Antiphased Cyclotron-Magnetoplasma Mode in a Quantum Hall System

Article (PDF Available)inPhysical review. B, Condensed matter 79(12) · December 2008with24 Reads
DOI: 10.1103/PhysRevB.79.121310 · Source: arXiv
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
arXiv:0812.1489v1 [cond-mat.mes-hall] 8 Dec 2008
Antiphased Cyclotron-Magnetoplasma Mode
in a Quantum Hall System
L.V. Kulik
1
, S. Dickmann
1
, I.K. Drozdov
1
, I.S. Zhuravlev
1
,
V.E. Kirpichev
1
, I.V. Kukushkin
1,2
, S. Schmult
2
, and W. Dietsche
2
1
Institute of Solid State Physics, RAS, Chernogolovka, 142432 Russia
2
Max-Planck-Institut f¨ur Festk¨orperforschung,
Heisenbergstr. 1, 70569 Stuttgart, Germany
(Dated: December 8, 2008)
An antiphased m agnetoplasma (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 m omentum d egeneracy 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 perturb ation 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 pro perties o f 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.
1,2,3
Within 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 r
c
= E
C
/~ω
c
is considered to be small. Here
E
C
= αe
2
/κl
B
is the characteristic Coulomb energy, ω
c
is the cyclotron frequency, and the
numerical coefficient α < 1 represents the averaged renormalization factor due to the finite
thickness of the 2DEG in experimentally a ccessible 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 “ra ising” operator
ˆ
K
s
=
P
npσ
n + 1c
n+1,p,σ
c
n,p,σ
on the
2
2DEG ground state |0i, where c
n,p,σ
is 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).
4
Yet, Kohn’s theorem doe s 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 cop hased
(CP) oscillation of spin-up a nd 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 r
c
, Kohn’s mode ( t he CP mag netoplasmon) has the energy
3,5
E
s
(q)=~ω
c
+νe
2
q/2κ+O(E
C
q
2
l
2
B
) (1)
at small q (ql
B
1). The AP mode is a state orthogonal to
ˆ
K
s
|0i. It has the energy E
a
(q) =
~ω
c
+O(E
C
q
2
l
2
B
) calculated to first order in r
c
.
3
Both Coulomb shifts,
s, a
= E
s, a
(0) ~ω
c
,
thus vanish if calculated up to r
c
. 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.
6
It 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 excitations
7,8,9
opened 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/Al
0.3
Ga
0.7
As quantum well (QW) with an electron density of 1.2 ÷ 2.4 × 10
11
cm
2
.
The mobilities were 3 ÷ 5 × 10
6
cm
2
/V·s - very high for such narrow QWs. The electron
densities were tuned via the o pto -depletion effect and were measured by means of in-situ
photoluminescence. The experiment was performed at a temperature of 0.3 K. The QWs
were set on a rotating sample holder in a cryostat with a 15 T magnet. The angle between
the sample surface and the magnetic field was varied in-situ. By continuously tuning the
3
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 0.02 W/cm
2
. A two-fiber optical system was employed in the
experiments.
11
One fiber transmitted the pumping laser beam to the sample, the second
collected the scattered light and guided it out o f the cryostat. The scattered light was
dispersed by a Raman spectrograph and recorded with a charge-coupled device camera.
Spectral r esolution o f the system was about 0.03 meV.
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.
11
Most accurately this procedure was
performed for the narrowest 18 nm QW where the non-linearity was fairly small. Besides, it
is exactly the 18 nm QW where the correlation shift reaches its largest value, as it is a ff ected
by the QW width through the renormalization factor α. Therefore, hereafter we will only
address the 18 nm Q W.
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 · 10
5
cm
1
for the
spectra in Fig. 1. Kohn’s resonance is well broadened because of linear q-disp ersion (1),
and because the momentum is effectively integrated in the range of q 0.6 ÷ 0.8 · 10
5
cm
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 a ny 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
4
FIG. 1: ILS spectra of Kohns (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, wh en 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 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 latt er 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,
8,12
and here we employ a similar approach to calculate the AP
shift at ν =3.
Our technique is a variation of the standard perturbative technique,
13
although it has
some special features. The first is the usage of the excitonic representation,
12,14
where 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 a re created by action of the interaction Ha miltonian 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 a lso has to be taken into a ccount up to the second order corrections.
Because of the two-fold degeneracy of the q =0 MP states we have to employ two single-
5
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
i = Q
01
|0i and |X
i = Q
12
|0i, where Q
mk
= Q
mk q
q=0
, and Q
mk
= Q
mk q
q=0
,
and the exciton operators ar e defined, e.g., as
12,14,15
Q
mk q
=
1
p
N
φ
X
p
e
iq
x
p
c
k,p+
q
y
2
,
c
m,p
q
y
2
,
(2)
(Q
mk q
differs by changing to in the r.h.s.); q is measured in units of 1/l
B
, N
φ
is the LL
degeneracy number. The commutation rules of exciton operators define a special Lie algebra.
Considering
ˆ
H
int
as 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
H
int
=
e
2
2κl
B
X
n
1
, n
2
, q
m
1
, m
2
ˆ
H
↓↓
n
1
n
2
q
m
1
m
2
+ 2
ˆ
H
↓↑
n
1
n
2
q
m
1
m
2
+
ˆ
H
↑↑
n
1
n
2
q
m
1
m
2
, (3)
where
ˆ
H
↓↑
n
1
n
2
q
m
1
m
2
= V (q)h
m
1
n
1
(q)h
m
2
n
2
(q)Q
m
1
n
1
q
Q
m
2
n
2
q
, 2πV (q) is the dimensionless 2D
Fourier component of the Coulomb potential, h
mn
(q) = (m!/n!)
1/2
e
q
2
/4
(q
)
nm
L
nm
m
(q
2
/2)
[L
n
m
is the Laguerre p olynomial, q
±
=
i
2
(q
x
±iq
y
)]. Expressions for the first and the third
operators in parentheses in Eq. (3) differ from the expression for
ˆ
H
↓↑
...
by replacement of
Q
-operators’ indexes: m
2
n
2
m
2
n
2
, and m
1
n
1
m
1
n
1
correspondingly. Besides, we may
define that
ˆ
H
↑↓
...
ˆ
H
↓↑
...
. As a result of a consistent perturbative study we find that the
correct zero-order MP states C
|X
i + C
|X
i and the correlation shifts are obtained from
6
the equation
EC
σ
=
X
σ
C
σ
M
σσ
, (4)
where the quantities M
(1)
σσ
= hX
σ
|
ˆ
H
int
|X
σ
i E
(1)
0
δ
σ, σ
, calculated within the first-order
approximation, vanish (E
(1)
0
is the ground state energy calculated to the first order), whereas
the second-order approximation yields
M
↓↓
=
(e
2
/κl
B
)
2
4~ω
c
X
σ
1
, σ
2
X
n
1
, n
2
, q
m
1
, m
2
X
n
1
, n
2
, q
m
1
, m
2
*
0
"
Q
01
,
ˆ
H
σ
1
σ
2
n
1
n
2
q
m
1
m
2
#
ˆ
H
σ
1
σ
1
n
1
n
2
q
m
1
m
2
, Q
01
0
+
n
1
+ n
2
m
1
m
2
+N
1/2
φ
*
0
ˆ
H
σ
1
σ
2
n
1
n
2
q
m
1
m
2
Q
00
Q
11
,
ˆ
H
σ
1
σ
1
n
1
n
2
q
m
1
m
2
0
+
n
1
+ n
2
m
1
m
2
.
(5)
The LL number indexes n
i
, n
i
and m
i
, m
i
run from 0 to infinity, however only terms for
which n
1
+n
2
m
1
m
2
=n
1
+n
2
m
1
m
2
1 contribute to the total sum (5). (Other terms,
being not subject to this condition, have zero numerators.) The expression for another
diagonal matrix element M
↑↑
differs from Eq. (5) by replacements Q
01
Q
12
, Q
01
Q
12
,
Q
00
Q
11
, and Q
11
Q
22
, 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
to Q
12
in the
first term. Corresp ondingly, M
↑↓
is also obtained by omitting the second term and replacing
Q
01
with Q
12
. 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, not e that one solution of Eqs. (4) is a ctually known. Indeed, the
CP magnetoplasma mode in the zero order is written as
ˆ
K
s
|0i
p
N
φ
|X
i+
2|X
i
.
Therefore, substituting C
= 1, C
=
2 and E =
s
0 into Eqs. (4), we obta in two
necessary identities: M
↓↑
M
↓↓
/
2 and M
↑↑
M
↓↓
/2.
16
Another root of the secular
equation, det|M
σ
1
σ
2
Eδ
σ
1
, σ
2
| = 0, is just the correlation shift for the AP mode and thus
expressed in terms of the o nly matrix element (5): E =
a
= 3M
↓↓
/2. Second, considerable
simplifications occur in the calculations associated with Eq. (5). It is evident that the
ˆ
H
↑↑
...
terms commuting with Q-operators in Eq. (5 ) do not contribute to the result. However,
due to Kohn’s theorem, the
ˆ
H
↓↓
...
operators do not contribute either. Indeed, consider our
ground state as a direct product of two fully polarized ground states: |0i|0 ↓i⊗|0↑i . Here
|0↓i is the ν =1 ground state with a positive g-factor, and |0 ↑i 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 the
ˆ
H
↓↓
...
operators act only o n t he ν = 1 ground
state and, taken together, yield ze ro, because sum of these terms would constitute the q = 0
7
correlation shift of Kohn’s mode for the ν = 1 QH ferromagnet.
Substituting the terms
ˆ
H
↓↑
...
and
ˆ
H
↑↓
...
into Eq. (5) and calculating the commutators
according to commutation rules fo r exciton operators,
12
one finds
a
=
3m
e
e
4
2κ
2
~
2
Z
0
qdqV(q)
2
G(q) , (6)
where
G(q) =
X
n
2
=2
|h
1n
2
|
2
(h
2
00
2h
00
h
11
)
n
2
1
+
|h
0n
2
|
2
(h
2
00
2h
00
h
11
)|h
01
h
1n
2
|
2
n
2
|h
01
h
0n
2
|
2
n
2
+ 1
+
X
n
1
=1
|h
1n
1
h
1n
2
|
2
n
1
+n
2
2
+
|h
1n
1
h
0n
2
|
2
|h
0n
1
h
1n2
|
2
n
1
+n
2
1
|h
0n
1
h
0n
2
|
2
n
1
+n
2
#
.
(7)
We emphasize that this result for
a
includes 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 ...h
00
h
11
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
e
e
4
2
~
2
11.34 meV units. This value is nearly 2/3 of the corre-
lation shift for the ν = 2 cyclotron spin-flip mode
SF
= 0.1534,
12
which is in surprisingly
good agreement with the experimental 1 dependence. Finally, substituting V (q) = F (q)/q
into Eq. (6 ) , one obta ins 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 procedure
17
. The calculation result looks quite satisfactory compared to the ILS
data, if one takes into account that under specific experimental conditions the quantity r
c
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 pra ctice the only tool for direct study of cooperative
excitations in a correlated 2DEG , suffer f rom 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
E
s
/~ ω
c
+
2
p
/2ω
c
(Ω
p
to denote t he 2D plasma frequency). Therefore the CP magneto-
plasma modes are actually classical plasma oscillations irrelevant to any quantum effects.
Contrary to this, homogeneous but antisymmetric modes, namely t he 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, r epresents therefore a
purely quantum effect. In particular, it includes exchange corrections, which can be taken
8
FIG. 3: Main picture: correlation shifts for the AP magnetoplasma mode (solid dots) and for the
|S; S
z
i = |1; 0i component of the cyclotron spin-flip mode (open dots). The solid line shows the
2|
(ν=2)
SF
| dependence. In the inset, theoretical values for |
SF
| and |
a
| at ν = 2 and ν = 3
found for the self-consistently computed f ormfactor F (q) (triangles) and for the s tr ict 2D limit
F (q) = 1 (diamonds). C orresponding functions 2|
(ν=2)
SF
| are shown by dashed and dot-dashed
lines. C ir cles and the solid line represent the experimental data. I llustration of single-electron
transitions involved in the |S; S
z
i=|1; 0icomponent 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) approa ch. 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 dependence shown in F ig . 3.
The authors thank A. Pinczuk and A.B. Van’kov for useful discussion and acknowledge
suppo r t from the Russian Foundation for Basic Research, CRDF, and DFG.
1
Yu.A. Bychkov, S.V. Iord an s kii, and G.M. Eliashberg, J ETP Lett. 33, 143 (1981).
2
Yu.A. Bychkov and E.I. Rashba, Sov. Phys. JETP 58, 1062 (1983).
3
C. Kallin and B.I. Halperin, Phys. Rev. B 30, 5655 (1984).
4
W. Kohn, Phys. Rev., 123, 1242 (1961).
5
K. W. Chiu and J. J. Quinn, Phys. Rev. B 9, 4724 (1974).
6
T. Ando, A. B. Fowler, F. Stern, Rev. Mod. Phys. 54, 437 (1982).
7
M. A. Eriksson, A. Pinczuk, B. S. Dennis, S. H. Simon, L. N. Pfeiffer, and K. W. West, Phys.
Rev. Lett. 82, 2163 (1999).
8
L. V. Kulik, I. V. Kukushkin, S. Dickmann et al., Phys. Rev. B 72, 073304 (2005).
9
9
A. B. Van’kov, L. V. Kulik, I. V. Kukushkin et al., Phys. Rev. Lett. 97, 246801 (2006).
10
Positive value of
a
would not remove the degeneracy but only shifts the degeneracy point to
a non-zero value of q r
c
/l
B
. Such a feature would be physically unjustified.
11
L. V. Kulik and V. E. Kirpichev, Phys. Usp. 49, 353 (2006) [UFN 176, 365 (2006)].
12
S. Dickmann, I.V. Kukushkin, Phys. Rev. B 71, 241310(R) (2005).
13
L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Butterworth-Heinemann, Oxford, 1991).
14
S. Dickmann, Phys. Rev. B 65, 195310 (2002).
15
A.B. Dzyubenko and Yu.E. Lozovik, Sov. Phys. Solid State 25, 874 (1983) [ibid. 26, 938 (1984)].
16
Calculation of all elements M
σ
1
σ
2
can 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.
17
M. S-C. Luo, Sh.L. Chuang, S. Schmitt-Rink, and A. Pinczuk, Phys. Rev. B 48, 11086 (1993).
  • [Show abstract] [Hide abstract] ABSTRACT: The magnetoplasmon spectrum in a 2D electron plasma is investigated theoretically taking into account the spin—orbit interaction in the Rashba model. The expression is derived for the polarization operator in the semiclassical range of magnetic fields (in which the Fermi energy eF is much larger than the Landau quantum). Approximate analytic formulas are obtained for spin—plasmon branches. The dependences of plasma wave frequencies on the wavevector and magnetic field are calculated from a numerical solution of the dispersion equation. In addition, the magnetic field and frequency dependences of the plasmon absorption intensity are constructed. The absorption intensity at the peak is found to be sensitive to the spin—orbit interaction constant.
    Article · Jan 2010
  • [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.
    Article · Feb 2010
  • [Show abstract] [Hide abstract] ABSTRACT: We report on the observation of a new spin mode in a quantum Hall system in the vicinity of odd electron filling factors under experimental conditions excluding the possibility of Skyrmion excitations. The new mode having presumably zero energy at odd filling factors emerges at small deviations from odd filling factors and couples to the spin-exciton. The existence of an extra spin mode assumes a nontrivial magnetic order at partial fillings of Landau levels surrounding quantum Hall ferromagnets other then the Skyrmion crystal.
    Full-text · Article · Apr 2010
Show more

    Recommended publications

    Discover more