Collective excitations in a magnetically doped quantized Hall
  ferromagnet
Collective excitations in a magnetically doped quantized Hall
ferromagnet
S. Dickmann1,2, V. Fleurov1,3 and K. Kikoin1,3,4
1Max-Planck-Institut für Physik Komplexer Systeme,
Nöthnitzer Str. 38,
D-01187 Dresden, Germany.
2Institute for Solid State Physics of RAS,
Chernogolovka 142432,
Moscow District, Russia.
3School of Physics and Astronomy,
Beverly and Raymond Sackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv 69978, Israel.
4Department of Physics,
Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel.
A theory of collective states in a magnetically quantized two-dimensional electron
gas (2DEG) with half-filled Landau level (quantized Hall ferromagnet) in the pres-
ence of magnetic 3d impurities is developed. The spectrum of bound and delocalized
spin-excitons as well as the renormalization of Zeeman splitting of the impurity 3d
levels due to the indirect exchange interaction with the 2DEG are studied for the
specific case of n-type GaAs doped with Mn where the Landé g-factors of impurity
and 2DEG have opposite signs. If the sign of the 2DEG g-factor is changed due to
external influences, then impurity related transitions to new ground state phases,
presenting various spin-flip and skyrmion-like textures, are possible. Conditions for
existence of these phases are discussed.
PACS: 73.43.Lp, 73.21.Fg, 72.15.Rn
http://arxiv.org/abs/0704.0658v3
I. INTRODUCTION
In a strong magnetic field the two-dimensional electron states in semiconductor
heterostructures 1 transform into Landau states with a completely discrete energy spec-
trum. This diamagnetically quantized two-dimensional electron gas (2DEG) possesses many
remarkable features including Quantum Hall effect.2 The role of impurities in the thermo-
dynamic, optical and transport properties of 2DEG is extremely important. Among many
facets of this problem we choose for discussion in this paper the formation of impurity related
collective excitations in a magnetically doped quantized 2DEG in the case of odd integer
filling factor ν = 2n+1. In a pristine state 2DEG with odd ν is in a Quantuzed Hall Fer-
romagnet (QHF) regime with nondegenerate ground state characterized by the total spin
quantum number S = Nφ/2 and maximum spin projection Sz = S. (Nφ is the magnetic-
flux-quanta number.) Different branches of the excitons are well distinguishable among the
low-energy excitations. They are classified as spin waves (spin excitons), magnetoplasmons or
multi-exciton states depending on the spin and orbital quantum numbers.3,4,5,6,7,8,9,10 Besides,
low-lying collective half-integer-spin fermionic states (trions, skyrmions,...) may be formed
in a QHF under certain circumstances.11,12,13,14,15,16,17,18 Magnetic impurities are character-
ized by their own spectrum of local spin excitations, and one can anticipate a strong interplay
between local and collective excitations in a magnetically doped QHF.
It is known that the influence of impurities on the discrete spectrum of quantized Landau
electrons in a 2DEG has many specific features. Even such a basic property, as the inter-
action of a 2DEG with neutral short range impurities is far from being trivial.19,20,21 Only
the Landau states with a finite probability density on the scatterer locations interact with
impurities. This means that the whole set of Landau states breaks down into two groups:
the major part of the Landau levels (LLs) is not affected by the impurity scattering, and
the states having a nonzero scattering amplitude on an impurity form a separate system of
bound Landau states in the energy gaps between the free LLs.
To be more specific, we consider a 2DEG formed in the n-type GaAs/GaAlAs heterostruc-
tures doped with transition metal (TM) impurities. The reason for this choice is that the
technology of (Ga,Mn)As epilayers is well developed, and the QHF regime is achieved ex-
perimentally in GaAs based heterostrures. As a rule, transition metal ions substitute for
the metallic component of the binary II-VI and III-V semiconductors.22,23,24 The influence
of isolated TM impurities on the spectrum of the Landau states was investigated in Ref. 25.
It was shown that the resonance scattering in the d-channel is in many respects similar to
that of the short range impurity scattering in the s-channel.19,20,21 The symmetry selection
rules for the resonance d-waves in the cylindrical (symmetric) gauge pick up the Landau
states with the orbital number m = 0 (in the symmetric gauge). These states are the same
states that are involved in the s-scattering by the impurities with a short range scattering
potential.21 Besides, this scattering is spin selective in magnetically quantized 2DEG.
It should be emphasized that in the problem under consideration the criterion of isolated
impurities acquires a specific feature. In fact the Mn concentration range, where our theory is
applicable, is limited from below by technological capabilities and from above by the obvious
requirement that the impurity induced disorder does not destroy collective excitonic states.
So, the relevant interval of bulk Mn concentrations is 1013 cm−3 < nMn<∼ 1015 cm−3. Here the
upper limit corresponds to the 2D concentration of 109 cm−2 which in our case is actually well
below the Landau band capacity Nφ at B ∼ 10T that equals to the the 2D electron number
on the upper (half-filled) LL. One may expect that the above mentioned classification of
excitonic states is valid only at the bulk concentration nMn<∼ 1015 cm−3, which is much less
than in the materials used for creation of dilute magnetic semiconductors.26
We calculate in this paper spectra of bound and continuous collective excitations related
to magnetic impurities. When studying the influence of magnetic impurities on the excitonic
spectrum of 2DEG, a distinction between the negative and positive signs of the gyromagnetic
ratio of 2DEG electrons g2DEG should be also mentioned. It will be shown that in the
conventional situation of negative g2DEG the interaction with magnetic impurity lowers the
ground state energy due to effectively antiferromagnetic character of the effective indirect
exchange. This results in formation of a set of bound and delocalized collective excitations
presenting combined modes classified by a change in the total spin number Sz. When g2DEG >
0, so that the g factors of both subsystems (2DEG electrons and impurities) have the same
sign, magnetic impurities may form bound states in the gap below the spin exciton continuum
and even initiate a global reconstruction of the QHF ground state.
II. MODEL HAMILTONIAN
Following Ref. 23, we describe the electron scattering on a TM impurity in semiconduc-
tor within the framework of the Anderson impurity model Hamiltonian 27 generalized for the
case of multicharged impurity states in semiconductors.28,29,30 According to this model, the
principal source of magnetic interaction is the resonance scattering of conduction electrons
on the d-electron levels of TM impurity in the presence of a strong on-site Coulomb interac-
tion U . Due to this interaction, the local moment of TM impurity survives in the crystalline
environment, and ‘kinematic’ indirect exchange interaction between the conduction and im-
purity electrons arises in the second order in the s-d-hybridization parameter, even in the
absence of a direct exchange.
The generic Hamiltonian describing the QHF regime in a magnetically doped semicon-
ductor is
Ĥ = Ĥd + Ĥs + Ĥt + Ĥsd. (2.1)
Here Ĥd =
i Ĥdi describes the TM impurities on the sites i, Ĥs is related to the band elec-
trons on the LLs, and Ĥt is responsible for hybridization between the impurity d-electrons
and Landau electrons. Eventually, it is this hybridization that generates coupling between
collective modes in 2DEG and localized spin excitations on the impurity sites. In our ex-
tremely weak doping regime both the direct and indirect interactions between magnetic
impurities are negligible. Each magnetic scatterer may be considered independently, and it
is convenient to choose the symmetric gauge A = (−B
x, 0) with the quantum numbers
λ = (n,m) for the Landau electrons hybridized with the atomic d-electrons centered around
the site i positioned in the center of coordinates. The Coulomb interaction is taken into
account in the impurity and in the band electron subsystems. Besides the direct Coulomb
interaction between the d and s electrons described by the last term in the Hamiltonian (2.1)
is added to the conventional impurity Hamiltonian (cf. Ref. 27) described by the first and
third terms. All additional interactions will be discussed below in detail.
Substitutional Mn impurity in GaAs retains all its five d electrons due to a special sta-
bility of the half filled 3d shell. In the p-type GaAs the electrically neutral state of Mn
in Ga position is Mn(3+)(3d5+hole), where the hole is bound on the relatively shallow ac-
ceptor level near the top of the valence band, whereas the occupied d-electron levels are
deep in the valence band.23,24,31,32 In the n-type heterostructures these acceptor states are
overcompensated, and the chemical potential is pinned on one of the lowest Landau levels in
the conduction band. Since we are interested only in the low-energy excitations above the
ground state of n-type system, Mn impurities will be considered as the Mn(3+)(3d5) ions in
the subsequent calculations.
A. Single-orbital model. Spin-selective hybridization.
One may significantly simplify the calculation of the spectrum of excitations by reducing
the general Hamiltonian (2.1) to the form, in which only the terms relevant to the calculation
of desired collective states are present. As a result of this simplification outlined below in
Subsection IIB one arrives at the single-orbital, single Landau band hybridization Hamilto-
nian, which explicitly takes into account the Hund rule governing the high-spin states 3d5
and 3d6 of the Mn 3d shell (the state 3d4 is proved to be irrelevant in our specific case of
Mn in GaAs lattice, see below Fig. 1). These impurity states are characterized by the max-
imum total spin quantum numbers S(d) = 5/2 at 3d5, and S(d) = 2 at 3d6, and the effective
Hamiltonian Ĥ is defined in the charge sector {|d5, N〉, |d6, N − 1〉} of states with variable
number N or N − 1 of the electrons on the highest n-th LL (of course in our case N ≈ Nφ).
The Hamiltonian now reads
ǫdσn̂γ0σ + Un̂γ0↑n̂γ0↓ +
nmσanmσ + Ĥ
Coul + Ĥt . (2.2)
where the impurity Hamiltonian Ĥd of Eq. (2.1) is represented in by the two first terms,
in which n̂γ0σ = c
cγ0σ, and c
is the creation operator for the d-electron at the orbital
γ0 with the spin z-component σ. The notation γ0 designates the only 3d-orbital with the
Y02 ∼ 3z2 − r2 symmetry, which effectively couples with the m = 0 state of the LL.25 The
parameter U characterizes Coulomb and exchange interactions determining the addition
energy for the transition 3d5 → 3d6. The third term in Eq. (2.2) is the Hamiltonian
of noninteracting Landau electrons where a†nmσ is the creation operator for the (n,m, σ)
Landau state. The most of interaction components are included in Ĥ ′Coul. This term does
not include only the d-d interaction parametrized by U and the last term Ĥt. The latter
generically is also the part of Coulomb interaction between impurity and Landau electrons
which intermixes impurity and Landau orbitals . However, in our case Ĥt acquires the form
of single-electron hybridization operator [see discussion after Eq. (2.13)],
Ĥt =
n0σcγ0σ +H.c. (2.3)
As was mentioned above, this operator is responsible for the resonance orbital-selective scat-
terings in QHF. It includes hybridization of the impurity electron with the 2DEG electrons
within the n-th LL. This means that only the influence of impurity on the intra-LL excita-
tions (of the spin-wave type) is taken into account. The hybridization with the states with
n′ 6=n describing the processes with energy change ~ωc or higher is omitted.
In the absence of interaction term Ĥ ′Coul, the Hamiltonian (2.2) acts in the subspace
|d5, s; vac〉, |d6, s+
; an0↑|vac〉 and |d5, s+ 1; a†n0↓an0↑|vac〉, (2.4)
where the fully polarized 2DEG without impurity is chosen to be the ‘vacuum’ state |vac〉 =↑
, ↑, ... ↑〉. Therefore a†nm↑|vac〉 = anm↓|vac〉 ≡ 0. We represent the total spin component as
+ s. Thus we characterize the states in the set (2.4) by the quantum number Sz.
It is important that only Sz = S
z + S
z is an exact spin quantum number in our system.
Separately, the Hamiltonian (2.2) commutes neither with the spin component S
z of the LL
electrons nor with the impurity spin component S
z . Equally, it does not commute with
the total spin S2 and with the spins (S(s))2 and (S(d))2 (see Appendix B). The number s
in the set (2.4) changes within the interval −5
≤ s < 5
. It is convenient to choose the
state |d5, 5
; vac〉 as a reference point (‘global vacuum’). This state is not mixed with any
other state of the system by the operator (2.3) so that it enters the set of eigenstates of the
Hamiltonian (2.2), although at g2DEG < 0 it is one of the excited states of the system.
Within a given ‘triad’ (2.4), i.e. at a given s, the operator (2.3) intermixes these basis
states. The corresponding non-diagonal matrix elements are 〈vac; s, d5|Ĥt|d6, s+ 12 ; an0↑|vac〉
and 〈vac|a†n0↑; s+ 12 , d
6|Ĥt|d5, s+ 1; a†n0↓an0↑|vac〉, where the bra- and ket-vectors are appro-
priately normalized. Therefore, for any given quantum number Sz =
+ s the magnetic
impurity scattering problem can be effectively described in terms of a single-orbital impurity
model that involves only one or two dγ0-electrons. The single-orbital basis
|s−; vac〉, |s0; an0↑|vac〉 and |s+; a†n0↓an0↑|vac〉 (2.5)
arises instead of the original multi-electron basis (2.4) where the indices (−, 0, +) label the
bare energies Es− , Es0 = Es− + U + ǫd↑ − εn↑ and Es+ = Es− + (gi − g2DEG)µBB. Here
giµBB = ǫd↑ − ǫd↓, and g2DEGµBB = εn↑ − εn↓ are Zeeman energies for impurity and 2DEG,
respectively. The two states Es± form a Zeeman doublet for a given s, while the state Es0
becomes resonant with the LL continuum.
As a result of this mapping, where only one component γ0 of the normalized multi-electron
states is responsible for the hybridization, the Ĥt operator can be redefined for each triad
(2.5) as
Ĥt(Sz) = V
β↑(s)c
↑a0 + β↓(s)c
+ H.c. (2.6)
It becomes thereby spin-selective. The shorthand notation a0 ≡ an0↑, b0 ≡ an0↓ is used here
and below; c
is the creation Fermi operator for the s± impurity states |s±; vac〉 = c†↑/↓|vac〉
and |s0; vac〉 = c†↑c
↓|vac〉. The Clebsch-Gordan coefficients βσ(s) reflect normalization of
eigenvectors (2.4) by replacing them with normalized single-orbital basis (2.5). For s =
−5/2,−3/2,−1/2, 1/2, 3/2 we have
, β↓ =
. (2.7)
The highest state in the bare Zeeman ladder |
; vac〉 ≡ |d5, 5
; vac〉 remains nonhy-
bridized.
Unlike the original Anderson model 27 the mixing coefficient, V ≡ Wn0 in our particular
case, arises as a non-diagonal component of the s-d Coulomb interaction (see Subsections
IIC and IID for further discussion).
B. Description of the employed simplifications
Here we list the simplifications which have allowed us to reduce Eq. (2.1) to the Hamil-
tonian (2.2) with following change Ĥt→ Ĥt, and to apply to our system.
The first simplification exploits the fact that the characteristic Coulomb energy of Landau
electrons EC = αe
2/κlB is considered to be small in the QHF regime as compared to the
cyclotron energy ~ωc. Here α is the average form-factor related to the finite thickness of
the 2DEG (0.3<∼α<1). In the EC≪ ~ωc limit one may neglect the LL mixing. Besides, it
implies that in our case the energies of collective excitations are smaller than ~ωc.
(a) (b)
FIG. 1: Allen reactions that involve an additional electron (a) or hole (b) in the impurity site.
ǫd↓ and εd↑ + U are addition energies for the 5-th and 6-th electrons in the 3d-shell of Mn ion in
accordance with Eqs. (2.8). The ground state with the energy E0 corresponds to the completely oc-
cupied lowest Landau subband. Spins of Mn 3d shell and occupied Landau subband are antiparallel
because of the different signs of g factors for Mn and 2DEG in GaAs.
The second simplification is related to the ‘deepness’ of the 3d-levels of a neutral Mn
impurity relatively to the bottom of conduction band in GaAs. We know from the previous
studies 25 that the scattering potential created by a TM impurity for the Landau electrons
is generated by the s-d hybridization. It has the resonance character, and the spin selective
scattering becomes strong when one of the impurity 3d levels is close to the LLs of conduction
electrons. The process of s-d hybridization may be represented by the so called ‘Allen
reactions’23,24,33 (see Fig. 1)
3d5 → 3d6 + h , (2.8a)
3d5 → 3d4 + e . (2.8b)
The first of these reactions describes hopping of an electron from the filled Landau subband
to the impurity d-shell, whereas the second one means hopping of an electron from the d
shell to a state in the empty Landau subband. It is known from the numerical calculations 32
that the addition energy for the 6-th electron in the Mn 3d shell (eCFR− state in terms of
Ref. 24) is in resonance with the states near the bottom of GaAs conduction band. It really
means that the values U and εd↑ well compensate each other in the sum εd↑+U . So, one may
retain only the processes (2.8a) in Ĥt and neglect contributions of the 3d
5 → 3d4 ionization.
The third major reduction of the Hamiltonian is the elimination of the impurity orbital
degrees of freedom due to the selection rules for the s-d hybridization matrix elements.25 This
orbital selectivity arises, first, because of symmetry reasons since only electrons with equal
axial m-numbers in d and LL states are hybridized. Second, a precondition of the selectivity
is related to the fact that the magnetic length lB is much larger than the radius rd of 3d-
electron state (in the energy scale this condition takes the form of inequality U, εdσ ≫ ~ωc).
The hybridization integral determined by the overlap of the d- and Landau wave functions
behaves as ∼ (rd/lB)m ≪ 1 for m 6= 0. All resonance scattering (hybridization) amplitudes
with m 6= 0 are thus negligibly small, and only the s-scattering term (m = 0) can be
retained in Ht. This explains why only one of the five 3d-orbitals, namely γ0, is involved in
the resonance interaction with the 2D Landau electrons.
C. Interaction Hamiltonian in excitonic representation
As it was mentioned above, the states of the system are characterized by the total spin
component Sz. For a given Sz we may deduce the effective Hamiltonian
Ĥ(Sz) = Ĥd + Ĥ(s)1 + Ĥt(Sz) + Ĥs−s + Ĥs−d (2.9)
with the single-orbital impurity term Ĥd = ǫd↑n̂↑+ ǫd↓n̂↓+Un̂↑n̂↓ (n̂σ= c
σcσ), and with the
hybridization term determined by Eq. (2.6).
The remaining terms in the Hamiltonian (2.9) are defined within the framework of the
single-LL approximation for the Landau electrons.3,4,5,6,9,10,11,12,16,18,34,35,36 Although only the
states withm = 0 in the LL are involved in the resonance scattering the complete basis for the
description of collective excitations includes all m-orbitals of the LL, and the corresponding
Schrödinger field operators should be taken in the form
Ψ̂↑(R) = c↑ψd(R) + ζs(z)
amϕm(r),
Ψ̂↓(R) = c↓ψd(R) + ζs(z)
bmϕm(r). (2.10)
Here the shorthand notation am = anm↑, bm = anm↓ is used. R = (r, z) is the 3D coordinate
with the reference point at the impurity site, ζs(z) is the size-quantized functions of s-
electrons in the layer, ϕm is the wave function of the n-th LL, where indexm in the symmetric
gauge changes within the interval (−n,−n + 1, ...Nφ − n− 1).
Using the above definitions and Eqs. (2.10) in the generic interaction operator
ĤCoul =
σ1,σ2=↑,↓
d3R1d
3R2Ψ̂
(R2)Ψ̂
(R1)W (R1 −R2) Ψ̂σ1(R1)Ψ̂σ2(R2) (2.11)
[where W (R) ≈ e2/κR at R ≫ rd], one may rewrite the s-s and s-d Coulomb interactions in
the excitonic representation (ER).6,9,18 This actually means that after substitution of Eqs.
(2.10) into formula (2.11) the latter can be expressed in terms of combinations of various
components of the density-matrix operators. These are so-called ER operators presented in
our case only by the intra-LL set, i.e. by the spin-exciton operators Q†q where an electron
is promoted from one spin-sublevel to another [see Refs. 6,9,18 and Appendix A where the
necessary ER equations are given with the reference to our case] and by operators A†q and B†q
acting within the sublevels a or b (see ibidem). As a result the Coulomb terms of Eq. (2.9)
can be written only by means of the intra-sublevel operators A†q and B†q [their definitions
are given by Eq. (A.4) in Appendix A],
Ĥs−s =
Wss(q)
A†qAq + 2A†qBq + B†qBq
(A0 + B0)
Wss(q) , (2.12)
Ĥs−d = (n̂↑ + n̂↓)
Wsd(q)(Aq + Bq) . (2.13)
The Coulomb vertices are presented also in Appendix A [Eqs. (A12) and (A13)].
We neglect in Eq. (2.11) the direct exchange s-d interaction terms (see the next sub-
section). The mixing operator Ĥt(Sz) in our model Hamiltonian (2.6) includes in fact off-
diagonal interaction terms from ĤCoul. Indeed, Coulomb interaction described by the terms
∼ c†↓n̂↑bm + H.c. and ∼ c
↑n̂↓am + H.c. induces transitions adding or removing one electron
to the d-center in accordance with the Allen reaction diagrams (2.8). These terms represent
the s-d hybridization formally conditioned by the d-center occupation; however, since in our
case the reaction (2.8b) is forbidden, they actually operate as ∼ c†↓bm +H.c. and ∼ c
+ H.c. in Eq. (2.6), respectively. [In terms of the d5↔ d6 transitions the hybridization is
taken just in the form of Eq. (2.3).]
The single particle Hamiltonian for LL electrons may be also written in the ER represen-
tation,
1 = Nφ [(εn − εZ/2)A0 + (εn + εZ/2)B0] , (2.14)
where εZ = |g2DEG|µBB and εn = (n + 1/2)~ωc.
D. Numerical estimates of the energy parameters
Before turning to our main task, i.e. to the calculation of excitation spectra, it is worth-
while to evaluate the characteristic energy parameters related to this problem. We estimate
the parameters of 2DEG in GaAs for the typical value B = 10T of magnetic field. In
this field EC ∼ 5meV characterizes the Coulomb interaction (A13). Below in our calcu-
lation this value is mostly presented by the spin-exciton mass, which can be estimated
empirically, i.e. the inverse mass is 1/Mx ∼ 2meV in energy units. The LLs’ spacing is
~ωc ≈ 16meV, and the Zeeman splitting between two Landau subbands is εZ ≈ 0.25meV
(because g2DEG = gGaAs ≈ −0.44). The Zeeman splitting for Mn ion is giµBB ≈ 1.1meV
(because gi = gMn ≈ 2.0). The hybridization constant V and the repulsion U are the other
important parameters characterizing the magnetic impurity. It is rather difficult to extract
them from the available experimental data. We can only roughly estimate the energy U
as a distance between the Mn-related peaks in the density of states of occupied and empty
states in the spectrum of bulk (Ga,Mn)As calculated with an account of the electron-electron
interaction.32 Such an estimate gives U∼ 4−4.5 eV. From the same calculations we estimate
the energy difference
∆ = ǫd↑ + U − εn + εZ/2, (2.15)
which determines the position of the Mn(d6) electron level above the bottom of Landau
band (see Fig. 1a) as ∆ . 0.1 eV. In order to estimate the parameter V , one should recollect
that the dominating contribution to hybridization integral is given by the matrix elements
of Coulomb interaction, having the form V c
↓c↓ (see above). This means that V ∼
d ζ(zd)/lB. Estimating the radius of the ψd function as rd ∼ 2 Å, and ζ(zd) ∼ 0.15 Å−1/2
(for the impurity located in the vicinity of the quantum well bottom), one gets V ∼ 20meV.
This gives |V |2/∆ ∼ 4− 8meV for the relevant kinematic exchange parameter. At the same
time the direct exchange turns out to be insignificant. Indeed, one can estimate from Eq.
(2.11) that the characteristic coupling constants for the terms ∼ c†↑c↓b†m1am2 and c
↓bm1am2
are of the order of Ur3d|ζ(zd)|2/l2B being therefore by the factor ∼ ∆/U smaller than |V |2/∆.
III. COLLECTIVE SPIN-FLIP STATES. NEGATIVE g2DEG-FACTOR
The Coulomb interactions Ĥs−s and Ĥs−d admix the LL states with m 6= 0 to the three-
state basis (2.5). Instead of triads (2.5), the basis
|s−; vac〉, |s0; am|vac〉 and |s+;Q†q|vac〉 (3.1)
contains spin-exciton continua Q†q|vac〉 attached to the spin-flipped impurity state s+. [The
definition of the spin-exciton creation operator is given by Eq. (A1).]
This set is complete only within the single-orbital approximation.37 At a given s it is
convenient to take the energy E0+(s) of the state |0〉 = |s+; vac〉 ≡ c†↑|vac〉 as the reference
point because this state is not affected by the hybridization within the framework of the
single-orbital model. This energy is defined as E0+(s) = 〈vac|; s + 1, d5|Ĥ|d5, s + 1; |vac〉
where the Hamiltonian Ĥ is given by Eqs. (2.2) and (2.3). For a given Sz =
+ s we have
E0+(s) = Evac − (52 − s)giµBB with Evac defined as the energy of the global vacuum state
|d5, 5/2; |vac〉. One can check with the help of expressions (B1) and (B4) in Appendix B
that the vectors |s0; am|vac〉 and Q†q|0〉 at q 6= 0 correspond to the definite total spin state
with S = Sz, whereas |s−; |vac〉 and Q†0|0〉 are not characterized by any definite number S.38
A. Secular equation
Following the above discussion the spin-flip operator may be represented in the form
X̂† = c
↓c↑ −
↓am +
f(q)Q†q. (3.2)
The normalizability condition 〈X|X〉 < ∞ for the bound spin-exciton state |X〉 = X†|0〉
then reads
m |Dm|2 +
|f(q)|2 <∞ and the sum
|f(q)|2 = Nφ
dq|f(q)|2
presenting the contribution of continuous spin excitons into the norm 〈X|X〉 becomes thereby
an essential characteristic of the spin-flip excitation. For the regular (normalizable) solutions
we expect f(q) ∼ N−1/2φ . Besides, singular states, for which the sum
|f(q)|2 diverges
also exist. These states form continuous spectrum of impurity-related spin-excitons.
The coefficients Dm and f(q) are determined from the equation
[Ĥ, X̂†]|0〉 = E|X〉 . (3.3)
where the energy E is counted from E0+(s). Before turning to the computation we specify
the energy levels of the basis states (3.1) at V = 0. The state |s−; vac〉 has the energy
E0−(s) = E0+(s)− giµBB. The doubly occupied impurity state |d0; am|vac〉 appears due to
the charge transfer with creation of a conventional ‘hole’ in the LL. Its energy is Ed,m(s) =
E0+ + Ed,m where
Ed,m = ǫd↓ + U + εZ/2− εn + ǫm + E∞ (3.4)
[cf. Eq. (2.15)]. Here E∞ = (1/Nφ)
Wss(q) [see Eq. (A11) for definition of E∞]. This
term appears due to the global electroneutrality requirement when calculating the energy
of the hole am|vac〉.4,5,9,11 The term ǫm = −(2/Nφ)
hm+n,m+n(q)Wsd(q) is the Coulomb
interaction energy of the hole am|vac〉 with the doubly occupied d-center [see Eq. (A2) for
functions hik].
Substituting operators (2.9) and (3.2) into Eq. (3.3), projecting the result onto the basis
vectors (3.1) and using Eqs. (2.6), (A3), (2.12), (2.13), (2.14) and (A5)-(A10) we obtain a
closed system of equations for the coefficients Dm, f(q). This system defines the eigenvalues
of Eq. (3.3) for a given s. The symmetry of the problem allows us to look for the solutions
in the form f(q) = fm(q)e
imφ. Below we limit ourselves to a study of the isotropic case
of m = 0. (Discussion of the case m 6= 0 may be found in Ref. 35.) As a result we get
Dm = D0δm,0 and our system for given Sz =
+ s acquires the simple form
E + giµBB = β↑(s)V
∗D0 ,
(E − Ed,0)D0 = β↑(s)V − β↓(s)V N−1/2φ
h∗nn(q)f(q) ,
(E − εZ − Eq) f(q) = −N−1/2φ hnn(q)β↓(s)V
∗D0 . (3.5)
The energy of the free exciton state Q†q|0〉 is εZ + Eq [see Eq. (A11)].
The collective states localized around a magnetic impurity are described by solutions of
Eq. (3.5) outside the free spin-wave band (i.e. in the energy interval E < εZ or E > εZ+E∞).
The corresponding eigenfunctions are characterized by the regular envelope function f0(q).
We arrive then at the secular equation
β2↓(s)
|hnn(q)|2
E − εZ − Eq
β2↑(s)
E + giµBB
E − Ed,0
|V |2
(3.6)
for the energy E. The first term in the l.h.s. of Eq. (3.6), including the sum of spin-exciton
propagators, presents the self energy, which usually arises in the Schrödinger or Lippmann-
Schwinger equation describing the perturbation introduced by a short-range potential into
the continuous spectrum. The prototype of this term in the theory of magnetic defects is
the self energy for localized spin waves in the Heisenberg ferromagnet with a single substi-
tution impurity.39 Specific features of our model are manifested by the energy dependence
of impurity related processes. First, instead of a constant term (inverse impurity potential)
in the r.h.s. of Eq. (3.6) we have the inverse resonance potential |V |2/(E − Ed,0), which
stems from the hybridization between LLs and the 3d-level of impurity electron.28,29 Second,
an additional term describing impurity spin-flip process in terms of the single-orbital model
arises in the l.h.s. of Eq. (3.6).
B. Spectrum of the localized states
First we carry out a simple study considering solutions of Eq. (3.6) in the absence of an
exciton band, i.e. by formally substituting Eq = 0 into Eq. (3.6). (This is instructive in
order to classify the bound collective states.) We obtain then a simple algebraic equation
with two roots. When solving this equation we use the sum rule
|hnn(q)|2 = Nφ and
neglect the energy dependence in the r.h.s. due to the condition Ed,0 ≈ ∆ ≫ E. Each
doublet is bound to its own reference energy E0+(s) in accordance with the corresponding
spin component S
z = s+ 1 of the Mn
(+2) ion. Due to the kinematic exchange (2-nd order
spin-flip processes) each state in the Zeeman grid (lower root of Eq. 3.6) acquires a partner
state (upper root of Eq. 3.6), except for the highest level with s = 5/2, which remains
intact, because the spin flip processes are kinematically forbidden for this state. The level
splitting is illustrated by the scheme in Fig. 2. We see that the kinematic exchange makes
the Zeeman states of impurity ion non-equidistant and an additional multiplet of excited
states arises as a prototype of the bound spin-excitons.
Having this classification in mind we turn to calculating the bound exciton states for a
finite dispersion of the free spin waves. According to the estimates of the model parameters
presented in Subsection IIC we solve Eq. (3.6) for the realistic conditions EC>∼ giµBB ≫ εZ
whereas the ratio between the energies EC and |V |2/∆ may be arbitrary.
All the generic features of impurity-related states may be seen in the case of a unit filling
where n = 0 (ν = 1) and we study this situation in detail. The solutions we are looking for
are localized in the energy interval |E−Ed,0| ≈ ∆ so one can neglect the energy dependence
in the r.h.s. of Eq. (3.6). A graphical solution of Eq. (3.6) is schematically shown in Fig 3.
Two intersection points labeled as E
0 and E
x correspond to two discrete solutions. Just
as in Fig. 2 this pair of solutions arises at any s except for s = 5/2. The lower solution with
the energy E
0 is the state of the Mn
(+2) ion with the spin component 〈Ŝ(d)z 〉 ≈ s shifted
downwards from the value Evac + (s − 52)giµBB by the effective exchange interaction with
the spin-wave continuum (in this case D0 > Nb). The upper solution corresponds to the
spin-flipped state of the Mn(+2) ion with 〈Ŝ(d)z 〉 ≈ s+ 1 dressed with the spin-wave localized
on the impurity. In this case 〈Ŝ(s)z 〉 ≈ Nφ2 − 1 and Nb > D0. This bound exciton state,
-4 s=-3/2
s=-5/2
s=-1/2
s=1/2
s=3/2
s=5/2
FIG. 2: A scheme of the Mn(+2) Zeeman level splitting due to the kinematic exchange in the absence
of exciton dispersion. The bare Zeeman ladder is shown on the left. Five of six levels in this grid are
shifted down (extreme right column), whereas the s = 5/2 level remains not renormalized. Each of
these five levels has its high-energy counterpart. The energy is measured in the giµBB units. The
following values of the input parameters are chosen: εZ = 0.2, |V |2/∆ = 2 and Ed,0 = ∆ ≫ 1. The
factors β2↓(s) and β
↑(s) are presented by Eq. (2.7).
described semi-phenomenologically in Ref. 35, is shallow compared with the main charac-
teristic energy parameter EC. Like in many other impurity-related states in 2DEG
21,25,40 its
energy is confined within the interval −giµBB < E(s)x < εZ in the logarithmic vicinity of the
bottom of the delocalized spin-exciton band. Due to this fact one may find the level position
analytically. Using the quadratic approximation for the exciton dispersion law Eq = q2/2Mx
and turning from summation to integration in the l.h.s. of Eq. (3.6) one has
q 6=0
|h00(q)|2
E − εZ − Eq
≈ β2↓Mx ln [γMx(|εZ − E|)] (3.7)
Here Mx is the spin-exciton mass defined as 1/Mx =
dpp3vss(p)e
−p2/2/2 ∼ EC [see Eqs.
(A11) and (A13), the lB=1 units are used.] and γ = 1.781.... Then the binding energy
E(s)x = εZ −
β2↓MxgiµBB
β2↓Mx|V |2
. (3.8)
is found from Eq. (3.7). This result is valid provided at least one of the two inequalities,
β2↓MxgiµBB ≪ β2↑ or β2↓Mx|V |2/∆ ≪ 1, holds, which is not too strict requirement due to
the exponential smallness of the second term in r.h.s. of Eq. (3.8).
The asymptotic value of the lower state E
0 is also easily found. In the case of strong
hybridization |V |2/∆ ≫ EC one gets E(s)0 ≈ −giµBB − 5|V |2/6∆. In this asymptotic limit
0 Z-gi BB
- /|v|2delocalized
spin excitons
FIG. 3: Graphical solution of the secular equation. The l.h.s. and r.h.s. of Eq. (3.6) are shown as
functions of the argument E by solid and dashed lines, respectively (E-dependence in the r.h.s. is
neglected). Filled area indicates possible values of the l.h.s. because it belongs to the interval of E
where the sum in Eq. (3.6) becomes indefinite.
the excitation energy does not depend on s. In the opposite limit |V |2/∆ ≪ EC we have
0 ≈ −giµBB − β2↑(s)|V |2/∆.
In the intermediate region |V |2/∆ ∼ EC Eq. (3.6) for E(s)0 can be solved numerically. It
is convenient to rewrite this equation in the dimensionless form
β2↓(s)
2/2qdq
F (s) − ξe(q)
β2↑(s)
F (s) + g
+ 1 = 0 (3.9)
where ξ = ∆/Mx|V |2 is the ratio of the characteristic Coulomb energy in the Landau band
and the characteristic kinematic exchange energy. The relevant energy parameters in (3.6)
are redefined as E = (|V |2/∆)F (s)(ξ), εZ + Eq = M−1x e(q) and giµBB = (|V |2/∆) g. Then
the system of localized levels Ẽ
0,x counted from the global vacuum energy is described by
the set of equations
0,x = −giµBB(3/2− s) + F
0,x (ξ)|V |2/∆ (3.10)
with s = −5/2,−3/2,−1/2, 1/2, 3/2. The family of lower roots F (s)0 (ξ) of Eq. (3.9) changing
smoothly from −β2↑ − g at ξ = ∞ to approximately −6/5 − 5gβ2↑/6 at ξ = 0 describe the
renormalization of the Zeeman grid of impurity spin-flipped states due to the kinematic
exchange with LL continuum. To illustrate this dependence we have found the solution
of Eq. (3.10) for g = 0.25 neglecting εZ and modeling the spin-exciton dispersion by the
function e(q) = 2 − 2e−q2/4I0(q2/4), which corresponds to the ideal 2D case.3,4,5 (At the
same time the parameter Mx may be considered as an empirical value.) The results of this
calculation are presented in Fig. 4.
0 2 4 6 8 10
-0.6 F0
(s)( )
FIG. 4: The lower root of Eq. (3.9) with g = 0.25. The numbers s are indicated near the curves.
See text for further details.
C. Delocalized impurity-related excitations.
We conclude this section by a brief discussion of the delocalized states (free spin waves
distorted by the resonance magnetic impurity scattering). These states are described by the
functions f(q) with a divergent norm in the expansion (3.2). Secular equation for these states
cannot be presented in the form (3.6) but there are solutions satisfying Eqs. (3.5) at any
energy within the spin-exciton band, εZ < E < εZ + E∞. These states are the ’counterparts’
of the levels E
x in the spin wave continuum. Let q0(E) be a root of equation εZ + Eq0 = E.
Substituting
f(q) = C
δ|q|,q0 + u(q)(1− δ|q|,q0) (3.11)
into Eqs. (3.5) one gets three equations for the coefficients D0, C and u(q). Turning
from summation to integration and using the rule
δ|q|,q0 = 2q0L/π (L
2 = 2πNφ, being
the 2DEG area) one finds the coefficient u(q) from the equation β2↑(E − εZ − Eq)u(q) =
−β2↓N
φ hnn(q)(giµBB + E). Then equation
/4 = 1 +
(E + giµBB)
qdq|hnn|2
E − εZ − Eq
Ed,0 − E
|V |2
(3.12)
for the spectrum is derived from Eq. (3.5) in the thermodynamic limit (L,Nφ → ∞). It can
be readily seen that the norm of the function (3.11) diverges as
|u(q)|2 ∼ Nφ.
IV. POSITIVE g2DEG-FACTOR. PINNING OF THE QHF SPIN
Experimentally the magnitude of the g2DEG factor in GaAs/AlxGa1−xAs structures can be
altered gradually by changing pressure or by varying Al content (x). It can be made very
small and even change its sign.14 The value of g2DEG may be effectively reduced also due
to optical orientation of nuclear spins changing the electron Zeeman splitting (Overhauser
shift).13,17 In this section we discuss the impurity-related reconstruction of the ground state
and the spectrum of spin-flip excitations at small but positive values of g2DEG. It will be
shown below that even a minute amount of magnetic impurities can drastically influence the
QHF state.
Keeping the previous notations, it is now convenient to redirect the magnetization axis
(ẑ → −ẑ), i.e. to make formal transformation gi → −gi instead of changing the sign of
g2DEG. It is clear that at least in the absence of the s-d hybridization the global vacuum
state |d5, 5/2; |vac〉 serves as the ground state, and all the spin-flips cost positive energy.
The localized states can still be found from Eq. (3.6) with redefined Zeeman energies,
giµBB → −giµBB and εZ → ε∗Z. The latter parameter actually takes the values ε∗Z =
g∗2DEGµBB>∼ 0.1K. Making change g → −g in Eq. (3.9), we denote the lower root of this
new equation as F
x . This root corresponds to the energy of the localized spin exciton with
changed impurity spin projection, δS
z ≈3/2−s, where s= 32 ,
. The total spin
component is Sz =−
−s (when presenting results, we return to the ‘normal’ coordinate
system where ẑ is directed along ~B), and the energy counted off the global vacuum level is
given by formula
Ẽ(s)x = giµBB(3/2− s) + F (s)x (ξ)|V |2/∆ . (4.1)
(It should be noted that now the new global vacuum is really below the old one by the energy
5giµBB.) Functions F
x (ξ) are presented in Fig. 5.
Other roots of the secular equation belong to the continuous spectrum. These states may
be analyzed following the approach described in Subsection IIIC. The special ‘resonance’
solution of Eq. (3.12) with gi substituted for −gi arises in this case at E=giµBB > ε∗Z. Then
u(q) = 0 and the norm 〈X|X〉 diverges not as ∼ Nφ but as L ∼ N1/2φ (see discussion in the
next section). As a function of s, the delocalized ‘resonance’ states form a set of equidistant
levels
Ẽ(s)res = giµBB
(4.2)
(again the energy of global vacuum is taken as the reference level).
When looking for the x-type solutions at E < ε∗Z but |E| ≪ 1/Mx , one may use Eq.
(3.7). Then one obtains for the localized spin-exciton energy 41 E
x = |V |2F (s)x (ξ)/∆ the
0 2 4 6 8 10
(s)( )
FIG. 5: The lower root of Eq. (3.9) with negative parameter g. Calculation is performed for
g=−0.25 and e(q)=2−2e−q2/4I0(q2/4). The values of s are indicated near the curves.
following equation
E(s)x ≈ ε∗Z −
β2↓MxgiµBB
β2↓Mx|V |2
, (4.3)
instead of (3.8). Here s = 3
has to be taken for the excitation from the ground state, then
β2↑ =
and β↓=1. The exponentially small energy E
(3/2)
x thus corresponds to formation of
bound spin exciton of large radius. However, for sufficiently small ε∗Z (or for a strong enough
kinematic exchange) the energy E
(3/2)
x becomes negative which means an instability of the
global vacuum |d5, 5/2; |vac〉 considered as the QHF ground state. This instability appears
provided
ξ < ξc1 , (4.4)
where ξc1 is determined by the equation
|V |2F (3/2)x (ξc1)/∆+ ε∗Z = 0 . (4.5)
The question arises, whether the condition (4.4) mean the global reconstruction of the
ground state and appearance of a new state with many spin excitons bound to the magnetic
impurity? To clarify this point, we discuss the limiting situation where ε∗Z → 0 but still
Z→∞. Then the ground state at any ξ is no longer the global vacuum because creation
of one spin exciton bound to the impurity lowers the energy of the system. The corresponding
energy gain compared to the global vacuum is presented as |V |2G1(ξ)/∆. [The subscript
‘1’ corresponds to one bound exciton; specifically, we have G1(ξ) = F
(3/2)
x .] To answer
the question, one should consider the situation with K captured spin excitons (then Sz =
K− Nφ+5
) and calculate the proper value GK(ξ) at arbitrary K. The latter is determined
by the competition between the antiferromagnetic kinematic exchange, which forces 2DEG
spins to reorient in the direction opposite to the impurity spin, and the Coulomb-exchange
energy appearing due to 2DEG inhomogeneity in a cluster of K spin excitons bound to
the impurity. This inhomogeneity energy is measured in 1/Mx units. Calculation of GK at
K & 1 (but K 6= 1) is beyond the abilities of our present approach but we can consider the
case of K≫1 and find the conditions under which such a massive pinning of 2DEG spins in
the vicinity of the impurity turns out more advantageous than binding of single spin exciton
(i.e. G∞>G1).
A. Skyrmionic states created by magnetic impurities
The state with K ≫ 1 can be described as a collective topological defect (skyrmion)
pinned to a magnetic impurity.36 A smooth inhomogeneity in the system of spins may be
presented as a continuous rotation in the 3D space. If one characterizes the local direction
of the spins by a unit vector ~n(r) with components nx = sin θ cosϕ, ny = sin θ sinϕ, and
nz = cos θ (ϕ and θ are the two first Eulerian angles) then the conditions θ|r=0 = 0 and
r=∞ = π inevitably result in the appearance of the topological invariant qT =
drρ(r)
where the density
ρ(r) =
~n · (∂x~n)× (∂y~n) (4.6)
is a vortex characteristics of the spatial twist. The value qT has to be an integer nonzero
number.42 Its physical meaning is the number of excessive (qT < 0) or deficient (qT > 0)
electrons in the system,11,12,16,18 i.e. qT = Nφ − N . In a perfect 2DEG and at nearly zero
Zeeman gap (ε∗Z→0) such a weakly inhomogeneous skyrmion state has the energy
Esk =
E∞qT +
(|qT| − qT) . (4.7)
This result is valid within the single Landau level approximation (see, e.g., Ref. 18). It is
enough to consider the case qT=±1, because any state with |qT|> 1 is merely a combination
of ‘singly-charged’ skyrmions. Due to the hybridization with the impurity the skyrmionic
state gains a negative kinematic exchange energy. The latter has to be taken into account in
combination with the Coulomb-exchange energy (4.7) and with the finite positive Zeeman
energy at g∗
EZ = ε
ZK, where K =
4πl2B
(1 + cos θ)dr (4.8)
(in the clean 2DEG the skyrmion energy is given by Esk+EZ).
One can conclude from symmetry considerations that the impurity is located at the cen-
ter of the topological defect. Then additional pinning energy may be found by means of
the conventional energy minimization procedure where the Euler angles are used as varia-
tional parameters. This energy is a difference between the energy of the global vacuum state
with a distant skyrmion and the ground state energy calculated in the presence of magnetic
impurity at the center of the topological defect (cf. Ref. 36 where similar procedure was
elaborated in the limit of potential impurity scattering). Namely, to calculate the contribu-
tion of magnetic impurity at K≫1, one should consider a domain around impurity which is
small in comparison with a characteristic area of the skyrmion, but contains a large enough
number of spin-flipped LL electrons involved in the formation of pinned topological defect.
Then the situation becomes similar to that considered in Sec. III: s-d hybridization of the
impurity electron with the m=0 electron in this domain generates the kinematic exchange
in accordance with Fig. 1, and leads to reconstruction of the spectrum in accordance with
Eq. (3.6). The shift of the energy with respect to the global vacuum is given by the value
5giµBB+ Ẽ
(−5/2)
0 , where Ẽ
(−5/2)
0 is determined by Eq. (3.10).
43 Hence we obtain that the
pining energy is Esk,pin=−giµBB − F (−5/2)0 (ξ)|V |2/∆, where F
(−5/2)
0 is shown in Fig. 4. In
the limit of strong pining (Esk,pin ≫ Esk) and ‘frozen’ impurity spin (g ≫ 1) this result agrees
with the pinning energy found earlier.36
The energy Esk,pin is calculated in the leading approximation, which does not depend on
the charge qT. However, it is instructive to obtain the correction related to the inhomogeneity
of the texture. It is known16,18 that the density (4.6) may be interpreted in terms of effective
magnetic field appearing in the Schrödinger equation due to this inhomogeneity. One may
introduce the renormalized magnetic length lB → l̃B as
− 2πρ(r). (4.9)
Taking into account that |V |2/∆ ∼ 1/l2B and ξ ∼ lB and rewriting Eqs. (4.6) and (4.9)
in terms of the Euler angles,42 one finds the correction to pinning energy due to the finite
radius R∗ of the skyrmion core (see Ref. 36 for a detailed calculation). The corrected energy
is determined by the value ρ(0) and has the form
sk,pin = −giµBB −
|V |2
(−5/2)
0 (ξ)− qT
(−5/2)
0 − ξ
(−5/2)
, qT = ±1 .
(4.10)
It is assumed here that g ≪ 1.
The skyrmion core radius R∗ is found by considering the competition between the Zeeman
energy (4.8) and the energy of Coulomb repulsion.11,44 Generally speaking, in our case in
order to find R∗ we should include the energy Esk,pin in the minimization procedure. However,
this correction only insignificantly influences the result due to the condition R∗ ≫ lB and
because of the rather strong e-e interaction resulting in the skyrmion formation. Using the
realistic estimate for the kinematic exchange energy |V |2/∆<∼EC the minimization yields
the same formula
(R∗/lB)
2 ln (l2BEC/ε
∗2) (4.11)
as in the case of ‘free’ skyrmions 44, where R∗3 = 9π2l2Be
2/ [64ε∗Zκ ln (EC/ε
Z)]. The number
of spin-flipped electrons turns out to be rather large
9π2e2
κε∗ZlB
[ln (EC/ε
1/3 ∼ 10− 20 (if ε∗Z ∼ 0.1K) . (4.12)
We first consider the regime where there is no skyrmions in the clean system but these
collective excitations could be created due to strong enough kinematic exchange interaction
between the LL electrons and magnetic impurities. This is the situation where the inequality
(4.4) is valid, and besides the condition |N−Nφ|≪Ni is realized, where Ni is the number
of impurities. The electroneutrality of the system requires that the topological defects are
created as skyrmion-antiskyrmion pairs. Two impurities are able to create a skyrmion-
antiskyrmion pair provided the pinning energy E
sk,pin+E
sk,pin exceeds the energy increase
due to the skyrmion-antiskyrmion gap. The latter in accordance with Eq. (4.7) includes the
Coulomb-exchange part equal to M−1x and twice the Zeeman energy [Eqs. (4.8) and (4.12)].
In addition, the energy of a skyrmion and an antiskyrmion pinned by two neighbouring
magnetic impurities has to be lower than the double energy of a pinned spin exciton. Thus
the condition G∞<G1 for creation of a pinned skyrmion-antiskyrmion pair can be rewritten
in the form
ξ<ξc,∞ , (4.13)
where the critical value ξc,∞ can be obtained with the help of Eq. (4.10):
g + F
(−5/2)
0 (ξc,∞) + ξc,∞/2 + EZ∆/|V |2 = F (3/2)x (ξc,∞) . (4.14)
Under the condition (4.13) an impurity acquires the localized magnetic moment K ∼ B−1/3
antiparallel to its own moment and exceeding it (when, e.g. K > 5/2 in the GaAs:Mn case).
B. Phase diagram of QHF ground state at g∗
There are two critical transitions in our problem: first, the global vacuum is destroyed
when ξ becomes less than ξc1 and single spin-flip exciton appears (this state may be charac-
terized as a ‘local pinning’); second, the massive pinning of 2DEG spins takes place when ξ
reaches the value ξc,∞. However, this scenario is somewhat changed if one takes into account
finite ratios Ni/Nφ. Indeed, up to this point we have supposed that the Zeeman energy ε
corresponding to the ‘global flip’ of all 2DEG spins is larger than any contribution to the
QHF energy due to the magnetic impurities. This global spin-flip actually represents the
spin configuration treated as the ground state in the previous section. When counted from
the global vacuum, its energy per one impurity is Es-f = giµBB+Nφε
Z/Ni+|V |2F
(−5/2)
0 (ξ)/∆.
Negative Es-f means that available magnetic impurities are able to polarize completely all
2DEG electrons even at positive g∗
. In agreement with the above discussion, one can
conclude that such a complete polarization takes place when
Es-f(ξ) < Emin(ξ) , (4.15)
where Emin = min{0, |V |2F (3/2)x (ξ)/∆+ε∗Z, giµBB+|V |2F
(−5/2)
0 (ξ)/∆+
M−1x +EZ}.
0.01 0.1
(g*c,
                          Localized skyrmions
   (massive pinning of the 2DEG spin)
       S
= (K-5/2)N
-N  /2,   K>>1
'Global vacuum':
= -(N +5 N
 / N  0 
bal p
  S z=
 (N -
5 N i 
     'Lo
cal pi
nning
    of 
the 2
spin:
    S z=
 -(N +
3N i )/
FIG. 6: Phase diagram illustrating the reconstruction of the QHF ground state at g∗
> 0 for
two cases: isolated impurity Ni/Nφ → 0 (see elucidating legend above the main picture) and finite
impurity concentration Ni/Nφ = 0.01. The calculation was carried out for the Zeeman parameters
g=giµBB∆/|V |2=0.25 and ε∗Z = 0.05(g∗2DEG/0.44)|V |2/∆ and for the spin exciton dispersion equal
to ξe(q) = 0.05(g∗
/0.44)+2ξ
1−e−q2/4I0(q2/4)
in |V |2/∆ units. Comments in the figure refer
to the Ni/Nφ = 0.01 case. See text for other details.
The phase diagram of our system at zero temperature is determined by the interplay
between Zeeman splitting, Coulomb interaction and kinematic impurity exchange energy,
and controlled by the impurity concentration. These factors are characterized by the di-
mensionless parameters g∗
, ξ and Ni/Nφ. One can construct this diagram by employing
the inequalities (4.4), (4.13) and (4.15). The results for both cases of infinitely small and
finite ratio Ni/Nφ are presented in Fig. 6 in the (g
, ξ) coordinates. We expressed the e-e
interaction values entering the skyrmion Zeeman energy [Eqs. (4.11) and (4.12)] in terms
of the parameter Mx: EC = M
x , e
2/κlB = M
x (8/π)
1/2. The phase diagram for the
Ni/Nφ → 0 case is explicated by the legend above the main picture. The ξ=ξc1(g∗2DEG) curve
in Fig. 6 separates states with unbroken global vacuum (the area above this line) and states
of ‘local pinning’ where each impurity is dressed by one bound spin-exciton. The dotted
line [ξc,∞(g
) curve] separates the state with ‘local pinning’ and the state of massive spin
reversal (K ≫ 1) determined by the pinned skyrmions (hatched area below this line).
In the more realistic case of Ni/Nφ=0.01, the curves ξc1(g
) and ξc,∞(g
) formally
remain the same since the parameter Ni/Nφ does not enter Eqs. (4.5) and (4.14). However,
in this case essential part of the (g∗
, ξ) area belongs to states where the 2DEG spins are
globally polarized in the ~B direction in spite of positive g∗
. This area filled by dark-grey
presents solutions of inequality (4.15). Unbroken global vacuum occupies only the blank
sector in the upper right corner of the phase diagram. At large ξ but fixed |V |2/∆, the line
separating the blank and dark-grey sectors tends to g∗
=0.088, which corresponds to value
ε∗Z=Ni|V |2/Nφ∆ being the result of the Es−f(ξ→∞)=0 equation. At the same time if the
ξ→∞ limit is realized owing to vanishing V , then both systems of the impurities and of the
2DEG become independent and at any positive g∗
the global vacuum presents certainly
the ground state. The light-grey area below the ξ=ξc1(g
) line but above the dotted line
corresponds to the singly spin-flip states with one exciton bound to impurity. The hatched
light-grey domain below the dotted line corresponds to the state with the localized skyrmions
created by strong kinematic exchange [Eq. (4.13)]. In our specific case of the Ni/Nφ = 0.01
ratio the dark-grey sector is not contiguous to this skyrmionic region. The total QHF spin
Sz in various states of the phase diagram is indicated in the picture.
Now we discuss the regime where free skyrmions are already available in the system
because the number of electrons well deviates from the quantum flux number. Namely, we
consider that |N−Nφ|>Ni (although still |N−Nφ|≪Nφ). In this case ‘excessive’ skyrmions
may be bound to an impurity. The result depends on the QHF phase. In the globally pinned
phase (dark-grey area) the binding is impossible since the effective interaction between the
impurity and the skyrmion is repulsive. In the state of local pinning (light-grey unhatched
domain) the binding also does not occur. Indeed, the binding energy would be equal to
Esk,pin (4.10) but due to the condition (4.4) this value is smaller than the spin exciton
delocalization energy −|V |2F (3/2)x /∆. At the global vacuum (blank sector) the binding takes
place and the binding energy is equal to the pinning energy (4.10). Certainly, the binding
takes place in the skyrmionic ground state (light-grey hatched sector). However, in contrast
to the |N−Nφ|<Ni case, now all Ni impurities bind skyrmions of the same charge qT, where
qT= ±1 if correspondingly N <>Nφ.
To conclude this section, it is worthy to remind that we have only considered the situation
where the g∗
>0 ground state is realized in the most symmetric phases when the pinned
spin K is equal to 0, 1 or K ≫ ∞. As it has been seen, there are only two critical
parameters ξc1 and ξc,∞ in this case. However, one might suppose that transition from
the local pinning (K =1) to the skyrmionic phase of massive pinning would proceed more
smoothly with diminishing parameter ξ. Namely, below the ξc1(g
) curve there should
be critical value ξ = ξc2(g
) at which the transition K = 1 → K =2 occurs. This value
would be the root of equation G1(ξc2) =G2(ξc2). The next critical point would correspond
to the K = 2 → K = 3 transition and so on. This sequence of values ξc1> ξc2 > ...ξcK > ...
where GK−1(ξcK) = GK(ξcK) should condense in the vicinity of ξc,∞. Actually this means
that the light-grey unhatched domain in Fig. 6 would present not only the singly spin-flip
2DEG state but a set of states with K = 1, 2, 3, ... spin-excitons localized at the impurity
where K is growing with diminishing ξ. In reality, for a finite ε∗Z, K reaches the value given
by Eq. (4.8) at ξ= ξc,∞. Such a ‘stratification’ of the light-grey unhatched sector would be
the only qualitative change of the phase picture of Fig. 6. Quantitative changes would be
presented by appropriate shifts of the ξc,∞ curve and of the boundary between the dark-grey
and light-grey areas. However, it is clear that these shifts would not be significant. The
corresponding crossover parameters ξ would at least remain of the same order as the ones
calculated with the help of Eqs. (4.14) and (4.15).
V. DISCUSSION
We have found that the interplay between the kinematic impurity exchange and the
Coulomb interaction in 2DEG results in the appearance of bound exciton states and in the
renormalizaton of impurity spin states, including the reconstruction of the QHF ground state
at g∗2DEG > 0.
Among the available experimental techniques, the inelastic light scattering (ILS) method
seems to be the most useful method for experimental study of the 2DEG spectra (see Refs.
7,10,45 and references therein). However, this tool has some special features, and it is
helpful to discuss our results from this point of view. Let us first consider the g2DEG<0 case.
When measuring the energy from the ground state level Ẽ
(−5/2)
0 , where the impurity has the
maximum spin projection [see Eq. (3.10)], one obtains ten levels of the localized excitations
0 1 2 3
FIG. 7: The case of g2DEG < 0. Energies of the excitations measured from the ground state
are plotted in units of |V |2/∆ as functions of ξ = ∆/Mx|V |2. Filled areas show energies of the
delocalized spin excitons. See text for details.
∆E related to the spin changes δSz = 0, 1, ...5
∆EδSz ,t = giµBBδSz + (|V |2/∆)
t (ξ)− F
(−5/2)
0 (ξ)
(t = 0, x), (5.1)
where s = δSz − 5/2, and the index t labels the type of the excited state [in Eq. (5.1) it is
considered that F
(5/2)
t ≡ 0]. Within the scope of the experiment where only the |δSz| ≤ 2
excitations seem to be observable as ILS peaks, we plot in Fig. 7. these five levels calculated
with the help of Eq. (3.9) for the parameters g = 0.25 and εZ = 0.05|V |2/∆. This calculation
is done for the sake of demonstration with the function Eq = 2Mx−1
1− e−q2/4I0(q2/4)
the fitting parameter Mx used to describe the spin-wave dispersion. In the available wide
quantum wells the inverse spin-exciton mass is relatively small.46 Hence the values ξ < 1
seem to be experimentally relevant, and the evolution of non-equidistant excitations ∆EδSz ,x
as a function of ξ (and therefore of B) should be observable in this interval.
The non-localized states discussed in Section IIIC actually present a transformation of
the x-type excitations when the spin exciton is detached from the impurity and falls in the
spin-wave continuum. The bottoms of continuous bands are shown as filled areas in Fig. 7.
The band edges are higher than the ∆EδSz ,x curves by the quantity −(|V |2/∆)F
(δSz−
x (ξ)
[see Eqs. (3.8) and (3.10)], and therefore the latter may be treated as spin-exciton binding
energy. However, it seems to be difficult to observe these states in the ILS spectra because
of comparatively small oscillator strengths, specifically, due to divergence of the envelope
function f(q).
Similar ILS picture should also take place for g∗
>0 in the phase of the 2DEG global
pinning (dark-grey area in Fig. 6). In the skyrmionic phase (light-grey hatched sector)
there are intra-impurity ILS transitions determined by Eq. (5.1). Besides, two new types
of resonances are expected: the first is the skyrmion delocalization with δSz = 0 and with
excitation energy equal to Esk,pin (4.10); another one is transition δSz = −1 where the
delocalized skyrmion leaves the impurity with the bound spin exciton. In the latter case the
transition energy is Esk,pin−|V |2F (3/2)x +ε∗Z.
In the global vacuum and local pinning states (blank and light-grey unhatched domains)
the ILS spectrum is determined by transitions between levels (4.1) and (4.2), so that, e.g., the
ILS transitions to the localized states from the global vacuum are determined by the energies
∆EδSz = ε
Z + giµBB(δSz−1) + |V |2F
(5/2−δSz )
x /∆ and correspond to nonzero spin change
δSz=1, 2, (3, 4, 5). At the same quantum numbers δSz there should also be resonance features
related to the impurity spin rotation, which cost the energy ∆EδSz ,res = giµBBδSz. These
resonances are in fact transitions to the continuous spectrum which should be noticeable on
the background of free spin waves contribution (ε∗Z< ∆E<E∞) [see the comment above Eq.
(4.2)].47 The ILS spectrum of excitations from the local-pinning ground state is presented:
first, by the δSz = −1 transition to the global vacuum (this energy is equal to ∆E−1 =
−|V |2F (3/2)x /∆ − ε∗Z); second, by the δSz = 1, 2, (3, 4) transitions to the localized spin-flip
states with energies ∆EδSz = giµBBδSz + |V |2
(3/2−δSz )
x −F (3/2)x
/∆ ; and third, by the
δSz = 0, 1, 2, (3, 4) transitions to the resonance states in the continuous spectrum with
transition energies ∆EδSz ,res=giµBB(δSz+1)− |V |2F
(3/2)
x /∆− ε∗Z .
Finally, we note that currently there are several possibilities for the experimental study of
skyrmion-like textures (e.g., see Refs. 13,14,17 and 48). However, for any method one of the
most serious obstacles impeding observation of spin-flip phases is the very narrow interval in
the vicinity of the g∗
=0 factor where the spin-flip reconstruction of the ground state or
skyrmion-like excitations are possible. From this point of view the minor magnetic doping
would become an additional fine tuning tool allowing to change the balance between EZ and
EC and to influence the skyrmion formation.
VI. ACKNOWLEDGMENTS
The work was mainly done during the authors’ stay at MPIPKS, Dresden. S.D. thanks
for the hospitality the Grenoble High Magnetic Field Laboratory and the Abdus Salam
International Center of Theoretical Physics (Trieste) where part of this work was also carried
out and acknowledges support of the Russian Foundation for Basic Research. V.F. and K.K.
acknowledge partial support of Israeli Science Foundation, grant No. 0603214212.
APPENDIX A: EXCITONIC REPRESENTATION
The excitonic representation is a convenient tool for a description of electron-hole states
in a 2DEG multiply degenerate in m. When acting on the vacuum state |vac〉 (in our case
this vacuum is defined in Subsection IIA), the exciton-creation operators form a system
of basis states diagonalizing the Hamiltonian including a considerable part of the Coulomb
interaction. Due to translational invariance of a clean 2DEG these exciton states are classified
by the 2D momentum q and the degeneracy turns out to be lifted. The exciton operators
for a single LL were first introduced in Refs. 6. The commutation rules for the same case of
single LL were found in Ref. 49 (see also Ref. 9 and references therein).
Unlike previous papers, where the ER technique was developed for the Landau gauge,
we use the symmetric gauge for bare one-electron states. In the symmetric gauge the spin-
exciton creation operator is expressed in terms of the am and bm Fermi operators,
Q†q = N
m,m′=0
h∗mm′(q)b
m−nam′−n , (A1)
(cf. the definition based on the Landau gauge 6,9,18,36). In this expression
hmk(q) = (m!/k!)
k−mLk−mm (q
2/2)e−q
2/4 (A2)
are the building block functions used in the ER technique, q−= iqe
2 ≡ i(qx−iqy)/
and Lk−mm are the Laguerre polynomials. Here and below all lengths are measured in the
magnetic length lB=1 units. The spin-exciton states are orthogonal and normalized,
〈vac|Qq1Q†q2 |vac〉 = δq1,q2 . (A3)
The operators (A1) together with the intra-sublevel operators
A†q =
m,m′=0
h∗mm′(q)a
m−nam′−n, and B†q =
m,m′=0
h∗mm′(q)b
m−nbm′−n . (A4)
form a closed Lie algebra. In order to check it we first obtain the commutation relations
[Q†q, a†m] = N
h∗m+n,k(q)b
k−n, [Q
q, bm] = −N
h∗k,m+n(q)ak−n, (A5)
[Aq, am] = −
hm+n,k(q)ak−n, [Bq, b†m] =
hk,m+n(q)b
k−n, (A6)
[Qq, a†m] = [Qq, bm] = [Aq, bm] = [Aq, b†m] = [Bq, am] = [Bq, a†m] ≡ 0 . (A7)
As a result we see that operators (A1) and (A4) really form a closed algebra with the
commutation relations 9,18,49
Qq1 ,Q+q2
= ei(q1×q2)z/2Aq1−q2 − e−i(q1×q2)z/2Bq1−q2 ,
e−i(q1×q2)z/2[A†q1 ,Q
] = −ei(q1×q2)z/2[B†q1 ,Q
] = −N−1φ Q
q1+q2
[A†q1,A
] = −
sin [(q1 × q2)z/2]A†q1+q2 ,
[B†q1 ,B
] = −
sin [(q1 × q2)z/2]B†q1+q2 .
Acting on the vacuum state the intra-sublevel operators result in
A†q|vac〉 = δq,0, and B†q|vac〉 ≡ 0 . (A9)
The excitonic basis Q†q|vac〉 determine the set of eigenstates of a clean 2DEG,
1 + Ĥs−s),Q†q
|vac〉 = (εZ + Eq)Q†q|vac〉 . (A10)
Here Eq stands for the Coulomb energy of the free spin wave defined by the equation 3,4,5
Wss(p)
1− ei(p×q)z
dppvss(p) [hnn(p)]
[1− J0(pq)] , (A11)
J0(pq) is the Bessel function.
The Coulomb vertices in the Hamiltonian (2.12), (2.13) are given by the equations
Wss(q) = vss(q)[hnn(q)]
2, Wsd(q) = vsd(q)hnn(q), (A12)
where 2πvss(q) and 2πvsd(q) are the 2D Fourier transforms of the average s-s and s-d
interaction potentials. One can present them as 1
vss(q) =
dz1dz2e
−q|z1−z2||ζ(z1)|2|ζ(z2)|2,
vsd(q) =
dze−q|z−zd||ζ(z)|2 . (A13)
(The impurity cite is assumed to be at the point Rd = {0, 0, zd} .)
APPENDIX B: SPIN OPERATORS
Bound spin excitons are characterized by the spin numbers Sz and S
2. The corresponding
operators have the form
Ŝz = Ŝ
z + Ŝ
z , (B1)
where
Ŝ(s)z =
(A0 − B0) ;
= NφQ†0Q0 +
Ŝ(s)z
+ Ŝ(s)z (B2)
Ŝ(d)z =
(n̂↑ − n̂↓) ;
(n̂↑ + n̂↓)−
n̂↑n̂↓ (B3)
are the spin operators for 2DEG (in the excitonic representation) and for magnetic impurity
(in terms of the single-orbital model), respectively. The total squared spin operator for the
system is defined as
+ 2Ŝ(s)z Ŝ
↑c↓ + c
↓c↑Q0
. (B4)
The operator Ŝz commutes with the Hamiltonian (2.9), while for Ŝ
2 one has
Ŝ2, Ĥ
≡ N1/2φ (giµBB − εZ) c
↓c↑Q0
+(β↑ − β↓)V
n̂↑b0 +N
φ Q0a0
− c†↑
n̂↓a0 +N
− H.c.
The difference between the g factors of the magnetic impurity and the host QHF and the
difference between the projection factors β↑ and β↓ measure the spin non-conservation.
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substantiation. In this case the state with Sz =
+s may be created, e.g., as a two-exciton
vector Q†q1Q
q2 |d5, s+2; |vac〉 which is admixed to the state |d5, s+1;Q
q|vac〉 ≡ |s+;Q†q|vac〉.
One may neglect this admixture because the bi-exciton binding energy Ebi−x is much smaller
than both giµBB and |V |2/∆ values, whereas the single exciton binding energy E∞ (A11) is
comparable with these two characteristic energies. It is known 8,9,15 that even in the ideal 2D
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takes place.
38 In terms of the single-orbital model the combinations |s−; vac〉+N
0|0〉 and |s−; vac〉−
0|0〉 present pure spin states with definite numbers S = Sz+1+O(1/Nφ) and S = Sz+
O(1/Nφ) respectively.
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πJσ̂zδ(z − zd)δ(r) (see Ref.36) gives for the bound exciton energy an equation similar to (4.3).
In terms of the present model J = |β↓V lB |2/ζ(zd)∆.
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http://arxiv.org/abs/0709.0541
B/2.4meV. See also: I.V. Kukushkin, J.H. Smet, D.S. Lyne Abergel, V.I. Fal’ko, W.
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47 Note that due to smallness of the F
(5/2−δSz)
x values in the region of the blank sector parameters
in Fig. 6, the δSz peaks for transitions to the continuous spectrum will probably almost coincide
with the δSz+1 peaks related to the mentioned transitions to the localized states.
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	Introduction
	Model Hamiltonian
	Single-orbital model. Spin-selective hybridization.
	Description of the employed simplifications
	Interaction Hamiltonian in excitonic representation
	Numerical estimates of the energy parameters
	Collective spin-flip states. negative g2DEG-factor
	Secular equation
	Spectrum of the localized states
	Delocalized impurity-related excitations.
	Positive g2DEG-factor. Pinning of the QHF spin
	Skyrmionic states created by magnetic impurities
	Phase diagram of QHF ground state at g2DEG*>0 
	Discussion
	acknowledgments
	EXCITONIC REPRESENTATION
	SPIN OPERATORS
	References
