# Antiphased Cyclotron-Magnetoplasma Mode in a Quantum Hall System

**ABSTRACT** An antiphased magnetoplasma (MP) mode in a two-dimensional electron gas (2DEG) has been studied by means of inelastic light scattering (ILS) spectroscopy. Unlike the cophased MP mode it is purely quantum excitation which has no classic plasma analogue. It is found that zero momentum degeneracy for the antiphased and cophased modes predicted by the first-order perturbation approach in terms of the {\it e-e} interaction is lifted. The zero momentum energy gap is determined by a negative correlation shift of the antiphased mode. This shift, observed experimentally and calculated theoretically within the second-order perturbation approach, is proportional to the effective Rydberg constant in a semiconductor material. Comment: Submitted to Phys. Rev. B

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**ABSTRACT:**The linear electromagnetic response of a uniform electron gas to a longitudinal electric field is determined, within the self-consistent-field theory, by the linear polarizability and the Lindhard dielectric function. Using the same approach we derive analytical expressions for the second- and third-order nonlinear polarizabilities of the three-, two- and one-dimensional homogeneous electron gases with the parabolic electron energy dispersion. The results are valid both for degenerate (Fermi) and non-degenerate (Boltzmann) electron gases. A resonant enhancement of the second and third harmonics generation due to a combination of the single-particle and collective (plasma) resonances is predicted.Physical Review Letters 03/2014; 113(2). · 7.73 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We have theoretically investigated magnetoplasma oscillations of a 2D electronic system with spin–orbit interaction (SOI) in the Bychkov–Rashba model. Accounting for SOI results in new branches of magnetoplasmons as compared with the spinless case where the Bernstein modes exist only. A remarkable feature of the problem in question is magnetic field controlled interaction of branches (multiple anticrossings). This is manifested in the spectra of plasmon absorption of electromagnetic radiation.Physica E Low-dimensional Systems and Nanostructures 01/2010; 42(4):944-947. · 1.86 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Using the inelastic light scattering technique we observe the spin excitations for a quantum Hall ferromagnet with charged defects. The lowest energy spin excitation branch behaves as the theoretically predicted collective spin precession (“cyclotron”) mode of the spin-texture liquid. A coupling between the spin-exciton and the “cyclotron” mode indicates that the existing theory of noninteracting spin texture liquid does not provide a fully adequate description for the ground state of a quantum Hall ferromagnet with charged defects.Physical review. B, Condensed matter 01/2013; 87(4). · 3.66 Impact Factor

Page 1

arXiv:0812.1489v1 [cond-mat.mes-hall] 8 Dec 2008

Antiphased Cyclotron-Magnetoplasma Mode

in a Quantum Hall System

L.V. Kulik1, S. Dickmann1, I.K. Drozdov1, I.S. Zhuravlev1,

V.E. Kirpichev1, I.V. Kukushkin1,2, S. Schmult2, and W. Dietsche2

1Institute of Solid State Physics, RAS, Chernogolovka, 142432 Russia

2Max-Planck-Institut f¨ ur Festk¨ orperforschung,

Heisenbergstr. 1, 70569 Stuttgart, Germany

(Dated: December 8, 2008)

An antiphased magnetoplasma (MP) mode in a two-dimensional electron gas

(2DEG) has been studied by means of inelastic light scattering (ILS) spectroscopy.

Unlike the cophased MP mode it is purely quantum excitation which has no classic

plasma analogue. It is found that zero momentum degeneracy for the antiphased

and cophased modes predicted by the first-order perturbation approach in terms of

the e-e interaction is lifted. The zero momentum energy gap is determined by a

negative correlation shift of the antiphased mode. This shift, observed experimen-

tally and calculated theoretically within the second-order perturbation approach, is

proportional to the effective Rydberg constant in a semiconductor material.

PACS: 71.35.Cc, 71.30.+h, 73.20.Dx

The unique symmetry properties of the quantum Hall (QH) electron liquid have stimu-

lated progress in the study of strongly correlated electron systems in perpendicular magnetic

field. In particular, it has been discovered that the simplest excitations of a 2DEG are ex-

citons consisting of an electron promoted from a filled Landau level (LL) and bound to an

effective hole left in the “initial” LL.1,2,3Within the exciton paradigm, the physics of this

many-particle quantum system is reduced to a two-particle problem. This can be solved in an

asymptotically exact way where the parameter rc= EC/?ωcis considered to be small. Here

EC= αe2/κlBis the characteristic Coulomb energy, ωcis the cyclotron frequency, and the

numerical coefficient α < 1 represents the averaged renormalization factor due to the finite

thickness of the 2DEG in experimentally accessible systems. The excitation energy in this

approach is the sum of two terms: (i) a single-electron gap (which is the Zeeman or cyclotron,

or combined one); and (ii) a correlation shift induced by the electron-electron (e-e) interac-

tion. Kohn’s renowned theorem dictates that in a translationally invariant electron system

one of the excitons [magnetoplasma (MP) mode] has no correlation shift at q = 0. This mode

is described by the action of Kohn’s “raising” operatorˆK†

s=?

npσ

√n + 1c†

n+1,p,σcn,p,σon the

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2DEG ground state |0?, where cn,p,σis the Fermi annihilation operator corresponding to the

state (n,p) with the spin index σ=↑,↓ (n is the LL number; p labels the inner LL number,

if, e.g., the Landau gauge is chosen).4Yet, Kohn’s theorem does not ban the existence of

another homogeneous MP mode that has a non-vanishing correlation shift. Precisely two

MP modes should coexist at odd electron fillings ν>1 when the numbers of fully filled spin

sublevels differ by unit, see the illustration in Fig. 1. The symmetric mode is a cophased

(CP) oscillation of spin-up and spin-down electrons, and the anti-symmetric one is an an-

tiphased (AP) oscillation of two spin subsystems. When calculated to first order in terms of

the parameter rc, Kohn’s mode (the CP magnetoplasmon) has the energy3,5

Es(q)=?ωc+νe2q/2κ+O(ECq2l2

B) (1)

at small q (qlB≪1). The AP mode is a state orthogonal toˆK†

?ωc+O(ECq2l2

s|0?. It has the energy Ea(q) =

B) calculated to first order in rc.3Both Coulomb shifts, ∆s,a= Es,a(0) − ?ωc,

thus vanish if calculated up to ∼ rc. So, within this approximation, both MP modes turn

out to be degenerate at q = 0.

Kohn’s MP mode has been a prime subject for the cyclotron resonance studies, and the

validity of Kohn’s theorem has been confirmed scores of times.6It is well established experi-

mentally that homogenous electromagnetic radiation incident on a translationally invariant

electron system is unable to excite internal degrees of freedom associated with the Coulomb

interaction, i.e. ∆s≡ 0. No similar experiments have been performed for the AP mode as

it is not active in the absorption of electromagnetic radiation. Recent development of Ra-

man scattering spectroscopy to the point when it became sensitive to the cyclotron spin-flip

and spin-density excitations7,8,9opened the opportunity to employ this spectroscopy in the

investigation of the AP mode. Here, we report on a direct observation of the AP mode for

a number of odd electron fillings and show that the theoretically predicted zero momentum

degeneracy for Kohn’s and AP modes is in fact lifted due to many particle correlations. We

also show that the second-order corrections to the excitation energies accurately reproduce

the observed effect. The correlation shift for the AP mode is non-vanishing and negative at

q = 0.10

Several high quality heterostructures were studied. Each consisted of a narrow 18÷20 nm

GaAs/Al0.3Ga0.7As quantum well (QW) with an electron density of 1.2 ÷ 2.4 × 1011cm−2.

The mobilities were 3 ÷ 5 × 106cm2/V·s - very high for such narrow QWs. The electron

densities were tuned via the opto-depletion effect and were measured by means of in-situ

photoluminescence. The experiment was performed at a temperature of 0.3K. The QWs

were set on a rotating sample holder in a cryostat with a 15T magnet. The angle between

the sample surface and the magnetic field was varied in-situ. By continuously tuning the

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angle we were able to increase the Zeeman energy while keeping the cyclotron energy fixed.

This reduced thermal spin-flip excitations through the Zeeman gap. The ILS spectra were

obtained using a Ti:sapphire laser tunable above the fundamental band gap of the QW.

The power density was below 0.02W/cm2. A two-fiber optical system was employed in the

experiments.11One fiber transmitted the pumping laser beam to the sample, the second

collected the scattered light and guided it out of the cryostat. The scattered light was

dispersed by a Raman spectrograph and recorded with a charge-coupled device camera.

Spectral resolution of the system was about 0.03meV.

Narrow QWs were chosen to maximize energy gaps separating the size-quantized electron

subbands. This mitigated the subband mixing induced by the tilted magnetic field. Yet, the

mixing effect was important and we put it under close scrutiny. The influence of the tilted

magnetic field on the cyclotron energy was studied for every QW by measuring the energies

and dispersions for the MP and Bernstein modes.11Most accurately this procedure was

performed for the narrowest 18nm QW where the non-linearity was fairly small. Besides, it

is exactly the 18nm QW where the correlation shift reaches its largest value, as it is affected

by the QW width through the renormalization factor α. Therefore, hereafter we will only

address the 18nm QW.

The ILS resonances for both CP and AP modes are shown in Fig. 1. They have quite

different properties. Kohn’s resonance is blue shifted from the cyclotron energy. Its small

momenta dispersion is given by Eq. (1). Experimentally q is defined by the orientation of

pumping and collecting fibers relative to the sample surface. It is 0.7 · 105cm−1for the

spectra in Fig. 1. Kohn’s resonance is well broadened because of linear q-dispersion (1),

and because the momentum is effectively integrated in the range of q ∼ 0.6 ÷ 0.8 · 105cm−1

due to the finite dimension of the fibers. On the contrary, the resonance for the AP mode

is red shifted and does not broaden. In fact, we did not see any appreciable change in

the AP mode energy upon varying the momentum transferred to the 2DEG via the ILS

process. This experimental finding agrees with the first order perturbation theory of Ref.3

which predicts a negligible (compared to the experimental resolution) change of the AP mode

energy at small q, defined by the light momentum. Variation of the AP shift in the accessible

range of magnetic fields and electron densities is also within the experimental uncertainty.

Since dimensional analysis of second order Coulomb corrections to the energies of inter-LL

excitations yields exactly an independence of the correlation shift on the magnetic field, we

assume that the origin of the AP shift should be sought within the second order perturbation

theory.8

The red shift for the AP mode at odd ν (QH ferromagnets) is filling factor dependent, it

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FIG. 1: ILS spectra of Kohn’s (CP) and antiphased (AP) magnetoplasma modes taken at ν = 3.

The arrow indicates the cyclotron energy. The picture illustrates two single electron transitions at

odd filling factors that, when coupled by the Coulomb interaction, give rise to two magnetoplasma

modes.

reduces at larger ν (Fig. 2). Interestingly its value falls on the same 1/ν curve that describes

the correlation shifts for the antisymmetric mode in another QH system, namely that for

the cyclotron spin-flip mode in a spin-unpolarized 2DEG at even ν (Fig. 3). These two kinds

of excitations differ by the total spin quantum number: S = 0 for the AP mode which is

a spinless magnetoplasmon, and S = 1 for the cyclotron spin-flip mode. The latter splits

into three Zeeman components with different spin projections along the magnetic field. As a

consequence, in the experimental spectra of Fig. 2 a single ILS resonance corresponds to the

AP mode, whereas the cyclotron spin-flip mode is represented by the Zeeman triplet. The

e-e correlation nature of red shift for the cyclotron spin-flip mode is confirmed theoretically

in our previous publications,8,12and here we employ a similar approach to calculate the AP

shift at ν=3.

Our technique is a variation of the standard perturbative technique,13although it has

some special features. The first is the usage of the excitonic representation,12,14where the

basis of exciton states is employed instead of degenerate single-electron LL states. Second,

in the development of the perturbative approach one is forced to use a non-orthogonal basis

of two-exciton states. These are created by action of the interaction Hamiltonian on the

single-exciton basis, when considering first-order corrections to the exciton states. The third

feature lies in calculating the exciton shift counted from the ground state energy, and the

latter also has to be taken into account up to the second order corrections.

Because of the two-fold degeneracy of the q=0 MP states we have to employ two single-

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FIG. 2: ILS spectra of the AP magnetoplasma mode (left) and three Zeeman components of the

cyclotron spin-flip mode (right) taken at odd (left) and even (right) filling factors. The arrows

indicate the corresponding cyclotron energies.

exciton states as a bare basis set. As a result, we come to a 2×2 secular equation. The bare

states are |X↓? = Q†

and the exciton operators are defined, e.g., as12,14,15

01|0? and |X↑? = Q†

12|0?, where Q†

mk= Q†

mk q

???

q=0, and Q†

mk= Q†

mk q

???

q=0,

Q†

mk q=

1

?Nφ

?

p

e−iqxpc†

k,p+qy

2,↑cm,p−qy

2,↑

(2)

(Q†

mkqdiffers by changing ↑ to ↓ in the r.h.s.); q is measured in units of 1/lB, Nφis the LL

degeneracy number. The commutation rules of exciton operators define a special Lie algebra.

Consideringˆ Hintas a part of the interaction Hamiltonian relevant to the calculation of the

second-order energy corrections, we present it as a combination of two-exciton operators

ˆ Hint=

e2

2κlB

?

m1, m2

n1, n2, q

?

ˆH↓↓

n1n2q

m1m2

†

+ 2ˆH↓↑

n1n2q

m1m2

†

+ˆH↑↑

n1n2q

m1m2

†

?

,(3)

whereˆH↓↑

n1n2q

m1m2

†

= V (q)hm1n1(q)hm2n2(−q)Q†

m1n1qQ†

m2n2−q, 2πV (q) is the dimensionless 2D

Fourier component of the Coulomb potential, hmn(q) = (m!/n!)1/2e−q2/4(q−)n−mLn−m

[Ln

operators in parentheses in Eq. (3) differ from the expression forˆH↓↑

m (q2/2)

mis the Laguerre polynomial, q±= ∓

i

√2(qx±iqy)]. Expressions for the first and the third

...

†by replacement of

Q†-operators’ indexes: m2n2→ m2n2, and m1n1→ m1n1correspondingly. Besides, we may

define that

ˆH↑↓

...

...

†≡ˆH↓↑

†. As a result of a consistent perturbative study we find that the

correct zero-order MP states C↓|X↓? + C↑|X↑? and the correlation shifts are obtained from

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the equation

ECσ=

?

σ′

Cσ′Mσσ′,(4)

where the quantities M(1)

approximation, vanish (E(1)

σσ′ = ?Xσ|ˆHint|Xσ′? − E(1)

is the ground state energy calculated to the first order), whereas

0δσ,σ′, calculated within the first-order

0

the second-order approximation yields

M↓↓= −(e2/κlB)2

4?ωc

?

σ1,σ2

?

m1, m2

n1, n2, q

?

m′

1, m′

n′

1, n′

2, q′

2

?

?

0

?????

?

Q01,ˆHσ1σ2

n′

m′

1n′

1m′

2q′

2

??

ˆHσ1σ1

n1n2q

m1m2

†,Q†

01

??????0

?

n1+ n2− m1− m2

+N−1/2

φ

0

?????

ˆHσ1σ2

n′

m′

1n′

1m′

2q′

2

?

Q00− Q11,ˆHσ1σ1

n1n2q

m1m2

†??????0

?

n1+ n2− m1− m2

.

(5)

The LL number indexes ni, n′

iand mi, m′

irun from 0 to infinity, however only terms for

which n1+n2−m1−m2=n′

being not subject to this condition, have zero numerators.) The expression for another

1+n′

2−m′

1−m′

2≥ 1 contribute to the total sum (5). (Other terms,

diagonal matrix element M↑↑differs from Eq. (5) by replacements Q01→ Q12, Q†

Q00→ Q11, and Q11→ Q22, whereas the non-diagonal element M↓↑differs from expression

(5) by the absence of the second term in parentheses and the change from Q†

01→ Q†

12,

01to Q†

12in the

first term. Correspondingly, M↑↓is also obtained by omitting the second term and replacing

Q01with Q12. Analysis shows that M↓↑≡ M↑↓, as it should be (both values are real).

Fortunately, the symmetry of the system and Kohn’s theorem simplify the calculations

a great deal. First, note that one solution of Eqs. (4) is actually known. Indeed, the

CP magnetoplasma mode in the zero order is written asˆKs|0? ≡?Nφ

Therefore, substituting C↓= 1, C↑=

necessary identities: M↓↑ ≡ −M↓↓/√2 and M↑↑ ≡ M↓↓/2.16Another root of the secular

equation, det|Mσ1σ2− Eδσ1,σ2| = 0, is just the correlation shift for the AP mode and thus

expressed in terms of the only matrix element (5): E =∆a=3M↓↓/2. Second, considerable

simplifications occur in the calculations associated with Eq. (5). It is evident that theˆH↑↑

?|X↓?+√2|X↑??.

√2 and E = ∆s≡ 0 into Eqs. (4), we obtain two

...

†

terms commuting with Q-operators in Eq. (5) do not contribute to the result. However,

due to Kohn’s theorem, theˆH↓↓

...

†operators do not contribute either. Indeed, consider our

ground state as a direct product of two fully polarized ground states: |0?≡|0↓?⊗|0↑?. Here

|0↓? is the ν=1 ground state with a positive g-factor, and |0↑? is the ν=2 QH ferromagnet

realized in the situation when the g-factor is negative but the Zeeman gap is larger than the

cyclotron gap. In Eq. (5) all terms with theˆH↓↓

...

†operators act only on the ν =1 ground

state and, taken together, yield zero, because sum of these terms would constitute the q = 0

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correlation shift of Kohn’s mode for the ν=1 QH ferromagnet.

Substituting the termsˆH↓↑

...

according to commutation rules for exciton operators,12one finds

†andˆH↑↓

...

†into Eq. (5) and calculating the commutators

∆a= −3m∗

ee4

2κ2?2

?∞

0

qdqV(q)2G(q),(6)

where

G(q) =

∞

?

n2=2

?|h1n2|2(h2

00−2h00h11)

n2− 1

∞

?

+|h0n2|2(h2

00− 2h00h11)−|h01h1n2|2

n2

−|h01h0n2|2

n2+ 1

+

n1=1

?|h1n1h1n2|2

n1+n2−2+|h1n1h0n2|2−|h0n1h1n2|2

n1+n2−1

−|h0n1h0n2|2

n1+n2

??

.

(7)

We emphasize that this result for ∆aincludes all contributions to the second-order correc-

tion. In Eq. (7) terms containing only squared moduli of the h-functions yield the direct

Coulomb contribution. Terms containing ...h00h11 are of exchange origin. (Thus the ex-

change contribution to the correlation shift is positive.)

In the strict 2D limit, V (q)=1/q, and the correlation shift (6)-(7) is equal to −0.1044 if

expressed in the 2Ry∗=m∗

ee4/κ2?2≈11.34meV units. This value is nearly 2/3 of the corre-

lation shift for the ν =2 cyclotron spin-flip mode ∆SF=−0.1534,12which is in surprisingly

good agreement with the experimental 1/ν dependence. Finally, substituting V (q) = F(q)/q

into Eq. (6), one obtains a numerical result for the correlation shift of the zero momentum

AP mode at ν =3, see Fig. 3. Here, the formfactor F(q) is calculated with the usual self-

consistent procedure17. The calculation result looks quite satisfactory compared to the ILS

data, if one takes into account that under specific experimental conditions the quantity rc

can only be considered as a “small parameter” with great reserve.

To conclude, we outline the general meaning of the presented results. It is known that op-

tical methods (including ILS), being in practice the only tool for direct study of cooperative

excitations in a correlated 2DEG, suffer from an inevitable disadvantage: small momenta of

studied excitations, are far off the interesting region corresponding to inverse values of mean

electron-electron distance. Besides, studying the symmetric MP spectra, one only comes to

the results well described by the classical plasma formula (1), which can be rewritten as

Es/? ≈ ωc+Ω2

plasma modes are actually classical plasma oscillations irrelevant to any quantum effects.

p/2ωc(Ωpto denote the 2D plasma frequency). Therefore the CP magneto-

Contrary to this, homogeneous but antisymmetric modes, namely the AP mode in a QH

ferromagnet and the cyclotron spin-flip mode in an unpolarized QH system are quantum

excitations even at zero q — related to the existence of both the spin-up and spin-down

subsystems. The correlation shift, measured in effective Rydbergs, represents therefore a

purely quantum effect. In particular, it includes exchange corrections, which can be taken

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FIG. 3: Main picture: correlation shifts for the AP magnetoplasma mode (solid dots) and for the

|S;Sz? = |1;0? component of the cyclotron spin-flip mode (open dots). The solid line shows the

2|∆(ν=2)

found for the self-consistently computed formfactor F(q) (triangles) and for the strict 2D limit

SF

|/ν dependence. In the inset, theoretical values for |∆SF| and |∆a| at ν = 2 and ν = 3

F(q) = 1 (diamonds). Corresponding functions 2|∆(ν=2)

lines. Circles and the solid line represent the experimental data. Illustration of single-electron

SF

|/ν are shown by dashed and dot-dashed

transitions involved in the |S;Sz?=|1;0?component of the cyclotron spin-flip triplet (ν = 2,4,6,...)

and in the AP mode (ν = 3,5,7,..) is given on the right.

into account neither by classical plasma calculations nor by the random phase approxima-

tion (RPA) approach. Quantum origin, common for both types of antisymmetric excitation,

seems to be a reason why both second-order correlation shifts are empirically well described

by the same 1/ν dependence shown in Fig. 3.

The authors thank A. Pinczuk and A.B. Van’kov for useful discussion and acknowledge

support from the Russian Foundation for Basic Research, CRDF, and DFG.

1Yu.A. Bychkov, S.V. Iordanskii, and G.M. Eliashberg, JETP Lett. 33, 143 (1981).

2Yu.A. Bychkov and E.I. Rashba, Sov. Phys. JETP 58, 1062 (1983).

3C. Kallin and B.I. Halperin, Phys. Rev. B 30, 5655 (1984).

4W. Kohn, Phys. Rev., 123, 1242 (1961).

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6T. Ando, A. B. Fowler, F. Stern, Rev. Mod. Phys. 54, 437 (1982).

7M. A. Eriksson, A. Pinczuk, B. S. Dennis, S. H. Simon, L. N. Pfeiffer, and K. W. West, Phys.

Rev. Lett. 82, 2163 (1999).

8L.V. Kulik, I.V. Kukushkin, S. Dickmann et al., Phys. Rev. B 72, 073304 (2005).

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9

9A.B. Van’kov, L.V. Kulik, I.V. Kukushkin et al., Phys. Rev. Lett. 97, 246801 (2006).

10Positive value of ∆awould not remove the degeneracy but only shifts the degeneracy point to

a non-zero value of q ∼ rc/lB. Such a feature would be physically unjustified.

11L.V. Kulik and V.E. Kirpichev, Phys. Usp. 49, 353 (2006) [UFN 176, 365 (2006)].

12S. Dickmann, I.V. Kukushkin, Phys. Rev. B 71, 241310(R) (2005).

13L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Butterworth-Heinemann, Oxford, 1991).

14S. Dickmann, Phys. Rev. B 65, 195310 (2002).

15A.B. Dzyubenko and Yu.E. Lozovik, Sov. Phys. Solid State 25, 874 (1983) [ibid. 26, 938 (1984)].

16Calculation of all elements Mσ1σ2can be performed by means of formula (5) and by similar

ones. This calculation, giving values satisfying the necessary identities, serves as a good check

for the correctness of our theory.

17M. S-C. Luo, Sh.L. Chuang, S. Schmitt-Rink, and A. Pinczuk, Phys. Rev. B 48, 11086 (1993).

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