Spin-Wave Relaxation in a Quantum Hall Ferromagnet

Abstract

We study spin wave relaxation in quantum Hall ferromagnet regimes. Spin-orbit coupling is considered as a factor determining spin nonconservation, and external random potential as a cause of energy dissipation making spin-flip processes irreversible. We compare this relaxation mechanism with other relaxation channels existing in a quantum Hall ferromagnet. Comment: Submitted to JETP Letters

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arXiv:0812.1703v1 [cond-mat.mes-hall] 9 Dec 2008
Spin-Wave Relaxation in a Quantum Hall Ferromagnet
S. Dickmann1and S.L. Artyukhin1,2
1Institute for Solid State Physics of RAS,
Chernogolovka 142432, Moscow District, Russia.
2University of Groningen, Broerstraat 5, 9712 CP Groningen, Netherlands
(Dated: December 9, 2008)
We study spin wave relaxation in quantum Hall ferromagnet regimes. Spin-orbit
coupling is considered as a factor determining spin nonconservation, and external
random potential as a cause of energy dissipation making spin-flip processes irre-
versible.We compare this relaxation mechanism with other relaxation channels
existing in a quantum Hall ferromagnet.
PACS numbers 73.21.Fg, 73.43.Lp, 78.67.De
1.
Last years are characterized by growing interest in spin relaxation (SR) in low-
dimension systems — first of all, in the relaxation in quantum dots studied within the
projects aimed at development of a computer employing spin memory. Yet, the relaxation
of an electron spin in lateral quantum dots manufactured on the basis of two-dimensional
(2D) heterostructures, should be in many respects similar to the SR of electrons localized in
the 2D layer in minima of a smooth random potential (SRP). In high magnetic fields this
single-electron relaxation corresponds to the situation occurring at low Landau level (LL)
filling: ν ≪ 1 or |ν−2n| ≪ 1 (n is an integer).1
The SR at different filing factors, ν>
∼1, has quite different nature representing in this
case a many-electron process. In particular, in a quantum Hall ferromagnet (QHF), i.e. at
ν = 1,3,... or ν = 1/3,1/5,..., the SR reduces to the relaxation of lowest collective ex-
citations, i.e. spin waves.2,3The SR observation would thereby be a good tool to study
fundamental collective properties of a strongly correlated 2D electron gas (2DEG). However,
in spite of much recent interest in the SR in a 2DEG, up to now only a handful of exper-
iments relevant to the SR in a QHF were performed: these are indirect results based on
the linewidth measurements in the electron spin resonance,4and a direct observation where
the photoluminescence dynamics of spin-up and spin-down states was studied.5Meanwhile,
availability of the new time-resolved technique of photon counting allows us to believe that
new direct experiments on observation of excitations’ relaxation in a 2DEG, in particular of
the spin wave relaxation (SWR), will become available in the near future.6
Theoretically the SWR in a QHF was studied in works 7,8. It is worth noting here that

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the SWR represents actually not spin dephasing but the energy relaxation due to the spin-flip
process. Indeed, any spin-flip means at least dissipation of the Zeeman energy ǫZ= |g|µBB
(g ≈ −0.44 in a GaAs structure). The latter is a part of the spin-wave (spin exciton, SE)
energy
Esw= ǫZ+ Eq,(1)
where Eqis the SE correlation energy depending on the 2D wave vector q.2,3At variance with
the relaxation channel of Ref. 7 where electron-phonon interaction was considered as the
mechanism making the relaxation irreversible, and contrary to the case of Ref. 8 where the
irreversibility was provided by an inter-spin-exciton interaction mechanism, we now study
smooth disorder field as the reason causing the energy transform. The SRP thereby deter-
mines an alternative relaxation channel competing with the ones studied earlier. Another
distinction of the present work from Refs. 7,8 consists in the study of not only the integer
QHF (at ν = 1,3,...) but also of the fractional one (ν = 1/3,1/5,...) as well. At the same
time we again consider the spin-orbit coupling (SO) as the cause mixing different spin states
and therefore providing the spin nonconservation. Actually, various SWR channels coexist
in parallel. We consider the total rate and find crossover regions of external parameters
(magnetic field, temperature, etc.) where one relaxation channel ceases to be dominant and
changes into another.
The SR channel due to SRP was already considered in the integer quantum Hall ferro-
magnetic case.1,9However, studied in these works instead of the SWR was a specific SR
when initially the total macroscopic spin?S of the system as a whole is turned away from
the equilibrium direction parallel to?B. (Relaxation of this Goldstone mode microscopically
reduces to annihilation processes of the so-called zero SEs, having exactly zero momenta.)
Contrary to this case, the spin perturbation determined by excitation of the spin waves
(non-zero SEs) represents an initial deviation where ∆S=∆Sz, so that?S is kept parallel to
?B and the total symmetry of system remains unchanged.
Concerning the origin of SRP, one should note that it has in the 2D layer the “direct”
component and the effective one. The former is the SRP determined by charged donors
located outside the spacer. The latter is essential in some kinds of quantum wells, being
determined by spatial fluctuations (in the plane of the layer) of quantum well width. These
fluctuations lead to fluctuations of the size-quantization energy and may be presented as an
SRP term in the single electron Hamiltonian. Both SRP components have approximately
the same amplitude ∆∼10K and correlation length Λ∼30 − 50nm.
2. The total Hamiltonian has form Htot=?
jH(j)
1+Hint, where j enumerates electrons,

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Hintis the e-e interaction, and the single-electron operator is
H1= ¯ h2ˆ q2/2m∗
e− ǫZˆ σz/2 + HSO+ ϕ(r).(2)
In this equation ϕ(r) is the SRP field; the SO Hamiltonian is specified for the (001) GaAs
plane,
HSO= α(ˆ q × ˆ σ)z+β (ˆ qyˆ σy−ˆ qxˆ σx) ,(3)
presenting a combination of the Rashba term and the crystalline anisotropy term10(ˆ q =
−i∇ + eA/c¯ h is a 2D operator, σx,y,zare the Pauli matrices). If the SRP is assumed to be
Gaussian, then it is defined by the correlator K(r) = ?ϕ(r)ϕ(0)?. By choosing ?ϕ(r)? = 0,
in terms of the correlation length Λ and the LL width ∆ the correlator is
K(r) = ∆2exp(−r2/Λ2).(4)
We first find the bare single-electron basis diagonalizing the Hamiltonian (2) without the
SRP field. To within the leading order in the HSOterms we obtain
Ψpa=
?
ψnp
v√n+1ψn + 1 p+iu√nψn − 1 p
?
,
Ψpb=
?
−v√nψn − 1 p+iu√n+1ψn + 1 p
ψnp
?
(5)
Here ψnpis the electron wave function in the Landau gauge, n is the number of the half-filled
LL in the odd-integer quantum Hall regime, i.e. in the ν=2n+1 case. Otherwise, if ν≤1, we
set n=0. u and v are small dimensionless parameters: u = β√2/lB¯ hωcand v = α√2/lB¯ hωc
(ωcand lBare the cyclotron frequency and the magnetic length, respectively). The single-
electron states thus cease to be purely spin states but acquire a chirality a or b. The spin
flip corresponds thereby to the a → b process now.
By analogy with previous works1,7,8,9(see also Ref. 11) we define the SE creation operator
Q†
abq=
1
?Nφ
?
p
e−iqxpb†
p+qy
2ap−qy
2,(6)
where ap and bp are the Fermi annihilation operators corresponding to states (5), Nφ is
the LL degeneracy number. In Eq. (6) and everywhere below we measure wave vector q
in the 1/lB units. If the ratio rc= (αe2/κlB)/¯ hωcis considered to be small (α < 1 is the
averaged formfactor which appears due to finiteness of the layer thickness), and the SRP
and SO terms in Eq. (2) are ignored, then the operator (6) acting on the ground state in
the odd-integer quantum Hall regime yields the eigen state of the total Hamiltonian: namely,
[Htot,Q†
state, Q†
abq]|0?=(ǫZ+Eq)Q†
abq|0?, is the asymptotically exact one to the first order in rc.
abq|0?, where |0?=|
Nφ
? ?? ?
↑,↑,... ↑?. This basic property of the exciton

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Now consider corrections arising due to the HSOterms. When presented in terms of basis
states (5), spin operators?Ψ†ˆS2Ψd2r and?Ψ†ˆSzΨd2r [where Ψ=?
invariant form up to the second order in u and v. However, the interaction Hamiltonian
Hint=1
2
which correspond to creation and annihilation of SEs in the system. It is exactly these terms
p(apΨpa+bpΨpb)] preserve
?dr1dr2Ψ†(r2)Ψ†(r1)U(r1−r2)Ψ(r1)Ψ(r2) acquires proportional to u and v terms
that lead to the “coalescence” channel of the SWR.8In the present work we study another
relaxation channel. Therefore, neglecting this SO corrections toˆHint, we focus on the SRP
term. Calculating?Ψ†ϕ(r)Ψd2r, we get the terms responsible for a spin-flip:
ˆ ϕ = N1/2
ϕ(q)(iuq+− vq−)Qq+ H.c.
φlB
?
q
(7)
(it is assumed here that q ≪ 1). ϕ(q) is the Fourier component [i.e. ϕ =?
At variance with integer QHF, the use of the excitonic basis Q†
model approach in the case of fractional quantum Hall regime. Generally, spin-flip excitations
qϕ(q)eiqr], and
q±=∓i(qx±iqy)/√2.
abq|0? presents only a
within the same Landau level might be many-particle rather than two-particle excitations
at fractional filling because the same change of the spin numbers δS = δSz= −1 may be
achieved with participation of arbitrary number of intra-spin-sublevel excitations (charge-
density waves). These waves are generated by the operator A†
νNφ
?
however, states of the Q†
spin-flip at fractional ν. On the other hand, a comprehensive phenomenological analysis3,12
q=N−1/2
φ
Q†
aaqacting on the
ground state |0? = |
?? ?
↑,.. ↑,.. ↑?.12It is trivial in the case of integer ν (A†
abq1A†
q|0? = δq,0|0?);
q2A†
q3...|0? type might constitute a basis set if one studies a
suggests that even the spin-flip basis reduced to single-mode (single-exciton) states would be
quite appropriate, at least for lowest-energy excitations in the case of fractional QHF. This
single-mode approach is indirectly substantiated by the fact that the charge-density wave has
a Coulomb gap12which is well larger than the Zeeman gap ǫZ. Hence for a fractional QHF,
just as in Ref. 3, we will consider the only state Q†
The commutation algebra for operators Q†
same as for integer filling,7,8,9. However, a difference arises in the calculation of expectation
abq|0? to describe the spin-flip excitation.
abq, A†
q′ and B†
q′′ = N−1/2
φ
Q†
bbq′′ is certainly the
?0|AqA†
but at ν<1 it is expressed in terms of the two-particle correlation function g(r) calculated
q′|0? which is needful for the following. This value is simply δq,0δq′,0at integer filling,
for the ground state:
?0|AqA†
q′|0? =
ν
Nφ
?
2πνg(q)eq2/2+1
?
δq′,q. (8)
Here g(q)=
1
(2π)2
?g(r)e−iqrd2r is the Fourier component. Function g(r) is well known, e.g., in
the case of Laughlin’s state.12,13If the ground state is presented in terms of the Hartree-Fock

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model, we get the expression 2πg=
?
Nφδq,0−e−q2/2?
which does not depend on ν. Besides,
at odd-integer filling factors this Hartree-Fock expression becomes Fourier component of the
exact correlation function. In the latter case one should also make the substitution ν→ ν−2n
in Eq. (8), i.e. formally set ν=1 there.
3. The operator (7) obviously does not conserve the number of SEs. However, if the
SWR is governed by this operator, the corresponding problem can not be solved in terms
of a single-exciton study. Indeed, the SE interaction with the SRP incorporates the energy
Ux-SRP∼qlB∆/Λ (the SE possesses the dipole momentum elB[q׈ z])2. The SE momentum
is estimated from the condition Eq<
to this inequality, the energy of annihilating exciton can not be transformed to anywhere.
∼T, and we therefore find that Ux-SRP≪ ǫZ, T. Due
By analogy with Ref. 8, we study a coalescence process where initial double-exciton state
|i?=Q†
energy:
abq1Q†
abq2|0? transforms to final single-exciton state |f?=Q†
abq′|0? having the combined
ǫZ+ Eq′ = 2ǫZ+ Eq1+ Eq2
(9)
(c.f. also the Auger magnetoplasma relaxation considered in Ref. 14). At the same time,
contrary to Ref. 8, there is no momentum conservation in this SWR channel. Thus the phase
volume where the Xq1+Xq2→Xq′ transition is possible turns out to be much larger than
that in the coalescence process of Ref. 8. This transition is governed by the Fermi golden
rule probability: wfi= (2π/¯ h)|Mfi|2δ(Ef− Ei), and our immediate task is to calculate the
matrix element Mfi=ν−3/2?f|ˆ ϕ|i?. (The factor ν−3/2appears due to the normalization since
norms of the |i? and |f? states are ν2and ν, respectively.)
We perform the calculation for relevant values of momenta q1,q2,q′≪ 1 which satisfy
the conditions Eq1,Eq2<
dimensional units). By employing exciton-operators’ commutation rules7and evident iden-
∼T<
∼1K. (These inequalities correspond to q1,q2,q′≪1/lBin usual
tities Qabq|0?≡Bq|0?≡0 and ?0|Aq|0?≡ν, we obtain with the help of Eqs. (7)-(8) that
?2
j=1
Mfi(q1,q2,q′)=2πν1/2
N1/2
φ
?
g(|qj−q′|)e(qj−q′)2/2
??
q
ϕ(q)(iuq+−vq−)δq1+q2,q+q′. (10)
Besides, within our approximation, g(q)eq2/2should be replaced with g(q)eq2/2???
in the case of Laughlin’s ground state describing the fractional QHF. So, for ν=1,1/3,1/5,...,
q→0. The
latter quantity is equal to −1/2π in the Hartree-Fock approach or −1/2πν when calculated
replacing the terms in square brackets with −1/πν, we obtain a simple result:
|Mfi(q1,q2,q′)|2=4πK(q)q2(u2+v2)
νN2
φ
?????
q=q1+q2−q′
.(11)