CYCLOTRON RESONANCE IN A TWO-DIMENSIONAL
ELECTRON GAS WITH LONG-RANGE DISORDER
M. M. FOGLER
School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton,
New Jersey 08540, USA
E-mail: fogler@ias.edu
B. I. SHKLOVSKII
Theoretical Physics Institute, University of Minnesota, 116 Church St. Southeast,
Minneapolis, Minnesota 55455, USA
E-mail: shklovskii@physics.spa.umn.edu
We show that the the cyclotron resonance in a two-dimensional electron gas has
nontrivial properties if the correlation length of the disorder is larger than the de
Broglie wavelength: (a) the linewidth depends dierently on the magnetic eld in
strong, intermediate, and weak magnetic eld regimes (b) at the transition from
the intermediate to the weak elds the linewidth suddenly collapses due to an
explosive growth of the fraction of electrons with the diusive classical motion and
a resulting very large quantum localization length.
The cyclotron resonance (CR) is one of the basic tools for studying the
electronic properties of physical systems in an external magnetic eld. A
very interesting example of such a system is a two-dimensional electron gas
(2DEG). The CR can be studied by measuring the transmission of an electromagnetic signal of some frequency ! through the 2DEG. The change in
the transmission is proportional to the real part of the dynamical conductivity averaged over the active and inactive circular polarizations. The active
polarization's contribution Re + (!) has a peak at ! close to the cyclotron
frequency !c of the external magnetic eld. The disorder related zero temperature width of this peak is the subject of this paper. The discussion follows
our recent publication, Ref. 1.
Although many aspects of the CR theory have been worked out initially
by Ando 2 and later by other authors, 3;4 the consistent description of this
phenomenon exists only for the case where the correlation length d of the
random potential U(x; y) acting on the electrons in the disordered 2DEG is
smaller than the inverse Fermi wave vector kF,1. In this case the eect of
the random potential is described by a single quantity, the transport time .
However, unlike the 3D case, the conventional Drude-Lorentz formula, which
gives the Lorentzian peak with the half width at half maximum (HWHM)
of ,1 , applies only in the uninteresting case !c 1. In the other limit
fogler: submitted to World Scientic on July 28, 1998
1
1/2
(2)
-1
(1)
2
4/3
0 (W/EF) (W/EF)
1 (
c
)
-1
Figure 1. Dependence of the CR linewidth on (!c ),1, the quantity inversely proportional
to the magnetic eld. Thick line: our results for the long-range potential with given W=EF
and [for the case kF d (EF =W )2=3 ]. Labels (1) and (2) correspond to the equation
numbers. Thin line: short-range potential with the same .
(!c 1) the p
CR lineshape is non-Lorentzian and has a much larger width 2
!1=2 = 0:73 !c = due to the formation of discrete Landau levels. This
behavior of !1=2 is illustrated by the thin line in Fig. 1. Here and below
!1=2 is dened by
Z !1=2
Z 1
1
d! Re + (! + !c) = 2 d! Re + (! + !c ):
0
0
Using such a \median" width instead of the conventional HWHM is more
adequate because the CR lineshape can be rather intricate for a long-range
random potential. For simplicity, we will focus on a model of a Gaussian
random potential whose correlator decays suciently fast at distances larger
than d and does not possess any other characteristic scales besides d. We will
assume that kF d (EF =W)2=3, where EF is the Fermi energy and W is the
rms amplitude of U. As one can see from Fig. 1 illustrating our results for
this case, the dependence of !1=2 on the magnetic eld is nonmonotonic.
Even more remarkable, !1=2 exhibits a rapid collapse to its classical value
of ,1 in the vicinity of the point !c (EF =W)2=3 1. The derivation of
these results is based on the picture of the \classical localization," 5;6 which is
a development of the drift picture of the electron motion used in early works
on the percolation theory of the quantum Hall eect. 5
If the random potential U is smooth, the motion of an electron on the
time scales important for the CR can be described classically, as a motion
of a single particle with energy EF . (We neglect the interaction). If the
fogler: submitted to World Scientic on July 28, 1998
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magnetic eld is not too low, the motion can be decomposed into a fast cyclotron gyration and a slow drift of the guiding center of the cyclotron
orbit. We call the magnetic eld strong if the cyclotron radius Rc = vF =!c
is smaller than d. In such strong elds the guiding center drifts along a level
line U() = const of the random potential, which is typically a closed loop
of size d. Recently it has been realized 6;7 that the drift approximation is
also valid in the intermediate eld regime 1 < Rc=d < (EF =W)2=3 [the same
as (EF =W )4=3 < !c < (EF =W)2 because (d=vF )(EF =W)2]. In this
regime the cyclotron gyration is still suciently fast, so that the guiding center remains practically \frozen" during one cyclotron period. Therefore, the
guiding center motion is determined by U0 , the random potential averaged
over the cyclotron orbit. The guiding center is still bound to one of the level
lines, but those are the level lines of U0 , not U. 6;7;8 A typical level line of U0
is still a loop of size d. The frequency !d of the guiding center
motion along
p
such a loop (the drift frequency) is equal to !d = (Rc =d) !c = for Rc d.
Obviously, the electrons on the periodic orbits are (classically) localized and
do not participate in the DC transport. However, a more2=3accurate analysis 6
reveals that a very small fraction of the order of e,(!c =!d ) of the trajectories
remains delocalized. Such trajectories form a stochastic web in the vicinity
of the percolation contour. The stochastic web rapidly grows with decreasing
magnetic eld and turns into a stochastic sea at Rc=d = (EF =W)2=3, where
!d = !c . In even lower magnetic elds (the weak eld regime) the stochastic
sea spans almost the entire phase space. Correspondingly,
the static conductivity is exponentially small, xx (0) / e,(!c =!d )2=3 when !d !c (the strong
and the intermediate eld regimes), rapidly blows up near !d = !c point (the
boundary of the intermediate and the weak eld regimes), and nally crosses
over to the Drude-Lorentz formula in weak elds. a
We found that the dynamical conductivity also exhibits a rapid change
near the !d = !c point: the aforementioned collapse of the CR linewidth.
Under the \classical localization" conditions the linewidth is due to inhomogeneous broadening. In dierent parts of the sample there exist dierent local
corrections ! to the the average gyration frequency !c of the velocity vector. Such corrections come from an additional centripetal force exerted on
the electron by the random potential. The linewidth !1=2 is given by
!1=2 [!]rms = const m!W d2 ; Rc d;
(1)
c
p
= !c =(); Rc d:
(2)
a These
results apply if the quantum localization eects can be neglected.
fogler: submitted to World Scientic on July 28, 1998
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Equations (1) and (2) show that !1=2 linearly increases as a function of
(!c ),1 in the strong-eld regime, reaches its maximum at Rc d, and then
decreases in the intermediate elds, see Fig. 1. The median width !1=2
describes the CR linewidth due to the majority of the electrons, which are
classically localized in these two regimes. As the eld weakens, fewer and
fewer electrons remain localized, and in the weak elds the situation is reversed: most of the trajectories are extended and ergodic. In this regime the
broadening is of the homogeneous type. The linewidth drops to its classical
value ,1 , which is much smaller than [!]rms . This drop can be understood
as the result of a motional narrowing. Remarkably, the onset of the motional
narrowing is very rapid (and hence can be characterized as the \collapse") because the stochastic web grows exponentially as the magnetic eld decreases.
After the collapse, i.e., in the entire weak-eld regime, !1=2 = const.
The nonmonotonic dependence of !1=2 on the magnetic eld with the
maximum at Rc d has been observed by Watts et al., 9 in agreement with
our theory. Arguably, the collapse of the CR linewidth has also been seen
(for the 400
A spacer sample). However, a decisive conrmation of the latter
prediction requires further experiments.
M. M. F. is supported by DOE Grant No. DE-FG02-90ER40542 and
B. I. S. by NSF Grant No. DMR-9616880. We thank A. Yu. Dobin,
M. I. Dyakonov, Yu. M. Galperin, and A. A. Koulakov for useful discussions.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
M. M. Fogler and B. I. Shklovskii, Phys. Rev. Lett. 80, 4749 (1998).
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More precisely, these are the level lines of some eective potential, which
is very close to U0 , see M. M. Fogler, Phys. Rev. B 57, 11947 (1998).
M. Watts, I. Auer, R. J. Nicholas, J. J. Harris, and C. T. Foxon, p. 581
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fogler: submitted to World Scientic on July 28, 1998
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