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†
√n + 1c†
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
at small q (qlB≪1). The AP mode is a state orthogonal toˆK†
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
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
The red shift for the AP mode at odd ν (QH ferromagnets) is filling factor dependent, it
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
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-
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†
q=0, and 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
n1, n2, q
= V (q)hm1n1(q)hm2n2(−q)Q†
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
operators in parentheses in Eq. (3) differ from the expression forˆH↓↑
mis the Laguerre polynomial, q±= ∓
√2(qx±iqy)]. Expressions for the first and the third
†by replacement of
Q†-operators’ indexes: m2n2→ m2n2, and m1n1→ m1n1correspondingly. Besides, we may
†. 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
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
the second-order approximation yields
n1, n2, q
n1+ n2− m1− m2
n1+ n2− m1− m2
The LL number indexes ni, n′
iand mi, m′
irun from 0 to infinity, however only terms for
being not subject to this condition, have zero numerators.) The expression for another
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†
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↑↑
√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
correlation shift of Kohn’s mode for the ν=1 QH ferromagnet.
Substituting the termsˆH↓↑
according to commutation rules for exciton operators,12one finds
†into Eq. (5) and calculating the commutators
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
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
found for the self-consistently computed formfactor F(q) (triangles) and for the strict 2D limit
|/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
|/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.
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9 Download full-text
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