Pis’ma v ZhETF, vol. 109, iss. 1, pp. 63 - 64

January 10

Light absorption properties related to long-living ensemble of spin

excitations in an unpolarized quantum Hall system

S. Dickmann¹⁾

Institute for Solid State Physics of RAS, 142432 Chernogolovka, Russia

Submitted 22 October 2018

Resubmitted 7 November 2018

Accepted 8 November 2018

DOI: 10.1134/S0370274X19010120

The cyclotron spin-flip excitation (CSFE) in the

exciton ensemble as a metastable system with a given

ν = 2 quantum Hall system, being the lowest-energy

number of excitons N.

one [1-3], has an extremely long lifetime. The latter is

It is noticeable that the CSFE represents a purely

theoretically estimated to be up to several milliseconds

electronic kind of magnetoexciton

[6] where the

[4]. In fact the CSFE relaxation found experimentally

quantum-mechanical average of distance between

in the unpolarized quantum Hall system created in a

positions of an electron promoted to the spin-up

real GaAs/AlGaAs heterostructure reaches 100 µs [5] at

sublevel of the first Landau level and an effective “hole”

finite temperature T ≃ 0.5 K, that seems to be a record

(vacancy in the spin-down sublevel of the zero Landau

value for a delocalized state excited in the conduction-

level) is equal to Δr = l2Bq × z [1], where q is the

√

band of mesoscopic systems. Such a slow relaxation sug-

magnetoexciton wave-vector, l_B

cℏ/eB is the

gests that ensemble of the excitations, being a kind

magnetic length. Thus this excitation possesses electric

of magnetoexcitons [6] and obeying the Bose-Einstein

dipole-momentum d_q = el^2Bq × z.

statistics, can experience at sufficiently high concentra-

I. Using the “excitonic representation” technique

tion a transition to a coherent state - Bose-Einstein con-

(see, e.g., [7]) we study an incoherent state of the CSFE

densate. Both the CSFE creation and the CSFE moni-

ensemble:

toring are performed by optical methods [3, 5]. In this

|ini, N〉 = Q^†q

Q^†q

...Q† |0〉,

(1)

connection, it is interesting to study the contribution to

N-1

∑

the light absorption related to the CSFE ensemble in

−1/2

where operator Q^†q = N_φ

e-iqx(p+qy/2)b^†pap+qy

the 2DEG. In the present work we perform a compara-

(first used in works [8]), acting on the ground state |0〉,

tive analysis of the absorption by the CSFE ensemble in

creates a magnetoexciton with 2D momentum q; |0〉 de-

incoherent and coherent phases. (This also strongly cor-

notes the ν = 2 ground state with a fully occupied zero

relates with the light emission if the resonant reflection

Landau level; a^†p is the operator creating an electron on

technique is used [5], therefore it is sufficient to consider

the upper spin sublevel of the zero Landau level with

only the absorption.)

spin-down, i.e., antiparallel to the magnetic field, and

The CSFE is a solution of the many-electron

b^†p creates an electron on the first Landau level with the

Schrödinger equation with the δS = 1 change of the

spin directed along the magnetic field (here and below

total spin as compared to the ground state where S = 0

p-numbers as well as the wave-vector q are measured in

[1]. Generally, this excitation is a triplet with S = 1 and

1/l_B units).

S_z = 1, 0, -1. All three components have equidistant en-

The perturbation operator responsible for the light

ergies gapped by the Zeeman value |gµ_BB|. The lowest-

absorption has the form

energy component corresponds to S_z = 1 because the

∑

g-factor is negative in the GaAs heterostructures. We

A=A V^†pa^†p,

(2)

will consider only these S = S_z = 1 magnetoexcitons

in our study. A noticeable concentration of such excita-

where V^†p is creation operator of a valence heavy-hole,

tions, N/N_φ ≲ 0.1 (N_φ is the total number of states in

and A is a certain constant. Operator (2) is uniquely

the Landau level), can be achieved experimentally [5].

determined by “verticality” of optical transitions condi-

In this letter we study the exciton ensemble only in the

tioned by the inequality Lk_photon∥ ≪ 1, where L is a

“dilute limit”, thus ignoring the CSFE-CSFE coupling.

characteristic of the electron 2D-density spatial fluctua-

Due to the very long CSFE relaxation time we study the

tions and k_photon∥ is the photon wave-vector component

parallel to the electron system plane. This condition ac-

¹⁾e-mail: dickmann@issp.ac.ru

tually is of met [5].

Письма в ЖЭТФ том 109 вып. 1 - 2

2019

S. Dickmann

The action of the

A operator on state |ini, N〉 re-

initial state (7)]. The square of the matrix element of

∑

sults in the A_i |f, q_i〉 sum, where each of N possible

the transition between the states |N〉 and |f〉 (provided

final states represents a combination of N-1 purely

their normalization) is



electronic magnetoexcitons with one magnetoexciton



|M_N |² =

〈〈f||A||N〉〉2 ≈ |A|2N,

(8)

formed by a valence hole:

∏

that is by factor N larger than, e.g., the matrix element

|f, q_i〉 = -Xq Q† |0〉.

(3)

(4) (double brackets ||...〉〉 denote the normalized states).

j=i

If we consider individual domains with areas dR,

∑

much smaller than the square of the characteristic scale

Here

X_q = N^-1/2φp e-iqx(p+qy/2)V^†pb^†

is the ex-

p+qy

of the spatial fluctuations of the random potential Λ²,

citon operator which, by acting on the ground state

but much larger than the area of the magnetic flux quan-

generates the valence hole and the b-sublevel electron.

tum 2πl^2B, then the total relaxation rate can be pre-

By neglecting any inter-excitonic coupling one finds the

sented as

transition matrix element squared

∫

2π|A|²N

|M_i|² = 〈q_i, f|A|ini, N〉2 ≈ |A|2.

(4)

R_II =

dRδ(ℏω+E_N,q-E_N-1,q-E_v-e,q), (9)

N_φℏ

Calculation of the relaxation rate is a summation pro-

where E_v-e,q is again the energy of the “valence magne-

cedure in accordance with the formula:

toexciton”. It can be shown that in the coherent phase

∑

all corrections of the first gradient approximation to the

2π

2π|A|² ∑

R_I =

|M_i|²δ(Dqi ) ≈

δ(Dqi ),

(5)

energies in the argument of the δ-function in (9) (i.e.,

ℏ

proportional to the electric field E(R) = -∇ϕ) mutu-

where

ally compensate each other. The second-order correc-

D_q = ℏω + E_q - E_v-e,q,

(6)

tions, ∼ E², are determined only by the E_v-e,q term.

ω is frequency of the probing laser beam, E_q to describe

This enables us to calculate the relaxation rate, R_II =

energy of the Q^†q|0〉 magnetoexciton, and E_v-e,q to de-

NK_II, and estimate the ratio of “oscillator strengths”

K_II/K_I ∝ B^3/2/E (where E ∼ Δ/Λ), which numerically

scribe energy of the “valence-magnetoexciton”

X^†q|0〉.

under conditions corresponding to the experiment [5] is

The summation (5) depends on the distribution of

approximately 10. In other words, in the coherent phase,

the quantum numbers q_i by their possible values in the

light absorption/emission in resonance reflection mea-

phase space. When the temperature is low enough (ac-

surements becomes about an order of magnitude more

tually at T < 1 K), this distribution is determined by

intense than in the incoherent phase.

the presence of a smooth random electrostatic poten-

The research was supported by the Russian Science

tial in the 2D heterostructure, so every magnetoexci-

Foundation: grant #18-12-00246. The author thanks

ton “gets stuck” at a certain point of the potential en-

I.V. Kukushkin and L.V. Kulik for the useful discussion.

ergy corresponding to the local minimum. The mini-

mum corresponds to zero group velocity, ∂E_q/∂q = 0,

Full text of the paper is published in JETP Letters

where the function E_q includes both the q-dispersion

journal. DOI: 10.1134/S0021364019010028

of the CSFE determined by the Coulomb coupling in

the many-electron system and the electrostatic energy

1. C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655

d_q∇ϕ(R) associated with the dipole moment of the ex-

(1984).

citation in the random external field ϕ(R). Our analy-

2. S. Dickmann and I. V. Kukushkin, Phys. Rev. B 71,

sis allows to find the rate R_I = NK_I, where “oscillator

241310(R) (2005).

strength” K_I depends on the magnetic field (increasing

3. L. V. Kulik, I. V. Kukushkin, S. Dickmann, V. E. Kir-

with the growth of B) and on parameters of the ran-

pichev, A. B. Van’kov, A. L. Parakhonsky, J. H. Smet,

dom potential: a characteristic amplitude of its spatial

K. von Klitzing, and W. Wegscheider, Phys. Rev. B 72,

fluctuations (Δ) and a correlation length (Λ). Actual

073304 (2005).

estimates are Δ ≃ 5-8 meV, Λ ≃ 50-60 nm.

4. S. Dickmann, Phys. Rev. Lett. 110, 166801 (2013).

II. In the coherent phase, all magnetoexciton have

5. L. V. Kulik

A.V. Gorbunov, A. S. Zhuravlev,

the same wave vector, that is, now the excitonic ensem-

V. B. Timofeev, S. Dickmann, and I.V. Kukushkin, Na-

ture Sci. Rep. 5, 10354 (2015).

ble has the form:

6. L. P. Gor’kov and I. E. Dzyaloshinskii, JETP 26, 449

|N〉 = (Q^†q)^N |0〉.

(7)

(1968).

7. S. Dickmann, V. M. Zhilin, and D. V. Kulakovskii, JETP

Considering this state as initial, we find the action of

101, 892 (2005).

operator (2) resulting in the final state |f〉 = |A|N〉 =

8. A. B. Dzyubenko and Yu. E. Lozovik, Sov. Phys. Solid

= -ANXq|N - 1〉 which has energy equal to EN-1,q +

State 25, 874 (1983); ibid 26, 938 (1984).

+ E_v-e,q [E_N,q to designate energy of the coherent

Письма в ЖЭТФ том 109 вып. 1 - 2

2019