Auger-like Relaxation of Inter-Landau-Level Magneto-Plasmon Excitations in the Quantised Hall Regime

Abstract

Auger relaxation in 2D strongly correlated electron gas can be represented as an Auger-like process for neutral magnetoplasmon excitations. The case of "dielectric" state with lack of free electrons (i.e. at integer filling $\nu$) is considered. Really the Auger-like process is a coalescence of two magnetoplasmons which are converted into a single one of a different plasmon mode with zero 2D wave-vector. This event turns out to be energetically allowed for magnetoplasmons near their roton minima where the spectrum has the infinite density of states. As a result the additional possibility appears for indirect observation of the magnetorotons by means of anti-Stokes Raman scattering. We find the rate of this process employing the technique of Excitonic Representation for the relevant matrix element calculation. Comment: 4 pages

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arXiv:0812.1744v1 [cond-mat.mes-hall] 9 Dec 2008
PHYSICAL REVIEW B Volume, Number1999
Auger-like Relaxation of Inter-Landau-Level Magneto-Plasmon Excitations in the
Quantised Hall Regime
S. Dickmann∗
Institute for Solid State Physics of Russian Academy of Sciences, 142432 Chernogolovka, Moscow District, Russia
Y. Levinson
Department of Condensed Matter Physics,
The Weizmann Institute of Science, 76100 Rehovot, Israel
(Received June 1999)
Auger relaxation in 2D strongly correlated electron gas can be represented as an Auger-like process for
neutral magnetoplasmon excitations. The case of “dielectric” state with lack of free electrons (i.e. at
integer filling ν) is considered. Really the Auger-like process is a coalescence of two magnetoplasmons
which are converted into a single one of a different plasmon mode with zero 2D wave-vector. This
event turns out to be energetically allowed for magnetoplasmons near their roton minima where the
spectrum has the infinite density of states. As a result the additional possibility appears for indirect
observation of the magnetorotons by means of anti-Stokes Raman scattering. We find the rate of
this process employing the technique of Excitonic Representation for the relevant matrix element
calculation.
Auger-type processes (APs) are believed to be the
dominant inter-Landau-level electron scattering mecha-
nism when emission of LO-phonons is suppressed off the
magnetophonon resonance conditions. Auger scattering
determines the population of Landau levels (LLs) in cy-
clotron resonance1,2, anti-Stokes hot luminescence3, and
Integer Quantum Hall breakdown phenomena4,5. One-
electron description of an AP is scattering of two elec-
trons at the same LLs resulting in deexcitation of one
of them to a lower LL and excitation of the other to a
higher LL. If this lower LL is partially filled in the ground
state of 2D electron gas (2DEG), then such a process re-
duces the total number of excited electrons, providing
the 2DEG relaxation. This simple picture is based on
LL equidistance and seems to correspond to real situa-
tion such as in experiments at ν < 11) or in the case
of large LL numbers of initially excited states4,5. On the
other hand it fails when Coulomb corrections to energy of
a free electron are significant and depend on LL number.
Moreover, near an integer ν the deficiency of unoccupied
states in the almost filled LL leads to the conclusion that
the usual AP relaxation would become very rare as it
would be a result of three-particle collisions among two
excited electrons and an effective hole (unoccupied state
at the LL filled in ground state).
It is meanwhile well known that strong Coulomb cor-
relations in the Quantum Hall regime renormalize dras-
tically the 2DEG excitation spectrum. The electron pro-
moted from n-th LL to (n + m)-th one and the effective
hole left at the n-th LL interact with each other; hence
they should be considered as a collective excitation. For
integer filling the spectrum, being of dielectric type (with
Zeeman gap |gµbB| for an odd ν, and with cyclotron gap
¯ hωcif ν is even), is represented by chargeless excitations,
namely: intra-LL spin-waves (m = 0), inter-LL cyclotron
excitations without spin-flip (so-called magnetoplasmons
(MPs) with m ?= 0), and those with spin-flip7,8. In this
representation an Auger-type process could be realized
as a conversion of two MPs with energy in the vicinity of
m¯ hωcinto one MP in the vicinity of 2m¯ hωc.
The lowest energy MP with m = 1 has pronounced
roton type minimum in the energy dependence ǫ(q)
on the 2D wave vector q.7,8,9,10.
the density of states is infinite and this is the rea-
son why the corresponding excitations, magnetorotons,
were detected by means of resonant combination back-
scattering11,12,13though this detection is only possible
due to breakdown of wave-vector conservation (see dis-
cussion in Refs.13,14). In the measured signal only one
other peak of the same MP mode just close to ¯ hωcis ob-
served. It corresponds to the MP with q near the origin,
and satisfying the momentum conservation this peak is
more intensive even though the MPs at q = 0 have much
lower density of states10. Important for a coalescence of
two MPs is the energetic possibility of their conversion
into some other excitation. We see that this process being
allowed for magnetorotons is forbidden for the MPs with
q = 0, since the energy of the final is essentially higher
than 2¯ hωcdue to Coulomb corrections8. Analogously the
coalescence is forbidden for two MPs which are in the
other “suspicious” phase region, namely, near the ǫ(q)
maximum (not observed experimentally as yet) where
the density of states is also infinite. The energy of one
“two-cyclotron” MP is essentially lower than the com-
bined energy of such two MPs near the maxima. Thus,
the mentioned experimental detection of magnetorotons
and the energetic possibility of the considered process are
the reasons explaining our special interest in the MPs co-
alescence near their roton minima. Moreover, it is prefer-
able to find out the generated “two-cyclotron” MP in the
state with small 2D wave-vector, because in this case the
generated MP could be detected by anti-Stokes Raman
Near this minimum

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scattering like in the experiments of Refs.11,12,13. This
is why we will present more detailed results exactly for
this case. We calculate the decay rate of such Auger-like
process.
We solve this problem for the case of “strong magnetic
field”, i.e. in the lowest-order approximation in the small
parameter Ec/¯ hωc, where Ec= e2/κlBis a characteristic
Coulomb energy for electron-electron (e-e) interaction in
2DEG, lBbeing the magnetic length, and κ the effective
dielectric constant. (For B = 10T
lB = 8.1nm and Ec = 14meV). It is well known that
in this approximation the problem of two-particle excita-
tion spectrum for ν = integer can be solved exactly7,8.
Now, our task is the calculation of the transition matrix
element.
Further we employ the so-called excitonic representa-
tion (ER), which is very advantageous for excitations
from a filled LL. Let us label a certain one-electron state
characterized by its LL and spin sublevel by a = (na,σa).
Then the excitations may be considered as effective ex-
citons with energies
¯ hωc = 17.3meV,
ǫab(q) = ¯ hωcm + |gµbB|δSz+ Eab(q),
where m = nb−na,
and the energy Eab has a Coulomb origin. It is of the
order of or smaller than Ec.
We restrict ourselves only to the case of ν = 1 con-
sidering only MPs with na= 0 and nb= 1,2 and with
σa = σb = +1/2. In this case we change the subscript
ab in Eq.(1) to 01 or 02, respectively. The analytical and
numerical calculation of the excitation spectra of these
01 and 02 MPs are presented in8in the strict 2D limit
(S2DL) when the thickness of the 2DEG d satisfies the
condition d ≪ lB. In fact the spectra depend on d but
their shape do not change qualitatively. The function
E01(q) has a roton minimum at q = q0≈ 1.92/lB:
E01(q) = ε0+ (q − q0)2/2M,
where in S2DL M−1≈ 0.28Ecl2
dependence E02(q) is also nonmonotonic, but in the range
0 < qlB < 2.5 does not change more than 0.07Ec. Of
special importance is the difference δ = E02(0) − 2ε0,
which “casually” is numerically small in the scale of Ec,
namely in S2DL δ ≈ 0.019Ec≃ 3 ÷ 4 K for B = 10 ÷
20T, but is positive19. The desired matrix element of the
considered conversion is
(1)
δSz= σb−σa,(σa, σb= ±1/2),
|q − q0| ≪ q0,(2)
Band ε0≈ 0.15Ec. The
M(q1,q2) =02?q1+ q2;1|H|q1,q2;2?01.
Here H is Hamiltonian, the initial state is a two 01MP
state, and the final one is a one 02MP state.
The total 2DEG Hamiltonian is H = H0+Hint, where
the Hamiltonian of the noninteracting electrons is
(3)
H0=
?
n,p,σ
[(n + 1/2)¯ hωc− |gµbB|σ]e+
n,p,σen,p,σ. (4)
Here en,p,σ is the electron annihilation operator at n-th
LL having σ as the ˆ z-component of spin, p = ky is the
intra-level Landau gauge quantum number. Within the
framework of strong magnetic field approximation it is
enough to keep in the interaction Hamiltonian, Hint, only
the terms which conserve cyclotron part of the energy,
or in other words, the terms which commute with H0.
The Coulomb part of the Hamiltonian may therefore be
written in the form
Hint= N−1
?
p,p′,q
n,m,l,k,σ1,σ2
e+
Vnmlk(q)exp[iqx(p′− p)] ·
n,p+qy,σ1e+
m,p′,σ2el,p′+qy,σ2ek,p,σ1, (5)
which provides automatically the cyclotron energy con-
servation rule n + m = l + k, because
Vnmlk(q) = (2π)−1V (q)hnk(q)h∗
lm(q)δn+m,l+k.(6)
We use now dimensionless length and wave-vectors mea-
sured in the units of lB and l−1
total number of magnetic flux quanta in the normaliza-
tion area L2, and V (q) is the 2D Fourier component of
the Coulomb potential averaged with the wave function
in the ˆ z direction (so that in S2DL: V (q) = 2πEc/q), and
B. N = L2/2πl2
Bis the
hnk(q) =
?+∞
−∞
dxχn(x + qy/2)eiqxxχk(x − qy/2) =
?1/2?iqx+ qysign(n − k)
?min(n,k)!
max(n,k)!
√2
?|n−k|
·
e−q2/4L|n−k|
min(n,k)(q2/2)
χn(x) is the normalized n-th harmonic oscillator func-
tion, Lj
nis Laguerre polynomial.
Now we define in ER the states in the matrix element
(3) in order to calculate the last one. Let a be the filled
LL, i.e. in our particular case a = (0,1/2). We des-
ignate ap ≡ ena,p,σawhile bp ≡ enb,p,σbfor every other
one-electron state b. The ER means a replacement of
operators e+
n,p,σand en,p,σ by a set of inter-LL “exci-
tonic” creation and annihilation operators for a ?= b (i.e.
nb?= na, or σa?= σb)
Q+
abq=
1
√N
?
p
e−iqxpb+
p+
qy
2ap−
qy
2, Qabq= Q+
ba−q,
(7)
and intra-LL “displacement” operators Aq and Bq(see
Ref.15). We do not write here the latter ones, because
they, being required for total ER of Hamiltonian (5), are
not used directly for matrix element (3) calculation.
Some commutation rules for operators (7) are the
same as the ones obtained in Ref.15minding the case
a = (n,1/2), b = (n,−1/2). We derive the additional
ones considering a ?= b ?= c:
[Q+
=e−iΘ12
N1/2Q+
Here Θ12 = Θ(q1,q2) = (q1× q2)z/2 =
where α is an angle between q2 and q1. Note that the
bcq1,Qabq2] = 0, [Q+
bcq1,Q+
abq2]
acq1+q2.(8)
1
2q1q2sinα,

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3
considered operators were employed earlier in some other
form as applied to “valley-wave” excitations16and, also,
to spin-waves15,17, when m = 0, |δSz| = 1.
The operator Q+
Here |0? is the ground state where the level a is fully
occupied, whereas b is empty: a+
equivalent to identities A+
q|0? ≡ δ0,q|0?
Qabq|0? ≡ 0.
The choice of the state b depends on a type of problem.
In our case the states entering the matrix element (3) are
abqcreates a abMP: |q;1?ab= Q+
abq|0?.
p|0? = bp|0? ≡ 0. This is
andB+
q|0? ≡
|q;1?02= Q+
As above 01 and 02 stand for ab with a = (0,1/2) and
b = (1,1/2) or b = (2,1/2) respectively. These states are
orthogonal and are eigenstates of the Hamiltonian H in
the limit N → ∞, i.e.
Hint|q;1?02= [E0+ E02(q)]|q;1?02+ ···,
Hint|q1,q2;2?01=
[E0+ E01(q1) + E01(q2)]|q1,q2;2?01+ ··· ,
where E0is the Coulomb ground state energy (Hint|0? =
E0|0?) and the dots correspond to some states having
a norm of the order of Ec/N. The states (9) are the
correct initial and final states in the scattering problem
for a low density gas of MPs, but the scattering matrix
element (3) has to be calculated with higher accuracy,
since M ∼ N−1/2.
Instead of the value (3) it is more convenient to cal-
culate the conjugate one M∗substituting in Eq. (3) the
expressions (9).After one has done the ER transfor-
mation of the Hamiltonian (5) in terms of operators (7)
together with Aqand Bq. then taking into account the
properties of the ground state |0? and the commutation
rules (8) one finds that the only term of Hamiltonian
which contributes to the matrix element (3) is
02 q|0?,
|q1,q2;2?01= Q+
01q1Q+
01q2|0?, (9)
?
q
V1120(q)Q+
01qQ12q.
Using again as tools the properties of operators (7) and
of the state |0? we obtain for q1?= q2
M(q1,q2) = N−1/2?u(q1)e−iΘ12+ u(q2)eiΘ12
− v(q1)e−iΘ12− v(q2)eiΘ12?,
where
u(q) = (25/2π)−1q2V (q)[2 − (qlB)2/2]e−(qlB)2/2,
v(q) = l2
B
0
(10)
?∞
dppu(p)J0(pql2
B).
We returned here to dimensional quantities (also suit-
able redefinition is V (q) → l2
find analytic expression of v(q), involving polynomials,
exponentials, and modified Bessel functions.
The depopulation rate of a 01MPs due to their coales-
cence is
R =1
2
q1,q2
δ [E01(q1) + E01(q2) − E02(q1+ q2)]
BV (q)). In S2DL one can
?
2π
¯ h|M(q1,q2)|2n(q1)n(q2) ·
(11)
where n(q) are occupation numbers of 01MPs. We con-
sider the occupancy for 02MPs to be small and do not
take into account corresponding stimulated processes and
02MP decay. (Note that the terms with q1= q2give no
essential contribution to this sum).
Because of the mentioned reasons, we will consider a
special situation when the 01MPs occupy states near the
magnetoroton minima and calculate the rate of depopu-
lation due to creation of 02MPs only with small q. These
02MPs can be detected by anti-Stokes Raman scattering
like in experiments11,12,13. For this purpose we have to
sum in Eq.(11) with the restriction |q1+q2| < ˜ q and we
will show later that ˜ q ≪ l−1
we can put in |M|2q1 = −q2 and |q1| = |q2| = q0.
We also assume that n(q) = n can be considered to
be constant as long as due to energy conservation we
are dealing with the a narrow band determined by in-
equalities:ε0 < E01(q) < E02(˜ q) − ε0.
simplifications and replacing summation by integration
?
with |q| < ˜ q per unit area to be
R(|q| < ˜ q)
L2
2¯ h
B. Under these assumptions
Using this
q= Nl2
B
?d2q/(2π) we find the rate of 02MP creation
=n2l2
Bq0
[u(q0) − v(q0)]2
?M
δ
?1/2
˜ q2. (12)
The question to be considered is the role of the ran-
dom impurity potential U(r) which was neglected in
the above calculations.The distance between an ex-
cited electron and a hole in real space is l2
Refs.7,8,9).Assuming U(r) to be smooth (correlation
length Λ ≫ lB) one can find that the energy correc-
tion for a MP with the wave vector q. In the dipole ap-
proximation it is δE(q,r) = −¯ hqvdfor any abMP, where
vd = (ˆ z × ∇U(r))l2
tional energy leads to inhomogeneous broadening of the
MP energy. One can see that the random potential cor-
rection plays no significant role if
Bq × ˆ z (see
B/¯ h is the drift velocity. This addi-
|dEab/dq| ≫ l2
B|∇U|
(13)
which means that the electron-hole Coulomb interaction
is stronger than the force the electron and the effective
hole are subjected to in the random electric potential.
Evidently the other meaning of this condition is that the
exciton velocity has to be greater than the drift velocity
in the external field19. Alternatively, we have two inde-
pendent quasiparticles, electron and hole, whose motion
is determined mainly by the random potential and the e-e
interaction has to be considered only as a perturbation6.
For q ≃ l−1
ening δE ≃ ∆(lB/Λ), where ∆ is the random poten-
tial amplitude (i.e. ∇U ∼ ∆/Λ). With typical values
Λ = 50nm and ∆ = 1meV one finds δE = 0.2meV.
This is small compared to the width of the MP band
and small compared to ǫ0but of the same order as the
energy δ relevant for energy conservation. At the same
time, since in the dipole approximation the inhomoge-
neous broadening of the level Eab(q) do not depend on
ab it gives no contribution to the delta-function argument
in Eq.(11). Higher order corrections to δE are of the or-
der of ∆(lB/Λ)2≃ 0.04meV and small compared even
Bone can estimate the inhomogeneous broad-

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4
to δ. As a result we conclude that the role of the random
potential is negligible compared to the e-e interactions.
However the role of the random potential is crucial
in determining the cutoff ˜ q. The momentum of a 02MP
detected by anti-Stokes Raman scattering is defined from
momentum conservation as q = k2?−k1?, where k1?and
k2?are the “in-plane” wave vector components of the
incident and scattered photons. In the case of no disorder
the cutoff ˜ q is defined by the uncertainty of k2?−k1?, i.e.
by the spectral resolution and the geometry of the optical
experiment. This uncertainty is < 104cm−1according
to Refs.11,12,13and the cutoff ˜ q actually comes from the
disorder which violates momentum conservation.
In the approximation of S2DL one may estimate
dE02/dq ≃ Ecq2l3
of q due to disorder can be found from Eq.(13) giving
˜ q ≃ (∆/Ec)1/2(ΛlB)−1/2. This value does not depend on
the magnetic field and for the used numerical parameters
˜ q ∼ 105cm−1. The substitution into Eq.(12) gives
R(|q| < ˜ q)/L2∼ 0.05 ·n2∆
Bfor qlB ≪ 118, and the uncertainty
¯ hlBΛ
∝ B1/2
(14)
(it is taken into account that u(q0)−v(q0) ≈ −0.062EC).
Now let us estimate the total decay of 01MPs sup-
posing that the most of them are concentrated in the
vicinity of the roton minima. Generally, a more com-
plicated summation (12) has to be fulfilled in this case,
because the allowed phase region where 02MPs can be
generated is not small. Indeed, the very weak depen-
dence E02(q) in its initial spectrum portion leads to the
only condition q<
∼l−1
However to obtain the approximate total rate of the co-
alescing 01MPs the formula (14) can be exploited again.
Estimating the 01MP density near their roton minima as
N ≃ nq0(2MδE)1/2(because the roton minima broaden-
∗Electronic address: dickmann@issp.ac.ru
B
for allowed O2MP wave-vectors.
ing due to inhomogeneity is |q − q0| ∼ (2MδE)1/2) and
setting dN/dt equal to decay rate (14) with ˜ q ∼ l−1
find the characteristic relaxation time τ = ndt/dn which
turns out to be
B
we
τ ∼ 102¯ h(∆lB/E3
CΛ)1/2/n ∼ 1/nps
(therefore τ ∝ 1/B). This value should be for real exper-
iments compared with time characteristics of other possi-
ble relaxation channels, for example when the conditions
of magnetophonon resonance are satisfied.
The value n remains indefinite because it depends on
the specific manner of 01MPs excitation. We think the
photoluminescence excitation technique is likely to be
more appropriate for it, as far as therewith the excita-
tion would occur in two independent steps: namely, by
generation of an electron at the 1-th LL and a hole in
the valence band, and by recombination of some elec-
tron from the filled LL with the hole. As a result 01MPs
with various q-s can appear. This technique should be
more effective for magnetoroton excitation in compari-
son with Refs.11,12,13though in itself it does not permit
to detect the magntorotons. Nevertheless, if one simul-
taneously could find 02MPs by means of anti-Stokes Ra-
man scattering or by means of hot luminescence from
the 2-nd LL, it would be an indirect confirmation of
the presence of 01MPs near their roton minima. Note
also that the appropriate consideration of kinetic rela-
tions shows that the occupation number for 02MPs could
be expected to be of the order of n2once the quasi-
equilibrium 2×01MP↔02MP is established.
S.D. thanks for hospitality the Department of Con-
densed Matter Physics of Weizmann Institute of Science,
where the main part of this work was done. The work
is supported by the MINERVA Foundation and by the
Russian Fund for Basic Research (Project 99-02-17476).
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