and rotation of
elongated 2ē bubbles by an angle ∆θi = (θik − θik0 ) + [δφi (θik ) − δφi (θik0 )] = ∆θia + ∆θiε when
defect charge hops from site k to k 0 . Overlap between the initial and the final state of the
bubbles is p(∆θi ) = hΨ∗ik |Ψik0 i = 1 − α(ai )(∆θi )2 , and the conductivity
Y
σs(h,v) ∝
p(∆θi ),
(6)
i
where the product is over all bubbles that rotate or change
of the charge
shape as a result
hopping. Rotation of elongated bubbles ∆θia = (w/Rik ) sin θik + (1 + s) π4 leads to a strainindependent conductivity when defect moves by w in the x̂ (s = 1) or ŷ (s = −1) direction.
A strain-dependent rotation ∆θiε = ∓γi cos(2θik )∆θia has different sign for hedgehogs and
vortices.
The ratio of resistivities due to vortices and hedgehogs, %(v) /%(h) , is calculated using
(6) and is plotted in Fig. 4 as a function of strain. A ratio of macroscopic resistances
R(v) /R(h) is further amplified in the van der Pauw geometry [30]. Experimentally measured
strain dependence of magnetoresistance for RIQHE states on the 2LL and 3LL is plotted in
Fig. 4c. For each case the ratio RR /RL is extracted, where RL and RR are the low-ν (L) and
high-ν (R) resistances bordering the RIQHE state. Good quantitative agreement between
R(v) /R(h) and RR /RL for the B1 state strengthens our conclusion that 1ē (vortex) excitations
dominate the low-ν side of B1 , while 3ē (hedgehog) excitations dominate the high-ν side.
Thus our theory explains both the symmetry and unusually large magnitude of strain’s
effect on resistivity, in addition to providing an explanation to the observed metal-insulator
transition.
Methods
Samples are fabricated from modulation-doped 30-nm wide AlGaAs/GaAs/AlGaAs quantum wells grown on (001) GaAs by molecular beam epitaxy. Devices are cleaved along (110)
crystallographic direction. Eight Ohmic contacts are formed along the edge of a sample by
annealing InSn alloy at 420◦ C for 3 min. Devices attached to a multilayered piezoelectric
7
transducer with epoxy with either (110) (yy) or (11̄0) (xx) aligned with the transducer
polling direction. A thin layer of metallized plastic foil is inserted between the sample and
the transducer, the foil is grounded during the experiments in order to screen possible nonuniform electric fields. A shear strain induced in the sample ε = εxx − εyy ≈ 2.5 · 10−7 Vp ,
where Vp is the voltage on the transducer and εxx ≈ −2εyy for the (xx) poling direction.
The actual strain εtot = εth + ε, where thermally-induced strain εth depends on the cooldown
conditions. The 2D electron gas is illuminated with a red LED at 10 K for 20 min. Several
samples from two different wafers with mobility ∼ 1 · 107 cm2 /Vs and density 2 − 3 · 1011
cm−2 at 0.3 K were investigated, all show similar results. Measurements are performed in a
van der Paw geometry using low frequency (17 Hz) lock-in technique with excitation current
Iac = 10 nA.
Acknowledgements
Research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under Awards DE-SC0008630 (L.P.R.), DESC0010544 (Y.L-G), DE-SC0006671 (G.C., J.D.W. and M.J.M.).
Authors contribution
L.P.R. conceived and designed the experiments, T.L. performed measurements. G.S. and
Y.L-G proposed explanation of the experimental results and put forward the theory. J.W.
and M. J. M. designed and grew wafers for the experiments, and G.C. provided data from
unstrained samples. Y.L-G and L.P.R and M. J. M. contributed to writing the paper.
Additional information
Supplementary information is available in the online version of the paper. Correspondence
should be addressed to L.P.R. (experiment) or Y.L-G. (theory).
Competing financial interests
The authors declare no competing financial interests.
8
m a g n e tic fie ld (T e s la )
(h /e 2)
0 .2 2
2 .8
2 .7
2 .6
2 .5
2 .4
2 .3
s trip e
B
B
1
V
p
= +300V , ε> 0
V
p
= −300V , ε< 0
2
R
0 .2 4
x y
0 .2 6
0 .2 0
(x x )
(y y )
0 .1 0
0 .1 0
0 .0 5
0 .0 5
0 .2
( k Ω)
0 .0 0
0 .0 0
-2 0 0
0
2 0 0
-2 0 0
0
2 0 0
-2 0 0
V p (V )
0
2 0 0
-2 0 0
V p (V )
0
2 0 0
V p (V )
R
x x
V p (V )
0 .1
0 .2 5
0 .2 0
0 .1
0 .0 1
y y
( k Ω)
0 .0
R
0 .0 0
-3 0 0
0
3 0 0
V p (V )
0 .0
4 .0
4 .1
4 .2
4 .3
4 .4
4 .5
4 .6
4 .7
4 .8
4 .9
5 .0
fillin g fa c to r
FIG. 1. Strain dependence of re-entrant QHE phases near filling factor ν = 9/2. Colored
regions mark IQHE states (grey), RIQHE phases B1 , B2 (yellow) and the stipe phase (green). In
the inserts resistance of the peaks marked by arrows are plotted as a function of strain for Rxx
(blue) & Ryy (red). Strain dependence of the stripe phase at ν = 9/2 is also shown, note the break
in the vertical axis. The data is taken at 100 mK.
9
(a)
𝑻𝟏 ∝ (𝒓𝟐𝒙 − 𝒓𝟐𝒚 )
(b)
𝑻𝒗𝟐 = (𝒂 × 𝒓)𝟐
𝑻𝒉𝟐 = (𝒂 ∙ 𝒓)𝟐
FIG. 2. Symmetry of a bubble crystal and topological charge defects. (a) Uniaxial strain
induces anisotropy of the electron wavefunction and uniaxial deformation of the bubble crystal,
which is described by the symmetry invariant T1 . (b) Bubble crystal is topologically deformed in
the vicinity of charged defects, the symmetry of deformations for 3ē and 1ē defects is described
by T2v and T2h invariants correspondingly. Rotation of elongated bubbles under uniaxial strain is
shown schematically by dotted lines. Electron density profile in the vicinity of defects is shown in
surface plots.
10
0 .1 3 1
0 .5 3 7
1 e
3 e
U
1
0 .5 3 6
(a )
0 .4
d e fe c t d e n s ity
ρd × 4 L 2
2
1 0 0
1
R ( Ω)
(b )
4 .3 0
4 .2 6
(e )
7 0
4 .2 2
1 0 0
(f)
2 0 0
0
T (m K )
4 .5
0
1 0 0
2 0 0
T (m K )
4 .3
4 .1
a n a ly tic a l T L, th e m e ltin g p h a se tra n sitio n
n u m e ric a lly c a lc u la te d T L
R
x x
( k Ω)
9 0
2 0
(c )
2 .3
2 .4
2 .5
R IQ H E
R IQ H E
s tr ip e s
0 .8
IQ H E
te m p e ra tu re (m K )
4 .7
1 .2
1
f illin g f a c to r
4 .9
1
2
1 0 0
0 .1
4 .3 4
0 .8
0 .1 2 8
d a ta
a c tiv a tio n
V R H
1 0
0
1 6 0
0 .6
1 / √ρd × 4 L
1 0 0 m K
7 0 m K
(K )
(d )
3 e
E (K )
3 e
1 e
(K )
1 e
0 .1 3 4
U
0 .5 3 8
1 0
2 .6
0 .4
0
2 .7
m a g n e tic f ie ld (T e s la )
FIG. 3. Phase diagram of the melting transition. (a) Activation energy for an isolated defect
is calculated for round bubbles (Eq. S8, dashed lines) and textured defects using analytical theory
(Eq. 3, solid lines) or full numerical calculations (dotted lines). Note that topological deformations
reduce activation energies by a factor of 10. (b) Density of defects ρd is calculated for 70 mK and 100
mK. The melting phase transition temperature TL corresponds to the defect density where defect
separation ≈ 2L. (c) Analytically and numerically calculated phase boundary TL (B) is plotted on
top of the experimentally measured temperature dependence of Rxx and coincides with the sharp
increase of conductance at the boundary of isolating and conducting phases. (d) Numerically
calculated energy of defect-defect interaction vs defect separation shows a crossover from constant
to logarithmic dependence at ≈ 2L. (e,f) Temperature dependence of longitudinal resistance in the
middle of the RIQHE phase. Fits to the T -dependence with activation and variable range hopping
give unrealistic Ea = 3.1 K and ξ = 70 nm.
11
1.5
(h)
(v)
1.0
1
0.1
0.5
-5
(a)
-5.0x10
-5
0.0
strain (exxeyy)
-5
5.0x10
(b)
-5.0x10
-5
0.0
5.0x10
strain (exxeyy)
(xx)
(yy)
L
10
1
R
R /R
(xx)
(yy)
10
R /R
(v)
 /
(h)
(xx)
(yy)
0.1
-200
0
200
-200
0
Vp (V)
I1
I1
200
-200
0
200
-200
Vp (V)
B1
0
200
-200
0
Vp (V)
Vp (V)
I3
I3
Vp (V)
300
200
100
0
-100
-200
-300
200
*
-200
0
200
Vp (V)
B2
Ryy (au)
Rxx (au)
Vp (V)
(c)
2.28
2.32
3.26 3.31
filling factor
4.2
4.3
2.55 2.60
*=1/2
2.55 2.60
4.7
4.8
filling factor
FIG. 4. Strain dependence of resistivity in the presence of topological defects. (a)
Ratio of resistivities for the transport dominated by vortex ρ(v) and hedgehog ρ(h) charge defects
is calculated using Eq. 6 and the corresponding ratio of resistances is plotted for a square van der
Pauw geometry [30] in (b). (c) Magnetoresistance near RIQHE states on 2nd & 3rd LLs measured
with different strains (voltages on the piezo transducer Vp ). Heights of the resistance peaks RL
and RR on the left and right sides of RIQHE states are extracted from two peak Lorentzian fits,
the corresponding ratio RR /RL is plotted at the top of each data set.
12
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(Springer, Berlin, 2002) pp. 98–138.
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crystalline phases of two-dimensional electrons in a strong magnetic field,” Phys. Rev. B 73,
165324 (2006).
[16] K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and
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(1995).
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[18] V.L. Berezinskii, “Destruction of long-range order in one-dimensional and 2-dimansional sys-
13
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
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610–616 (1972).
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nature of the reentrant integer quantum Hall states in the second Landau level,” Phys. Rev.
Lett. 108, 086803 (2012).
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Rev. Lett. 30, 1343–1346 (1973).
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Manfra, Loren N. Pfeiffer, Kenneth W. West, and Leonid P. Rokhinson, “Effect of strain on
stripe phases in the quantum Hall regime,” Phys. Rev. Lett. 106, 016804 (2011).
I. L. Aleiner and L. I. Glazman, “Two-dimensional electron liquid in a weak magnetic field,”
Phys. Rev. B 52, 11296–11312 (1995).
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sup-1
Supplementary Materials
New topological excitations and melting transitions in quantum Hall
systems
Tzu-ging Lin, George Simion, John D. Watson, Michael J. Manfra, Gabor A. Csathy, Yuli
Lyanda-Geller, and Leonid P. Rokhinson
THERMALLY-INDUCED STRAIN
0 .7 0
2 0 0 0
0 .6 5
( p p m /v o lt)
2 5 0 0
0 .6 0
p
1 5 0 0
p i e z o c o e f f i c i e n t εp / V
t h e r m a l l y - i n d u c e d s t r a i n εt h ( p p m )
I.
1 0 0 0
5 0 0
0
0 .5 5
0 .5 0
0 .4 5
(b )
(a )
-5 0 0
0 .4 0
-8 0
-6 0
-4 0
-2 0
c o o ld o w n v o lta g e V
0
0
2 0
( v o lts )
4 0
-8 0
-6 0
-4 0
-2 0
c o o ld o w n v o lta g e V
0
0
2 0
4 0
( v o lts )
FIG. S1. Thermally induced strain. Thermally induced strain can be controlled by applying
voltage V0 on the transduced during cooldown. The piezo coefficient has weak dependence on V0 .
A PZT-based multilayer piezoelectric transducer has very anisotropic thermal expansion
coefficients (+3/−1 ppm/deg near room temperature) along and perpendicular to the polling
direction. The residual thermally induced strain εth (at Vp = 0) can be controlled by voltage
V0 applied to the transducer during cooldown. A dependence of εth (V0 ) is plotted in Fig. S1,
measured with a bi-axial strain gauge. There is almost no thermal expension/contranction
and no hysteresis in the εp (Vp ) characteristic below 10 K.
II.
THEORY OF TOPOLOGICAL EXCITATIONS IN THE QUANTUM HALL
REGIME
In this section we present detailed theory of charge excitations in a bubble crystal, demonstrate how topological defects of different symmetry arise, and calculate transport properties
of a 2D gas in the presence of these topological defects. We also investigate how strain affects
charge transport in the presence of charged topological defects.
sup-2
A.
Charge Density Wave Phases
We use a conventional Hartree-Fock (HF) method [10, 11, 23, 24] to describe charge
density wave phases in the nth partially filled Landau level[23]:
Z
1
d2 q
HHF =
[VH (q) − VF (q)]|ρ(q)|2 ,
(S1)
2
(2π)2
where ρ(q) is the projected electronic density onto the uppermost LL. Unless otherwise
noted, we express distances in units of magnetic length λ, wavevectors in units of 1/λ, and
energies in units of e2 /κλ, where κ is the background dielectric constant. The Hartree and
exchange potentials are:
2π −q2 /2
e
[Ln (q 2 /2)]2 ,
q
Z 2 0
dq
0
VF (q) = 2π
VH (q)e−i(q×q )·ẑ ,
2
(2π)
VH (q) =
(S2)
(S3)
where Ln is the nth Laguerre polynomial.
A charge density wave state was proposed as the ground state for 2D systems in the lowest
Landau level [24] even prior to the discovery of the quantum Hall effect. This prediction
is relevant to high Landau levels where different phases compete[12] and bubble or stripe
phases become possible ground states[5, 11, 13–15]. In a bubble phase guiding centers of
electron cyclotron orbits form a triangular lattice. A Wigner crystal, a triangular lattice
with one electron per lattice cell (M = 1), is energetically favorable at small effective filling
factors ν ∗ = ν − nf < 0.2, where nf is the number of filled Landau levels and ν is the filling
factor. For larger ν ∗ bubble phases with M > 1 can be formed. HF calculations [11, 13–15]
set a limit M ≤ nf + 1, while density matrix renormalization group method restricts the
size of bubbles to M ≤ nf [31, 32]. A conventional picture of bubble phases is that crystals
with M electrons per bubble exist within a certain range of filling factors and the first order
phase transitions occur between M and M ± 1 phases. For ν ∗ ≈ 0.5 a stripe phase becomes
the ground state.
Considering bubble phases, it is convenient to express the HF
as a sum
PHamiltonian P
of effective interactions between the guiding centers: HHF = 1/2[ i6=j U (Rij ) + i U (0)],
where i and j labels the nodes of a triangular lattice. An effective interaction U is given by:
Z
d2 q ∗
U (Rij ) =
ρ (q)[VH (q) − VF (q)]ρj (q) exp (iq · Rij ) ,
(S4)
(2π)2 i
where ρi represents projected density of a bubble located at the site i. We surmise that HF
approach captures physics of the quantum Hall systems even at low n, particularly for the
3rd LL (n = 2) and 2nd LL (n = 1) [13–15].
B.
Charge Defects and elongations of bubbles in Bubble Crystals
In order to address anomalous strain dependence of longitudinal resistance observed experimentally we have to consider charge excitations of the bubble crystal. As we have
discussed in the main text, simple melting of the bubble crystal away from the center of the
bubble phase at elevated temperatures does not explain the observed symmetry change.
sup-3
Prior to this work bubbles were almost exclusively treated as entities with uniform charge
density. Ettouhami[33] suggested that guiding centers of electrons in two-electron (2ē) bubbles are spatially separated even in an ideal bubble crystal with no charge defects. We find
that if superposition between wavefunctions of electrons in the same bubble and their phase
factors due to magnetic translations are properly taken into account, round shape of bubbles
is energetically favorable in an ideal 2ē bubble crystal. We find, however, that in the vicinity
of charged defects, i.e. bubbles lacking one electron (1ē) or with one extra electron (3ē),
two-electron bubbles become elongated. The wavefunction of a bubble with two guiding
centers separated by 2a can be expressed as:
Ψξ,a (r1 , r2 ) = p
ψa (r1 , r2 ) + ξψ−a (r1 , r2 )
2 (1 −
2e−2a2 a2 ) (1
+ |ξ|2 ) + 4 (1 − 2a2 ) e−2a2 <e(ξ)
,
(S5)
where
i
i
Ψa (r1 , r2 ) = u−2 (r1 + a)u−1 (r2 − a)e 2 (r1 −r2 )×a − u−1 (r1 − a)u−2 (r2 + a)e 2 (r2 −r1 )×a , (S6)
and u−2 and u−1 are single electron wavefunctions with angular momenta m = −2 and −1
in third Landau level (n = 2) in the symmetric gauge [26]. This is a trial wavefunction
similar in spirit to the variational wavefunction proposed by Fogler and Koulakov for round
bubbles[11]. The direction of a characterizes spatial orientation of the elongated electron
bubble, and complex parameter ξ permits nontrivial combinations of a and −a elongations
that may potentially emerge in the presence of a magnetic field. Electron density of such
an elongated bubble, projected on the n = 2 LL, is
h
i−1
q2
2
2
ρξ,a (q) = e− 4
1 − 2e−2a a2 1 + |ξ|2 + 2 1 − 2a2 e−2a <e(ξ)
q2
|ξ|2 q 2
−iq·a
2
iq·a
2
+e
1 + |ξ| −
× e
1 + |ξ| −
2
2
2
− e−2a +(q×a)·ẑ 2a2 − (q × a) · ẑ 1 + |ξ|2 − 2 1 − 2a2 <eξ
ξq 2
2
− 2ξ(q × a) · ẑ + i 1 − |ξ| q · a
+
2
2
− e−2a +(a×q)·ẑ 2a2 − (a × q) · ẑ 1 + |ξ|2 − 2 1 − 2a2 <eξ
ξ ∗q2
∗
2
− 2ξ (a × q) · ẑ + i 1 − |ξ| q · a
,
(S7)
+
2
where ā = ax + iay , and a∗ = ax − iay .
We consider 1ē and 3ē charge defects, the lowest energy charged excitations of a 2ē bubble
crystal. The wavefunction of a 1ē defect is u−2 (r), and has a round shape. A 3ē defect has
internal structure with nonuniform density distribution, and its wavefunction is a Slater
determinant made of u−2 , u−1 , and u0 . Exact structure and shape of these defects can
be determined by minimizing the cohesive energy, however, as our numerical study shows,
energetics and electron transport are primarily determined by elongations of 2ē bubbles in
the vicinity of defects. Thus, the detailed structure of 3ē bubbles is not essential and we
model it with all three guiding centers being at the same point. Using the wavefunctions for
elongated 2ē bubbles (S5), expression for charge density (S7), and wavefunctions for 1ē and
3ē defects, we can write an effective Hartree-Fock Hamiltonian for the bubble crystal with
defects.
sup-4
æ m=-2
æ m=-1
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ m=-2
æ m=-1
æ
æ
æ m= 0
æ
æ
ææ
ææ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
FIG. S2. Numerically calculated topological textures surrounding 1ē and 3ē defects.
Red and blue circles show positions of electrons with angular momentum m = −2 and m = −1
correspondingly, green circle is a 3ē defect.
C.
Numerical simulation of the problem
Full treatment of the effective Hartree-Fock Hamiltonian is an extremely complex endeavor, and we introduce certain simplifications for our numerical simulations. In particular,
for ν ∗ far away from the center of the 2ē bubble phase we consider only one type of defects,
1ē defects at low ν ∗ (close to the Wigner crystal) and 3ē at high ν ∗ (close to the phase
transition to the 3ē bubble crystal). We justify this simplification a posteriori by demonstrating that the energy to form a 1ē defect is lower than the energy to form a 3ē defect at
low ν ∗ and higher at high ν ∗ . In order to calculate the minimal energy of the system in the
presence of defects we use the following construction. The initial configuration consists of
a unit cell with one defect of a given type surrounded by 2ē bubbles placed at the nodes of
the bubble crystal. The number of 2ē bubbles in a unit cell is determined by the density
of defects. Similar to the treatment of vacancies in a Wigner crystal[25] we apply periodic
boundary conditions repeating this unit cell, and taking into account interactions off all
bubbles and defects both within the unit cell and between different unit cells. Extra charge
on defects cannot be fully screened by elastic displacement of surrounding bubbles[25] or
bubble elongations because each unit cell has a finite size. Therefore, there is always a residual interaction that makes energetically favorable to position defects as far away from each
other as possible, justifying a periodic arrangement of defects. Displacement of 2ē bubbles
and magnitude and direction of their elongations are computed iteratively by minimizing
total energy (with up to 300 steps), for different defect densities.
Absolute energy minimum corresponds to the bubble crystal with no defects. For finite
defect density numerical simulations clearly demonstrate that around 1ē defects elongated 2ē
bubbles form hedgehog textures, while around 3ē defects they form vortex textures, Fig. S2.
In Fig. 3a dotted lines show ν ∗ -dependence of a single defect calculated at a low defect
density 0.0025. For comparison, dashed lines show energies of round defects (a = 0) when
only re-adjustments of 2ē bubbles positions are included. Elongations of 2ē bubbles lowers
defect energies by an order of magnitude. In Fig. 3d we use numerical minimization to
plot dependence of energy per defect on defect separation. At high defect densities energy
depends logarithmically on defect separation, while at smaller densities energy dependence
on inter-defect distance saturates. The crossover is found to be at ≈ 21 bubble crystal lattice
constants. At large separations, 2ē bubbles close to the boundary of unit cells are positioned
sup-5
at the sites of the ideal bubble crystal and exhibit almost no elongation. Interactions between
defects are almost completely (but still not fully) screened by elongated bubbles and, hence,
there is almost no energy dependence on defect separation. Elongations of boundary bubbles
steeply increase at defect separations smaller than 21w, when dependence of energy becomes
logarithmic.
D.
Analytical model
In order to gain insight into physics behind these numerical results and to understand the
effect of strain on conductivity, we construct an analytical model based on a small parameter
a/w, a ratio of a characteristic magnitude of elongation of 2ē bubbles, a ≈ λ, to the lattice
constant of the bubble crystal w ≈ 8λ. Writing an asymptotic expression for the effective
interaction of bubbles at different sites of the 2ē bubble lattice and taking into account
elongation of bubbles, it is convenient to express the interaction in terms of elongations ai
and effective dipole moments µi , which represent rotated and re-scaled ai . Retaining terms
−3
up to Rij
for energy change due to elongations we obtain
(µi − µj ) · R̂ij µi · µj − 3(µi · R̂ij )(µj · R̂ij )
+
2
3
Rij
Rij
u(ai , ξi ) + u(aj , ξj ) v(ai , ξi ) cos(2φi − 2θij ) + v(aj , ξj ) cos(2φj − 2θij )
+
+
.(S8)
3
3
Rij
Rij
δU2 (Rij ) =
Here θij is the phase of Rij , φi and φj are the phases of elongation vectors ai and aj for
bubbles located at sites i and j respectively, R̂ij = Rij /Rij , and
r
1−
µ=
2<eξ
1+|ξ|2
2
2R(a)
,
2<eξ
e2a2 − 2a2 1 + η1 (a) 1+|ξ|
2
2a2
2e
a 2a2
e
+ 2a2 1 + |ξ|2 − 2η2 (a)<e(ξ)
,
− 2a2 1 + |ξ|2 + 2η1 (a)<e(ξ)
2a2
− 2a2 + 2 1 + |ξ|2 + 2η3 (a)<e(ξ)
2e
v(a, ξ) = 3a
,
e2a2 − 2a2 1 + |ξ|2 + 2η1 (a)<e(ξ)
u(a, ξ) =
(S9)
(S10)
(S11)
where the following notations have been used
1 − 2a2
,
e2a2 − 2a2
1 − 2a2
η2 (a) = 2a2
,
e + 2a2
3 − 2a2
η3 (a) = 2a2
,
e − 2a2 + 2
η1 (a) =
(S12)
(S13)
(S14)
and R(a) represents vector a rotated by an angle φk = arg(1 − |ξ|2 + 2=mξ). The above
analytical expressions originate from the Hartree term of the potential (S2), contribution
from the exchange potential (S3) behaves as exp (−R2 /4) and is neglected.
sup-6
A a/w expansion of the interaction energy (S4) between a 1ē defect at site k and a 2ē
−3
bubble at site i up to the Rik
terms:
δU1 (Rik ) =
µi R̂ik u(ai , ξi ) + v(ai , ξi ) cos(2φi − 2θik )
+
.
2
3
Rik
2Rik
(S15)
Similarly, interaction energy between a 3ē defect and a 2ē bubble is:
3µi R̂ik 3 [u(ai , ξi ) + v(ai , ξi ) cos(2φi − 2θik )]
+
.
(S16)
2
3
Rik
2Rik
P
n
It is worth noting that due to the symmetry of a triangular lattice j cos(φi − θij )/Rij
= 0,
where the summation runs over all sites of an ideal crystalPand n is a positive integer. Also,
3
the following result is used in what follows: α = ζ1 = w3 j 1/Rij
≈ 11.03.
Effective Hamiltonian of a single charge defect in a 2ē bubble crystal in the presence of
elongations is derived using (S4) and (S8)-(S16), and is given by (3) of the main text:
"
!
#
X
µ
R̂
v
i
i ik
∓
H± =
+
cos (2ϕi ) + Uf± (i))
2
3
R
2R
ik
ik
i6=k
δU3 (Rik ) =
+
1 X µi · µj − 3(µi · R̂ij )(µj · R̂ij )
,
3
2 i6=j; i,j6=k
Rij
(S17)
where (+) is for 3ē and (−) is for 1ē defects, ϕi = φi − θi,k . The following notations are
introduced:
vi
±
Uf (i)
where
Uf± (ai , ξi )
=
= v(ai , ξi ),
= Uf± (ai , ξi ),
1
α
∓
3
3
w
2Rik
1
u(ai , ξ) + U (0, ai , ξi ) ,
2
(S18)
(S19)
(S20)
and U (0, ai , ξi ) is defined in (S4). The first two terms of (S17) represent an effective onebody (one-bubble) energy, the third term describes a formation energy at the bubble site i
due to the presence of a defect, and the fourth term represents an effective interaction due to
misalignment of elongated bubbles orientations. In the presence of multiple defects located
at sites k,
"
"
#
#
X
P
µ
R̂
v(a
,
ξ
)
i
i
ik
i
H ± = i6=k
∓
+
cos(2ϕi ) + Uf± (ai , ξi )
2
3
R
2R
ik
ik
k
+
1 X µi · µj − 3(µi · R̂ij )(µj · R̂ij )
.
3
2 i6=j; i,j6=k
Rij
(S21)
When charged defects are introduced into 2ē bubble crystal two effects contribute to
the lowering of the total energy: displacement of 2ē bubbles surrounding the defect from
the sites of an ideal bubble crystal and elongations of the 2ē bubbles. In the numerical
solution these two effects are included simultaneously. Finding an analytical solution of our
sup-7
model we also incorporate both effects, but treat them separately. First, we calculate how
much energy it costs to create a charge defect in an ideal bubble crystal. Next, we allow
displacement of round 2ē bubbles in order to lower the total energy. Finally, we lower the
energy by introducing elongations of 2ē bubbles. While this procedure is approximate, it
captures essential physics and allows us to understand numerical results.
1.
Energy of defects in a perfect triangular crystal in the absence of elongations
Consider a 2ē bubble crystal with a microscopic number Nd defects with charge 2 − σd
(σd = ±1). At a fixed filling factor the total number of electrons 2N on the top LL is fixed.
We consider defects to be many bubble crystal constants w apart. The ground state is a
triangular lattice of N 2ē bubbles. When charged defects are present, the total number of
bubbles has to change to N + Nd σd /2, as we keep the total number of electrons the same.
Assuming all bubbles to be still arranged in a triangular lattice, we find that the change in
lattice constant is δw = −wσd Nd /(4N ). Then the energy of the lattice with defects is:
1
σd Nd (w+δw)
(w+δw)
N 1− 1−
2
+ Nd d
,
(S22)
2
2 N
where w
2 is the energy required to add one 2ē bubble into the bubble crystal with lattice
constant w, w
d is the energy of a defect embedded in an otherwise ideal triangular lattice.
From (S4) we obtain that w
2 is
X
(2)
2 =
ζk ak /w2k+1 + Ũ (0),
(S23)
k=0,1,2
√
where Ũ (0) = U (0)/2 = 833 π/2048 is the formation energy, a0 = 4, a1 = 14, a2 = 765/4.
Since w ≈ 8λ we restrict asymptotic expansions to three terms. This approach is justified
by the comparison with numerical results. Similarly, interaction energy between a defect
P
P
(d)
(d)
and 2ē bubbles is Ud = k=0,1,2 bk /r2k+1 , making d = k=0,1,2 ζk bk /w2k+1 +ud , where the
√
formation energy of a 1ē defect is u1 = 0, while for the 3ē defect u3 = 77463 π/65536. For
1ē defect b0 (1) = 2, b1 (1) = 13/2, b2 (1) = 117/2, and for 3ē defect b0 (3) = 6, b1 (3) = 45/2
and b2 (3) = 531/2. Parameters ζ are ζ0 = −4.2, ζ1 = 11.03 and ζ2 = 6.76.
Using expressions for 2 and d we find the total energy of Nd defects
σd Nd
∂2
1
1
Nd Ed = N 1 − 1 −
2 (w) +
δw + Nd d (w) − N 2 (w)
(S24)
2
2 N
∂w
2
and energy per defect:
Ed = −
2.
σd ∂2
σd − 2
w+
2 + d .
8 ∂w
4
(S25)
Energy of defects in a distorted crystal in the absence of elongations
We now consider decrease of the total energy when 2ē bubbles are allowed to adjust
their positions while retaining their round shape. We approach this problem in the spirit of
Fisher, Halperin and Morf[25]. Due to the presence of charged defects 2ē bubbles at lattice
sup-8
sites Ri experience displacements u(Ri ) from their equilibrium positions. It is assumed
that u(Ri ) w. In the framework of elasticity theory, the energy associated with such
displacements up to the second order in u(Ri ) is given by
1X
Παβ (Ri , Rj )uα (Ri )uβ (Rj )
Ed ({ui }) =
2 i,j
X
X
2
−
δVα1 (Ri )uα (Ri ) −
δVα,β
(Ri )uα (Ri )uβ (Ri )
(S26)
i6=i0
i6=i0
where Π is the spring constant matrix,
δVα1 =
∂
[U2 (r) − Ud (r)] ,
∂rα
(S27)
and
1 ∂2
[U2 (r) − Ud (r)] .
=
2 ∂rα ∂rα
Here the potential energy describing the bubble lattice U2 is given by
2
δVα,β
U2 (Rij ) =
4
14
765
+ 3 +
,
5
Rij Rij 4Rij
(S28)
(S29)
the interaction energy of 2ē bubble at site i with a 1ē defect at site k Ud=1 is given by
U1 (Rik ) =
2
13
117
+
+
,
3
5
Rik 2Rik 2Rik
(S30)
the interaction energy of the 2ē bubble at site i with a 3ē defect at site k Ud=3 is given by
U3 (Rik ) =
6
45
531
+
+
.
3
5
Rik 2Rik 2Rik
(S31)
Multipole terms in expansions (S29-S31) appear in magnetic field as a result of the shape
of the electron wavefunction in the second LL. Because w is large compared to magnetic
length, and Rik is several w, we restrict these asymptotic expansions to three terms.
A spring constant matrix is determined in terms of its Fourier-transform, which is related
2
to the Fourier-transforms of the potentials δVα1 and δVα,β
by
2
(q) =
δVα,β
1X
Vγ,γ δαβ + Ac Πdαβ
2 γ
and
δVα1 (q) = −iAc
X ∂Πdγγ
γ
∂qa
,
(S32)
(S33)
where Ac is the area of the elementary cell of the bubble lattice. Assuming a neutralizing
background and writing the potential in the form
X ak
,
(S34)
V (r) =
2k+1
r
k
sup-9
we obtain an explicit expression for Π:
A2c Παβ (q)
qα qβ X Ξk ak
= 2π
+
q
w2k+1
k
4k + 6
2
q δαβ +
qα qβ ,
2k − 1
(S35)
where constants Ξ0 = 0.26 , Ξ1 = 2.07 and Ξ2 = 6.34. The change of energy of the lattice
in terms of Fourier-transformed quantities is given by
Z
dq
1
Ed ({ui }) =
Παβ (q)uα (q)uβ (q)
2
4π 2
Z
Z
dq
dqdk 2
1
−
δVα (q)uα (q) −
δV (q − k)uα (k)uβ (−q).
(S36)
2
4π
16π 4 α,β
Minimization of this expression assuming
qα
uα (q) = 2 f 1 + cq + dq 2
q
(S37)
yields a decrease in activation energy of the defects caused by displacements u(Ri ). The
obtained energy reduction is very close to the numerical results shown in Fig. 3a of the main
text, where dashed curves show the energy needed to create 1ē and 3ē electron defects in the
absence of elongations. The corresponding displacements are also close: the bubbles nearest
to the defect are displaced by ∼ 0.1w in numerical simulation compared with ∼ 0.08w
from analytical results, and for the displacement of the next nearest neighbors we obtained
∼ 0.03w numerically vs ∼ 0.02w analytically.
3.
Energy of defects in the presence of elongations
In our analytical approach we minimise elongation-dependent energy (S17) and combine
it with the result of displacement minimization (S36). Assuming that a/R 1, minimization of the first term in (S17) provides the zeroth order result. The ground state of
a bubble crystal with 1ē defect is obtained when φi − θij = 0, π, and for a 3ē defect at
φi − θij = π/2, 3π/2. These states are “vortex” (antivortex) and “2D-hedgehog” (antihedgehog) textures. Vortex and antivortex texture of a 1ē defect are almost degenerate at relevant
temperatures; similarly degenerate are headgehog and antihedgehog energies.
Minimization (S17) in the limit of sufficiently large Rik provides the values of the separation vector ai and the mixing parameter ξi . The effect of the interaction term is evaluated
approximating a bubble located far away from the defect as being surrounded by bubbles
with the same parameters a and ξ. The result of such minimization procedure is
4
8
1 3 27/6 w4
1 3
w2
ai = 1 1 3
−
(S38)
4
2 6 3 2 α 2 Rik
35/2 α 3 Rik
4/3
2w2 1
3ē
ξi = −1 + i 2
(S39)
2α 3 r
4/3
2w2 1
1ē
ξi = −1 +
,
(S40)
2
2α 3 r
and corresponding activation energies are plotted in Fig. 3a of the main text (solid lines).
Although these energies quantitatively differ from those obtained numerically (dotted lines),
sup-10
qualitatively both approaches (i) result in the decrease of activation energy by an order of
magnitude compared to the case of round bubbles (dashed lines), and (ii) predict hedgehog
textures around 1ē defects and vortex texture around 3ē defects.
An important insight obtained from the analytical model is that textures associated with
an isolated defect are extended over a finite distance L ∼ 10.5w from the charge defect. As
distance Rik from the center of the charge defect increases, bubble elongation ai decreases.
At Rik ∼ L elongations ai = 0 (round bubbles) become energetically favorable. This happens
for defect density that corresponds to a change in the dependence of activation energy on
the defect separation in our numerical simulation shown in Fig. 3d. We now discuss the
significance of these findings.
4.
Defect crystal, melting and insulator to metal transition
At low defect density ρd < 4L2 textures from different defects do not overlap. At a
given T and ν ∗ an equilibrium density of defects is established by electrons coming from
and leaving to Ohmic contacts, a process accompanied by the formation or destruction of
charge defects. Other channels of equilibration, such as annihilation of defects, are very
slow. We estimate a characteristic energy barrier to be ∼ 10K for hedgehog defects, ∼ 15K
for vortex defects, and ∼ 0.5K for recombination of the vortex and hedgehog, rendering such
processes ineffective at experimentally relevant temperatures T < 0.15 K. These energies
were numerically evaluated for ν = 0.22 taking into account symmetries of defect textures.
The density equilibrium density of defects at small density is given by activational dependence.For a given temperature, there is equilibrium density of defects and the corresponding
defect crystal.
Due to residual Coulomb interactions, defects form a superlattice superimposed on the
bubble crystal, somewhat similar to the Wigner[3] or Abrikosov[17] lattice, an arrangement
confirmed by numerical simulations When defect separation becomes < 2L textures of neighboring defects start to overlap. It is important to realize that in our analytical model the
two-body interaction (S17) represents an XY-model [27]. This is transparent if we take a
continuum limit ϕi → ϕj , where only cos (ϕi − ϕj ) term is important. For the XY-model the
vortex and hedgehog textures, which minimize the 1/R2 interaction of the bubble system
with defects, constitute topological excitations. We note that XY-model physics characterizes dipole-dipole interactions of (S17) even if continuum limit is not applied [34]. Thus, for
ρd > 4L2 energy (S17) includes interaction between elongated bubbles belonging to different
defects, described by the XY model. Proceeding within the framework of our analytical
model and calculating energy caused by such interaction, we take the lower bound on the
”exchange constant” J, which is the µi · µj term in (S17). We assume that J is defined by
a characteristic magnitude of elongations a ∼ λ in the region where topological excitations
overlap, which sets the lower bound to J ∼ e2 a2 /κL3 . Then, as in the XY-model, energy is
logarithmic,
E = πJ ln(L/w),
(S41)
where L is the size of the system and the core of topological excitations is taken to be of the
order of the bubble crystal lattice constant w. It is this logarithmical dependence that arises
in numerical simulations, Fig. 3d. Thermodynamical properties of the system are defined
by the free energy, which includes the entropy of topological excitations and is given by[27]
F = (πJ − 2T ) ln(L/w).
(S42)
sup-11
Such system must exhibit the Berezinski-Kosterlitz-Thouless (BKT) transition [18, 19] at
TKT = πJ/2. However, the thermodynamic transition in the RIQHE regime differs from
conventional BTK transitions, e.g., discussed in [28], due to the finite size of topological
defects. For ρd < 4L2 there is no overlap between neighboring defects (their textures), and
thus, the system cannot undergo the BKT transition. Temperature TL , at which textures
from neighboring defects begin to overlap (ρd = 4L2 ) is much higher than TKT ≈ 5 mK,
estimated analytically or extracted from the slope of numerically obtained curves in Fig. 3d.
Thus, TL TKT in both analytical and numerical calculations and TKT itself is not observed in our experiments. The unbinding of topological defects at TL required to avoid the
divergence of logarithmic interactions (S42) constitutes melting of the defect rather than the
bubble crystal. The TL ’s at different filling factors obtained in our calculations are rather
close to the experimentally observed temperatures of metal-insulator transitions.
5.
The role of residual charge impurities on the metal-insulator transition temperature
The difference between analytically and numerically obtained TL are primarily due to
the simplified energy minimization procedure in the analytical model. Quite remarkable,
however, is about 30% difference between numerically calculated TL and experimentally
measured metal-insulator transition temperatures. We attribute this difference to the role
of disorder not accounted for in our models.We will devote a separate paper to simulations
of disorder, but will now briefly sketch some of our results.
The 2D electron gas is separated from the donor layers by symmetric spacer layers of
approximately 100nm. Ionized impurities in the doping layer produce a smooth random
potential in the quantum well. For large spacers this potential has little effect on the bubble
crystal: our numerical simulations show that the ground state of the quantum Hall liquid
in the range of filling factors corresponding to 2ē bubble crystal in the presence of ionized
impurities located further than 4λ from the 2D gas is still a (slightly deformed) 2ē bubble
crystal.
However, even undoped GaAs has a residual acceptor density ∼ 1014 cm−3 and a smaller
concentration of residual donors. Numerical simulations show that depending on the filling
factor and on separation from the 2d layer, certain charged impurities within ∼ 3λ of the
quantum well lead to formation of charge 2ē complexes. Negatively charged impurities form
a 2ē charged complex with a 1ē bubble defect, and 2ē bubbles surrounding bubble defect
elongate and form a hedgehog. Positively charged impurities form a 2ē charged complex with
a 3ē bubble defect, and 2ē bubbles surrounding bubble defect elongate and form vortex. In
contrast to unbound 1ē or 3ē defects , these complexes are strongly localized by charged
defects and do not participate in the transport. However, the presence of these complexes
increases the overall density of topological defects, lowers the density of the unbound defects
needed for melting transition discussed above, and the observed TL .
E.
.
The effect of strain on electronic transport
sup-12
1.
Effect of strain on textures of elongations
In order to evaluate the effect of strain on the elongation of bubbles we consider bubbles located relatively far from the core of the defect so we can neglect the term inversely
3
proportional to Rij
in the formation energy (S20). We limit interactions to the nearest
neighbors and assume that elongations ai vary slowly with the distance from the core. Due
to large distance between the bubble and the defect, φi can be considered constant for the
bubble and its nearest neighbors. Small deviations δφi of φi change the total energy, which,
expanding (S17), we express as δE = βi (δφi )2 , where
r
1−
βi =
2<eξ
1+|ξ|2
2
a
2
.
2 2a2
2<eξ
Rik e − 2a2 1 + η1 (a) 1+|ξ|
2
(S43)
Addition of a shear strain εxx = −εyy = ε into the model is done similarly to the case
of stripe phase [22] using the “deformed coordinate system” obtained by the transformation
of coordinate system x → x(1 + ε), y → y(1 − ε). In this coordinate system the lattice is
identical to the unstrained one, but the interaction potential becomes anisotropic:
q
∂VHF
VHF (q) → VHF
qx2 (1 − ε)2 + qy2 (1 + ε)2 = VHF (q) + ε
q cos(2φq )
(S44)
∂q
The linear in strain contribution in the real space Hamiltonian can be written as Hε =
−εUfε (a, ξ) cos(2φi ) with
Z
d2 q ∗
∂VHF
ε
Uf (a, ξ) =
ρξ,aêx (q)
q cos(2φq )ρξ,aêx (q) .
(S45)
2
(2π)
∂q
P
n
Terms of the form j f (Rij
) are unaffected by the presence of strain due to triangular lattice
symmetry. Strain induces small rotations of the elongation of bubbles δφε i = εγi sin(2φi ),
where γi = β1i Ufε .
The angles describing elongation vectors become strain-dependent
π
3π
±
φi =
+ θik ±
+ εγi sin(2θik ) ,
(S46)
4
4
where (+) corresponds to the hedgehog and (−) the vortex texture.
2.
Hopping and conductivity in the presence of strain
When charge defect moves on the bubble lattice the textured pattern has to readjust for
the new center of the defect. A superposition matrix element between the initial state of
a bubble characterized by an elongation vector a and the rotated one with an elongation
vector a + δa is given by
p(δa) = hΨ∗a (r1 , r2 )|Ψa+δa (r1 , r2 )i =
2
2
δa2
δa2
2
2
ia×δa− δa2
2
iδa×a−2b2
1−
1 + |ξ| − <eξ
− 2e
b 1 + |ξ| − <eξ 1 − 2b
e
2
2
i−1
h
2
2
× 1 − 2e−2a a2 1 + |ξ|2 + 2 1 − 2a2 e−2a <e(ξ)
,
(S47)
sup-13
where b = a + δa/2. For small rotations δφ the rotation probability |p|2 becomes p(δφ)2 =
1 − p0 (a)(δφ)2 , where
i
n
h 2a2
4e + 1 |ξ|2 + 1 + 4<eξ + 2a2 <eξ |ξ|2 + 4<eξ + 1
p0 (a) = 2a6 |ξ|2 + 2<eξ + 1
h
i
h
i
2
2
2
2
+ a4 e2a 3 |ξ|2 + 1 − 2<eξ |ξ|2 + 1 + 4(<eξ)2 + e2a a2 |ξ|2 + 1 − 2(<eξ)2
o
2
2
2
2
4a2 2
4
− 2a |ξ| + 3<eξ + 1 |ξ| + 4<eξ + 1 − e a |ξ| + 1 2|ξ| + <eξ + 2
i−2
h 2
2
2a
2
2
.
(S48)
× e − 2a
1 + |ξ| + 2 1 − 2a <e(ξ)
If a defect is displaced by a small distance δ along the x̂ direction, the angle θi,k changes
by sin θik /Ri,k δ, while if the defect moves along ŷ direction the change is − cos θi,k /Ri,k δ.
Combining these expressions with the phase of elongation vector (S46) we find the change
in angles φi for a defect hoping event
w sin θi,k
,
Ri,k
w cos θi,k
δy φ±
.
i = −[1 ± εγi cos(2θi,k )]
Ri,k
δx φ±
i = +[1 ± εγi cos(2θi,k )]
Introducing these quantities into matrix elements of (S48) we find
w2
π
2
2
|p±
(δφ
)|
=
1
−
,
p
[1
±
2εγ
cos(2θ
)]
sin
θ
+
(1
+
s)
i
i
i,k
i,k
s
2 0
Rik
4
(S49)
(S50)
(S51)
where s = −1 corresponds to the hoping of the defect along x̂ and s = 1 along ŷ direction.
The conductivity σxx(yy) is proportional to the square of the matrix element of position
(displacement) operator rx(y) , which is proportional to the product of all |p(δφi )|2 . The ratio
between conductivities due to the motion of 1ē and 3ē defects is given by
2
w2
θi,k + (1 + s) π4
2 p0 [1 − 2εγi cos(2θi,k )] sin
σ1e Y 1 − Ri,k
.
=
(S52)
2
w2
π
σ3e
1
−
p
[1
+
2εγ
cos(2θ
)]
sin
θ
+
(1
+
s)
2
0
i
i,k
i,k
4
i
R
i,k
We see that although strain leads to very small rotations of the bubbles and large overlap
between initial and final states during defect motion, the large number of bubbles participating in the defect motion (> 300) leads to a giant amplification of the effect of strain.
Thus our model explains both the symmetry aspect and unusually large magnitude of the
strain effect on resistivity observed experimentally.
3.
Temperature range for mobile topological defects
We observe effects of strain even above TL . After melting of the defect crystal, topological
defects determine the resistivity as long as the bubble crystal is viable. The bubble crystal
itself is going to melt, e.g., due to dislocations. We estimate the melting temperature of the
bubble crystal using the Thouless formula [25, 29]:
√ √
√
4e2 πns
4 2e2 π
= √
,
(S53)
Γ=
4
Tc κ
3Tc wκ
sup-14
where ns is the bubble density, w is the Wigner lattice constant, e is the electron charge,
κ is the dielectric constant, Tc is the melting temperature. The factor 4 comes from the 2ē
charge of the bubbles. The dimensionless parameter Γ = 78 − 130 according to [29, 35–37].
The estimated Tc ∼ 400 mK > TL .
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(2001).
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levels: stripes, bubbles and the Wigner crystal,” Physica E 12, 43 – 45 (2002).
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