Antiphased CyclotronMagnetoplasma Mode in a Quantum Hall System
Abstract
An antiphased magnetoplasma (MP) mode in a twodimensional 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 firstorder perturbation approach in terms of the {\it ee} 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 secondorder perturbation approach, is proportional to the effective Rydberg constant in a semiconductor material. Comment: Submitted to Phys. Rev. B
arXiv:0812.1489v1 [condmat.meshall] 8 Dec 2008
Antiphased CyclotronMagnetoplasma 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
MaxPlanckInstitut f¨ur Festk¨orperforschung,
Heisenbergstr. 1, 70569 Stuttgart, Germany
(Dated: December 8, 2008)
An antiphased m agnetoplasma (MP) mode in a twodimensional 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 ﬁrstorder perturbation approach in terms of
the ee 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 secondorder 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
manyparticle 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 singleelectron gap (which is the Zeeman or cyclotron,
or combined one); and (ii) a correlation shift induced by the electronelectron (ee) 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 nonvanishing 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 spinup a nd spindown electrons, and the antisymmetric 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 spinﬂip
and spindensity 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 secondorder corrections to the excitation energies accurately reproduce
the observed eﬀect. The correlation shift for the AP mode is nonvanishing 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 insitu
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 insitu. By continuously tuning the
3
angle we were able to increase the Zeeman energy while keeping the cyclotron energy ﬁxed.
This reduced thermal spinﬂip 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 twoﬁber 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 chargecoupled device camera.
Spectral r esolution o f the system was about 0.03 meV.
Narrow QWs were chosen to maximize energy gaps separating the sizequantized 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 nonlinearity 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 qdisp 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 interLL
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 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, 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/ν curve that describes
the correlation shifts for the antisymmetric mode in another QH system, namely that for
the cyclotron spinﬂip mode in a spinunpolarized 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 spinﬂip 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 spinﬂip mode is represented by the Zeeman triplet. The
ee correlation nature of red shift for the cyclotron spinﬂip 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 singleelectron LL states. Second,
in the development of the perturbative approach one is forced to use a nonorthogonal basis
of twoexciton states. These a re created by action of the interaction Ha miltonian on the
singleexciton basis, when considering ﬁrstorder 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 twofold 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 spinﬂip 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
secondorder energy corrections, we present it as a combination of twoexciton 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
−
)
n−m
L
n−m
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 zeroorder 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 ﬁrstorder
approximation, vanish (E
(1)
0
is the ground state energy calculated to the ﬁrst order), whereas
the secondorder 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 nondiagonal 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+
√
2X
↑
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, detM
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 Qoperators 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 gfactor, and 0 ↑i is the ν = 2 QH ferromagnet
realized in the situation when the gfactor 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 secondorder correc
tion. In Eq. (7) terms containing only squared moduli of the hfunctions 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
/κ
2
~
2
≈11.34 meV units. This value is nearly 2/3 of the corre
lation shift for the ν = 2 cyclotron spinﬂip mode ∆
SF
= −0.1534,
12
which is in surprisingly
good agreement with the experimental 1/ν 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
electronelectron 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
+Ω
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 spinﬂip mode in an unpolarized QH system are quantum
excitations even at zero q — related to the existence of both the spinup and spindown
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 spinﬂip mode (open dots). The solid line shows the
2∆
(ν=2)
SF
/ν dependence. In the inset, theoretical values for ∆
SF
 and ∆
a
 at ν = 2 and ν = 3
found for the selfconsistently 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
/ν are shown by dashed and dotdashed
lines. C ir cles and the solid line represent the experimental data. I llustration of singleelectron
transitions involved in the S; S
z
i=1; 0icomponent of the cyclotron spinﬂip 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 secondorder correlation shifts are empirically well described
by the same 1/ν 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
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C. Kallin and B.I. Halperin, Phys. Rev. B 30, 5655 (1984).
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5
K. W. Chiu and J. J. Quinn, Phys. Rev. B 9, 4724 (1974).
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T. Ando, A. B. Fowler, F. Stern, Rev. Mod. Phys. 54, 437 (1982).
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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 nonzero 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 (ButterworthHeinemann, 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. SC. Luo, Sh.L. Chuang, S. SchmittRink, and A. Pinczuk, Phys. Rev. B 48, 11086 (1993).
 CitationsCitations16
 ReferencesReferences24
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