Page 1

arXiv:cond-mat/0312427v5 [cond-mat.mes-hall] 29 Jan 2004

Goldstone-Mode Relaxation in a Quantized Hall Ferromagnet

in the Presence of Smooth Random Potential

S. Dickmann

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

We discuss the spin relaxation of a strongly correlated two-dimensional

(2D) electron gas (2DEG) in the quantized Hall regime when the filling factor

is close to an odd-integer. As the initial state we consider a coherent deviation

of the spin system from the B direction and investigate a break-down of this

Goldstone-mode state due to the spin-orbit (SO) coupling and smooth disor-

der. The spin relaxation (SR) process is considered in terms of annihilation

transitions in the system of spin excitons (magnons).

PACS numbers: 73.43.Lp, 75.30.Ds, 71.70.Ej

Great bulk of recent SR measurements and theoretical studies deal with 2D electrons

confined in quantum wells or dots.1–10However, during the last two decades only a few of

works have been devoted to the SR in the quantized Hall regime proper.1–6Specifically, we

imply conditions under which a 2DEG is in a strong perpendicular magnetic field (B≥10T)

and in the absence of holes and magnetic impurities. We exclude from the consideration the

effects of electrons in the edge states (cf. Ref. 6), and restrict our study to an odd-integer

filling.

In the studied problem the relevant SR time is actually not a spin dephasing time but

a time of Zeeman energy relaxation conditioned by a spin-flip process. Indeed, any spin-flip

means actually a reduction of the Zeeman energy |gµBB∆Sz| (?B ? ˆ z, ∆Sz= Sz−S0is the

Szcomponent deviation from the equilibrium value S0, g ≈ −0.44). The mechanism, which

makes the relaxation irreversible, has thereby to provide the energy dissipation. Another

necessary condition is a spin-flip mechanism non-conserving Sz.

In the present work the SO coupling is considered as the cause mixing spin states and the

disorder [to be more precise, the smooth random potential (SRP)] as a dissipation. A crystal

lattice is still implicitly assumed to be present as a “cooler” for 2DEG excitations. We will

suppose that all electron-phonon relaxation processes which do not change the 2DEG spin

1

Page 2

state occur much faster than the SR. [Thermodynamic relaxation times are estimated to be

<

∼1ns, i.e. well shorter than the SR time (see the corresponding estimate given at the end

of the paper).]

We thus study the case where the 2DEG is a quantum Hall “ferromagnet” (QHF); i.e.

the filling factor is ν = N/Nφ≃ 2κ+1, where N and Nφ= L2/2πl2

electrons and magnetic flux quanta (L2is the 2DEG area, lBis the magnetic length). In the

Bare the numbers of

high magnetic field limit, which really represents the solution to the first order in the ratio

rc= (e2/εlB)/¯ hωcconsidered to be small (ωcis the cyclotron frequency, ε is the dielectric

constant), we get the ground state with zeroth, first, second,... and (κ−1)-th Landau levels

fully occupied and with κth level filled only by spin-up electrons aligned along B. Only a

bare handful of experimental results on the SR in such a QHF were obtained, first indirectly

(the line-widths of the electron spin resonance (ESR) were measured in Ref. 1) and then

directly (in communication of Ref. 2 the photoluminescence dynamics of spin-up and spin-

down states was studied). The measured times are 5−10ns and exceed by 1-2 orders the SR

times observed in quantum wells where the SR process is governed by the spin interaction

with band holes.7

As the spin-system perturbation in the QHF we study a coherent deviation, when the

total S number is not changed. Namely, the initial state is a Goldstone mode which represents

a quantum precession of the vector S around the B direction: |i?=

for the QHF ground state andˆS−=?

ˆ σ±= (ˆ σx±iˆ σy)/2, where ˆ σx,y,z are the Pauli matrices]. The number N is assumed to be

macroscopically large: 0≪N<Nφ. The spin numbers of the |i? state are S=S0=Nφ/2 and

Sz=Nφ/2−N (i.e. ∆Sz|t=0=−N). The total Hamiltonian has the form Htot=?

Here Hintis the many-electron (Coulomb interaction) part of the Hamiltonian which has the

?ˆS−

?N|0?. Here |0? stands

jˆ σ(j)

− is the lowering spin operator [j labels electrons;

jH(j)

1+Hint.

usual form (see, e.g., Ref. 11), and H1is the single-electron part:

H1= ¯ h2ˆ q2/2m∗

e− ǫZˆ σz/2 + HSO+ ϕ(r), (1)

where ˆ q = −i∇ + eA/c¯ h is a 2D operator, ǫZ= |g|µBB is the Zeeman energy of one spin-

flipped electron, and ϕ(r) is the SRP field [r has components (x,y)]. The SO Hamiltonian

is specified for the (001) GaAs plane,

HSO= α(ˆ q × ˆ σ)z+β (ˆ qyˆ σy−ˆ qxˆ σx) , (2)

2

Page 3

and presents a combination of the Rashba term12(with the coefficient α) and the crystalline

anisotropy term13(see also Refs. 3–5, 8). The parameters α and β are small: α,β ≪ lB¯ hωc

(moreover, really α<β∼10−7K · cm<lBǫZ). This enable us to account HSOperturbatively.

If the SRP is assumed to be Gaussian, then it is determined by the correlator K(r) =

?ϕ(r)ϕ(0)?, where ϕ(r) is the SRP field. We choose also ?ϕ(r)? = 0, i.e. the SRP energy

is measured from the center of the Landau level.In terms of the correlation length Λ

and Landau level width ∆, the correlator is K(r) = ∆2exp(−r2/Λ2). In the realistic case

∆ ≈ 5 − 10K, and Λ ∼ 30 − 50nm. We will study the case

T<

∼T∗≪ ǫZ< ∆ ≪ e2/εlB< ¯ hωc, andΛ > lB

(3)

(T is the temperature which is actually assumed to be zero in the calculations; the value T∗

will be defined subsequently). All results which follow are obtained in the leading approxi-

mation corresponding to these inequalities.

The QHF is also remarkable for the following reason: when neglecting the last two

terms in Eq. (1) the spin excitons (SEs) are actually exact (to the first order in rc) lowest-

energy eigen states. The most adequate description of the SE states is realized by the SE

creation operators.3–5,14However, before writing them out we choose the bare single-electron

representation. As previously4,5we use the spinor basis which diagonalizes the first three

terms in the Hamiltonian (1) to the first order in u = β√2/lB¯ hωcand v = α√2/lB¯ hωc:

Ψκpa=

?

ψκp

v√κ+1ψκ+1p+ iu√κψκ−1p

?

,Ψκpb=

?−v√κψκ−1p+ iu√κ+1ψκ+1p

ψκp

?

,(4)

where ψκp= L−1/2eipyφκ(pl2

B+ x) is the wave function of an electron in the Landau gauge

(φκis the harmonic oscillatory function). We note that here and in the following we present

only the perturbation expansion to within the framework of the leading order in u and v.

The exciton creation operator is

Q†

abq=

1

?Nφ

?

p

e−iqxpl2

Bb†

p+qy

2ap−qy

2,(5)

where apand bpare the Fermi annihilation operators corresponding to the states (4). The

annihilation excitonic operator is Qabq≡

operators A†

orbital index κ is dropped since for our purposes the approximation of projection onto a

?

Q†

abq

?†

bbq(Aq= A†

≡Q†

ba−qand we employ also the “shift”

q= Nφ−1/2Q†

aaqand B†

q= Nφ−1/2Q†

−q, Bq= B†

−q.) In Eq. (5) the

3

Page 4

single Landau level is quite sufficient. In the following we drop also the “spin-orbit” index ab

at the operator (5). Such single-level exciton operators constitute a Lie sub-algebra which is

a part of an Excitonic Representation (ER) algebra (e.g., see Ref. 15 and references therein).

In our case the relevant commutation rules are as follows:

?Qq1,Q+

eiθ12[Aq1,Qq2]=−e−iθ12[Bq1,Qq2] = N−1

q2

?= eiθ12Aq1−q2−e−iθ12Bq1−q2, and

φQq1+q2,

(6)

where θ12= l2

B(q1×q2)z/2. Besides evidently [Qq1,Qq2]=[Aq1,Bq2]=0. The ground state

|0? is completely determined by the equations Aq|0?=δq,0|0? and Bq|0?=0. Single-exciton

states are normalized: ?0|Qq1Q†

In the limit ∆ → 0, HSO→ 0 and at rc≪ 1 the state

q2|0?=δq1,q2.

|N;1;q?=Q†

q

?

Q†

0

?N|0?

(7)

is the eigen state of the system studied. It has the spin numbers S = Nφ/2−1 and Sz=

Nφ/2−1−N [see below the expressions (8) which should be used to calculate S and Sz]. The

corresponding energy is (N+1)ǫZ+Eq, where Eqis the exchange part of the SE energy. The

small momentum approximation qlB≪ 1 is quite sufficient for our problem, and therefore

Eq=(qlB)2/2Mx,κ(general expressions for the 2D magneto-excitons can be found in Ref. 11).

Here Mx,κis the SE mass at ν = 2κ+1, namely: 1/Mx,0=(e2/εlB)?π/8, 1/Mx,1=7/4Mx,0,...

Spin operators in terms of the ER are invariant with respect to HSO:

ˆSz= Nφ(A0− B0)/2,

ˆS−= Nφ1/2Q†

0,

ˆS2= NφQ†

0Q0+ˆS2

z+ˆSz, (8)

At the same time in the basis (4) the operators ϕ(r) and Hintacquire corrections proportional

to u and v. Specifically, calculating?Ψ†ϕ(r)Ψd2r, where Ψ=?

the terms responsible for a spin-flip process:

p(apΨκpa+bpΨκpb), we get

ˆ ϕ = N1/2

φlB

?

q

ϕ(q)(iuq+− vq−)Qq+ H.c. (at qlB≪ 1). (9)

Here ϕ(q) is the Fourier component [i.e. ϕ=?

We stress one essential feature of the states (7). In spite of the existence of a formal

qϕ(q)eiqr], and q±=∓i(qx± iqy)/√2.

operator equivalence Q†

states. Indeed, in these states the system has different total spin numbers S = Nφ/2 and

0≡ lim

q→0Q†

qwe find that |N;1;0? and lim

q→0|N;1;q? present different

S = Nφ/2−1 respectively. [Sz=Nφ/2−1−N is the same for both.] So, the excitation of a

4

Page 5

“zero” exciton (with zero 2D momentum) corresponds to the transition Sz→Sz−1 without

any change of the total S number, but each “nonzero” SE changes the spin numbers by 1:

S →S−1, Sz→Sz−1. Let us introduce the notation |N?=

in the ER is actually a “Goldstone condensate” (GC) containing N zero excitons: |i?=|N?.

Our goal is to study the process of the GC break-down. The state |N? is certainly

degenerate and we will solve the problem in terms of the quantum-system transitions within

?

Q†

0

?N|0?. The initial state |i?

a continuous spectrum. The transition probability is determined by the Fermi Golden Rule:

wfi= (2π/¯ h)|Mfi|2δ(Ef−Ei). In our case evidently the final state |f? is obviously the state

where a part of the Zeeman energy has been converted into an exchange energy. Since it is

exactly the single-electron terms that constitute the perturbation responsible for the Mfi

matrix element, we could find that such a transition is the 2X0→Xq∗ process in the lowmost

order of the perturbative approach (we denote the zero exciton by X0and the nonzero one

by Xq). In other words the final state for this transition is |f?=|N−2;1;q∗?. The value q∗

is determined by the energy conservation equation Ef=Eiwhich reads 2ǫZ=ǫZ+E(q∗), i.e.

q∗=?2Mx,κǫZ/lB.

We point out that the SO interaction (2) alone does not provide a quantum fluctuation

from the GC to any state with a different electron density (in particular, to the state |N−

2;1;q∗?). This feature may be verified by means of direct analysis of the SO influence on the

2DEG spectrum (c.f. Ref. 4). Besides, this can be recognized from general considerations.

Indeed, each of the components ˆ qi(i=x,y) commutes with any component of the operator

ˆP=?

and one can also check that

j[¯ hˆ qj− (e/c)B×rj]. The latter commutes with the Hamiltonian?

?ˆP,Q†

order in HSO]. The operatorˆP hence plays the role of 2D momentum in the magnetic field.16

j¯ h2ˆ q2

j/2m∗

e+Hint

q

?

=qQ†

q[if Q†

qis defined by Eq. (5) to within the zero

Its permutability with the total 2DEG Hamiltonian reflects the spatial homogeneity of the

system under consideration in the “clean limit”. Since [HSO,ˆP] ≡ 0, the SO coupling by

itself cannot destroy the homogeneity in any order of the perturbation approach.

The transition is thereby determined by the operator (9).The corresponding ma-

trix element Mif = ??q∗;1;N −2||ˆ ϕ||N??[R(N)R(N −2;1;q∗)]−1/2was actually calculated

in Ref. 4. (The SRP plays the same role as the phonon field studied there.) Here and

in the following the notation R(...) stands for the norm of the state |...?.14The result

is |Mif|2=

N(N−1)

2Nφ(u2+v2)|q∗lBϕ(q∗)|2, and we obtain the rate of the i → f transition

5

Page 6

(2π/¯ h)?

q|Mif|2δ(q2l2

B/2Mx,κ−ǫZ)=N(N−1)/τNφ

(N ≥ 1), where

1/τ = 8π2(α2+β2)M2

x,κǫZK(q∗)/¯ h3ω2

cl4

B.(10)

Here K stands for the Fourier component of the correlator: K(q)=L2|ϕ(q)|2/4π2.

The quantum transition to the state |f?=|N−2;1,q∗? is certainly a first step in the SR

process. This state is thermodynamically unstable. In a time which is much shorter than τ

it turns to a state |N−2;1,q0?, where q0takes the lowest possible nonzero value. In fact,

relevant values of q0are determined by the SRP field. The SE interaction with the SRP

incorporates the energy Ux-SRP∼ql2

el2

B∆/Λ (the “nonzero” SE possesses the dipole momentum

B[q×ˆ z], see Ref. 11). The latter determines the inhomogeneous uncertainty of the SE

momentum δq ∼ Mx,κ∆/Λ (this follows from the equation δq∂Eq/∂q = Ux-SRP). Therefore

“quasi-zero” wave numbers in the range defined by inequalities

0 < q0<

∼Mx,κ∆/Λ,(11)

present the lowest physical limit for momenta of the nonzero SEs. Under the conditions (3)

we find that q0≪ q∗, and the SE exchange energy (q0lB)2/2Mx,κis smaller than the value

T∗=Mx,κ(∆lB/Λ)2. This in turn is negligible in comparison with ǫZ.

To solve the problem in a complete form we have obviously to study the general state of

the type:

|N;M1,M2,...,MK? =?Q†

q01

?M1?Q†

q02

?M2...?Q†

q0K

?MK|N?.(12)

All the wave-vectors q0kare assumed to satisfy the condition (11). We will also use for this

state a shorthand notation |N;M?, where M =?K

SEs. If M ≫1, we assume that 1 ≪ K ≪ Nφ. In the framework of our approach the state

(12) is an approximate eigen state of a QHF having energy (N+M)ǫZ and spin numbers

kMkis the total number of the nonzero

Sz=Nφ/2−N−M and S=Nφ/2−M. This value of Szis the exact one. It can be calculated

employing the representation of Eqs. (8) and the commutation rules (6). The same algebra

allows us to find thatˆS|N;M? = S(S+1)(|N;M?+|˜ ε?), where the norm of the state |˜ ε?

is small compared with the norm of |N;M?, namely: R(˜ ε)/R(N;M) = O(m3n/K). The

notations n=N/Nφand m=M/Nφare used for “reduced” quantum numbers. In the special

case when N = 0, the state |0,M? can be treated as a “thermodynamic condensate” (TDC)

which arises if M is larger than the critical number of nonzero SEs. The latter is estimated

6

Page 7

at Nφl2

B

?d2q

2π/{exp[(Eq+|Ux-SRP|)/T] −1} (e.g., c.f. Ref. 4), and in our case (3) it is at least

smaller than NφMx,κT. At the same time, M is determined by the spin S of the system,

therefore at a given M=Nφ/2−S>NφMx,κT we find that below some threshold temperature

the nonzero SEs necessarily form a TDC. For macroscopically large N and M the state (12)

hence features a coexistence of GC and TDC. It should also be noted that specific values q0k

as well as specific distribution given by Mknumbers have no physical meaning. The final

results should not depend on them but only on M and N.

We can now write the kinetic equations corresponding to the relevant spin transitions.

The rate dN/dt is determined by the 2X0→Xq∗(→Xq0) process (which presents a GC

depletion with a simultaneous “flow” to TDC) and by the X0+Xq0→Xq∗(→Xq′

The rate dM/dt is also formed by the 2X0→Xq∗(→Xq0) transition (which provides a TDC

evolution) and by the Xq0+Xq′

0) one.

0→Xq∗(→Xq′′

0) one (determining a TDC depletion). [Values

of q0,q′

0and q′′

0belong to the region (11).] The corresponding equations are derived again

with help of the Fermi Golden Rule and Eqs. (6) (with vanishing θ12in the latter):

dn/dt = −(2µnn+ µnm)/τ,anddm/dt = (µnn− µmm)/τ ,(13)

where

µnn

=

|?M;N−2|Qq∗Q−q∗|N;M?|2

R(N;M)R(N−2;M+1;q∗)=N4R(N−2;M+1;q∗)

?

k

R(N;M)R(N−1;M;q∗)

=

N2

|?M1,...Mk−1,...Mi−1,...MK;N|Qq∗+q0k+q0iQ−q∗|N;M?|2

R(N;M)R(N;M−1;q∗)

=

N2

N2

φR(N;M)

?

1+O

?

m

nNφ

??

,

µnm

=

|?M1,...Mk−1,...MK;N−1|Qq∗+q0kQ−q∗|N;M?|2

4M2N2R(N−1;M;q∗)

φR(N;M)

[1+O(K/Nφ)],

µmm

=

?

k<i

2M4R(N;M−1;q∗)

φR(N;M)

[1+O(K/Nφ)]

[R(N;M+1;q∗) is the norm of the Q†

appearing in these equations satisfy the conditions R(N;M +1;q∗)/R(N;M) = r, R(N +

q∗|N;M? state]. In this way we find that the norms

1;M)/R(N;M) = Nrnand R(N;M1,...Mk+1,...MK)/R(N;M) = Mkrm, so that

µnn= n2r/r2

n,µnm= 4mnr/rnrm,µmm= 2m2r/r2

m,(14)

where the factors r,rn,rmare determined by the equations

1 = (1 − n − 2m)/rn+ O(m3/nK),

1 = (1 − 2n − 2m)/r + 4mn/rnrm+ n2/r2

1 = (1 − 2n − 2m)/rm+ n2/r2

n+ 2m2/r2

n+ O(m2),

m+ O(1/Nφ).

(15)

7

Page 8

(Positivity of rn≈ 1−n−2m provides the physically obvious requirement |Sz| < S.) The

last terms in Eqs. (15) would just depend on the specific set of the Mk numbers. We

can therefore calculate rmand obtain the final result only in the m≪ 1 case. Meanwhile,

the values of m(t) are determined by the initial value n(0). According to Eqs. (13)-(15)

max(m)≈n(t∗)[1−n(t∗)]/√2 (t∗is the time at which m peaks), i.e. at least m2<1/32. [In

particular, at n(0)=0.5 max(m)≈0.1.] The problem (13)-(15) is thus solved to the leading

order in m. We should put r=rn=rm=1 in µmnand µmm, but rn=1−n and r=(1−n)2in µnn.

This yields the analytical result n(t)=1/[2n(0)(t/τ)2+2(t/τ)+1/n(0)] and m = n(t)n(0)t/τ.

The dependences are shown in Fig. 1. The vector S(t) at moments t = 0,τ,2τ,...

is depicted in the inset. The time (10) hence governs the breakdown of the GC. (Under

the realistic conditions above, τ ∼ 10−8− 10−7s.) However the SR occurs certainly non-

exponentially and the actual time is increased by a factor of Nφ/∆Sz(0).

Experimentally, the GC could probably be created by microwave pumping at the electron

frequency ǫZ/¯ h. This should cause to “rotate” the QHF spin without changing of the S

modulus. As to observing of the SR variation with time, one can think that the optical

technique2is relevant in the case.

I acknowledge support by the MINERVA Foundation and by the Russian Foundation

for Basic Research. I thank also the Max Planck Institute for Physics of Complex Systems

(Dresden) and the Weizmann Institute of Science (Rehovot) for hospitality.

1M. Dobers, K.v. Klitzing, and G. Weimann, Phys. Rev. B 38, 5453 (1988); M. Dobers et al.,

Phys. Rev. Lett. 61, 1650 (1988).

2V.E. Zhitomirskii et al., JETP Lett. 58, 439 (1993).

3S. Dickmann and S.V. Iordanskii, JETP Lett., 63, 50 (1996).

4S. Dickmann and S.V. Iordanskii, JETP 83, 128 (1996).

5S. Dickmann and S.V. Iordanskii, JETP Lett. 70, 543 (1999).

6B.W. Alphenaar, H.O. M¨ uller, and K. Tsukagoshi, Phys. Rev. Lett. 81, 5628 (1998).

8

Page 9

7S. Bar-Ad, and I. Bar-Joseph, Phys. Rev. Lett. 68, 349 (1992); V. Srinivas, Y. J. Chen, and C.

E. C. Wood, Phys. Rev. B 47, 10907 (1993).

8A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 (2000).

9I.A. Merkulov, Al. L. Efros, and M. Rosen, Phys. Rev. B 65, 205309 (2002).

10A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B 67, 195329 (2003).

11I.V. Lerner and Yu. E. Lozovik, Sov. Phys. JETP 53, 763 (1981)]; Yu. A. Bychkov, S.V. Iordan-

skii, and G.M. Eliashberg, JETP Lett. 33, 143 (1981); C. Kallin and B.I. Halperin, Phys. Rev.

B 30, 5655 (1984).

12Yu.A. Bychkov and E.I. Rashba, JETP Lett. 39, 78 (1984).

13M.I. D’yakonov and V.Yu. Kachorovskii, Sov. Phys. Semicond. 20, 110 (1986).

14The excitonic operators appeared originally in publications of A.B. Dzyubenko and Yu.E. Lozovik

Sov. Phys. Solid State 25, 874 (1983) [ibid. 26, 938 (1984)]; in the paper of J. Phys. A 24, 415

(1991) these authors considered the many-exciton states and calculated their norms [see also

Refs. 4 and 5 and the R(N) and R(N−2;1;q) norms calculated there].

15S. Dickmann, Phys. Rev. B, 65, 195310 (2002).

16The operatorˆP appeared as the momentum operator of a magneto-exciton in the work of L.P.

Gor’kov and I.E. Dzyaloshinskii in Sov. Phys. JETP, 26, 449 (1968). To calculate the commutator

with operator Q†

ondary quantization:ˆPx=(2πi¯ h/L)

qwe choose the Landau gauge [A=(0,Bx,0)] and presentˆP in terms of the sec-

?

κ,p,q,σδ′(q)a†

κp+q/2σaκp−q/2σandˆPy=(¯ h/lB)

?

κ,p,σpa†

κpσaκpσ.

(δ′is the derivative of the δ-function, the operator aκpσcorresponds to the Landau-gauge state

|κ,p,σ?).

9

Page 10

0.11 10100

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.10.20.30.40.5

0.1

0.2

0.3

0.4

0.5

3τ2τ

0

τ

Sz

S||

|∆ ∆Sz| / Nφ φ

|∆ ∆S| / Nφ φ

t / τ τ

FIG. 1. Time dependences of |∆Sz|/Nφ=n(t)+m(t) and of |∆S|/Nφ=m(t) are shown in the main

picture for n(0)=|∆Sz(0)|/Nφ=0.455. The vectors S(t) at equidistant moments of time are plotted

in the inset with step τ. The dotted line is the arc with radius S0=Nφ/2. The gap between the

dashed and dotted lines reflects the deviation of the spin modulus from the value of S0.

10