![Analytical solution for the Landau level DOS at the cyclotron energy ρ(E = 1) versus magnetic field B for a quantum well of width a = 140 Å, doped with Z = 1, z i = −200 Å, and various impurity densities n i = 0.8 (a), 1.2 (b), 1.4 (c), and 1.6 × 10 12 cm −2 (d). The full, broken and dotted curves refer to our calculation, the prediction of [32] (figure 3, diamonds) and the measurement of [4], respectively.](https://www.researchgate.net/profile/F-Bechstedt/publication/231004691/figure/fig4/AS:668558946213888@1536408282018/Analytical-solution-for-the-Landau-level-DOS-at-the-cyclotron-energy-rE-1-versus_Q320.jpg)
![Analytical solution for the Landau level DOS ρ(E) versus energy E in dimensionless units for a quantum well of width a = 75 Å, doped with Z = 1, z i = −200 Å, under a magnetic field B = 7 T and various impurity densities n i = 0.1 (a), 0.5 (b), 1 (c) and 1.5 × 10 12 cm −2 (d). The full and broken lines refer to our calculation and the measurement of [8], respectively.](https://www.researchgate.net/profile/F-Bechstedt/publication/231004691/figure/fig3/AS:349711870513155@1460389215592/Analytical-solution-for-the-Landau-level-DOS-rE-versus-energy-E-in-dimensionless-units_Q320.jpg)
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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 15 (2003) 1305–1323 PII: S0953-8984(03)34398-X
Density of states of a two-dimensional electron gas in a
perpendicular magnetic field and a random field of
arbitrary correlation
SGlutsch
1
,FBechstedt
1
and Doan Nhat Quang
2
1
Institut f
¨
ur Festk
¨
orpertheorie und Theoretische Optik, Friedrich-Schiller-Universit
¨
at Jena,
Max-Wien-Platz 1, D-07743 Jena, Germany
2
Center for Theoretical Physics, National Center for Science and Technology, PO Box 429,
Boho, Hanoi 10000, Vietnam
Received 28 February 2002, in final form 16 January 2003
Published 17 February 2003
Onlineatstacks.iop.org/JPhysCM/15/1305
Abstract
Atheory is given of the density of states (DOS) of a two-dimensional electron
gas subjected to a uniform perpendicular magnetic field and any random field,
adequatelytakingintoaccount the realistic correlation function ofthelatter. For
arandom field of any long-range correlation, a semiclassical non-perturbative
path-integral approach is developed and provides an analytic solution for
theLandau level DOS. For a random field of any arbitrary correlation, a
computational approach is developed. In the case when the random field is
smooth enough, the analytic solutionis found to be in very good agreement with
the computational solution. It is proved that there is not necessarily a universal
form for the Landau level DOS. The classical DOS exhibits a symmetric
Gaussian form whose width depends merely on the rms potential of the random
field.The quantum correction results in an asymmetric non-Gaussian DOS
whose width depends not only on the rms potential and correlation length of
therandom field, but the applied magnetic field as well. The deviation of
the DOS from the Gaussian form is increased when reducing the correlation
length and/or weakening the magnetic field. Applied to a modulation-doped
quantum well, the theory turns out to be able to give a quantitative explanation
of experimental data with no fitting parameters.
1. Introduction
Since the discovery of the quantum Hall effect [1] the properties of a disordered two-
dimensional electron gas (2DEG) subjected to a perpendicular magnetic field have been
extensively studied. The nature of the density of states (DOS) of the 2DEG is a problemof vital
importance for the understanding of many quantum phenomena observed in the system, e.g.,
cyclotron resonance, specific heat, magnetization, magnetocapacitance and magnetotransport.
0953-8984/03/081305+19$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1305
1306 SGlutsch et al
The problem has attracted a lot of attention from both experimentalists and theorists [2].
Nevertheless, over the years of research the common conclusion has seemed to be only that the
DOS in question is composed of disorder-broadened Landau levels with a significant number
of electronic states in between. To date, there has been little or no consensus regarding the
exact form of the Landau level DOS of the 2DEG and the magnetic-field dependence of the
linewidth.This conclusion belongs not only to the calculated DOS but the measured DOS as
well. Indeed, some experimentalists have reported a Gaussian lineshape [3–9], while others
claim that it is Lorentzian [10, 11]. Some report a broadening which is independent of the
applied magnetic field [3, 6, 10],while others report a broadening which is a square root [4, 5, 8]
or oscillating [7, 12, 13] function of the magnetic field. In addition, in order to explain their
experimental results several authors [3, 5, 8] have invoked a constant background DOS whose
origin is unclear.
The first calculation of the Landau level DOS was performed within a self-consistent
Born approximation (SCBA) [14–16]. However, based on a single-site picture, the SCBA
theory underestimates the disorder effect. Within the many-site picture, the random field is
characterized by a correlation function. This is shown [2, 17] to capture microscopic details
of the electron system, e.g. the actual origin ofthedisorder as well as the actual geometry of
thesample. Moreover, for true 2DEGs this is found to exhibit a very complicated dependence
on its variables in real space. Therefore, in theexisting literature one has to adopt a severe
approximation, replacing the true spatial correlation of a random field by a simple one with
some fitting parameters. The simplest form is a δ-function, which enables an exact solution to
be found for the lowest Landau level DOS [18]. However, this model describes a zero-range
electron–impurity interaction and none of the theories with this white-noise limit [19–22]
predict a remarkable DOS lying between Landau levels.
Thus, the key problem is to keep the correlation length of the disorder finite from the
outset [22–32] by assuming a Gaussian or exponential correlation function. Unfortunately,
this simplification cannot allow an exact DOS solution and various approximation schemes
have to be proposed. The Landau level DOS was evaluated by means of a cumulant
expansion [22, 24, 25] for the Green function, or an expansion in the inverse correlation length
fordifferent quantities of interest, e.g. the Green function [28], the partition function [29] and
thecorrelation function [30–32]. The validity of the approximations was often insufficiently
discussed and, instead, the disorder was in practice considered to be a perturbation, as in the
SCBA [22, 24, 30]. In addition, the theoretical prediction was found not to be in quantitative
agreement with experimental findings [32].
The fact that so many different experiments seemingly yield so many different results for
theLandau level DOS suggests that its exact form is very likely fixed by the very realistic
nature of the 2DEG under consideration, i.e. by the experimental conditions in which the 2D
electrons have been created and move. This means that, in order to quantitatively describe the
measured Landau level DOS and its width, one has to work with the true correlation function.
Recently, we have developed semiclassical approaches to a 1D and 2D electron in the presence
of a random field of any long-range correlation [33, 34]. These turn out to meet that demand
and to result in an analytic DOS solution. In the present paper, we will extend our methods to
incorporate also a high perpendicular magnetic field into the theory.
It should be noted that semiclassical calculations of the Landau level DOS of a 2DEG
have recently been carried out for a smooth disorder of arbitrary correlation, based on a
diagrammatic [35] or a path-integral technique [36]. Nevertheless, these theories were
developed essentially for the case of weak disorder and, hence, turn out to be accurate,
especially for high Landau levels. It is clear that, under strong magnetic fields and low
temperatures, the low-energy region is likely to be of more physical interest in comparison to
Density of states of a two-dimensional electron gas 1307
the high-energy one. Thus, the aim of this paper is to find another version of the semiclassical
approach, which may get rid of the assumptionofweak disorder and must then be useful for
low Landau levels.
In section 2, the calculation of the Landau level DOS of a disordered 2DEG proceeds
within asemiclassical non-perturbative path-integral approach to a smooth random field,taking
explicitly into account the realistic correlation function of the random field via its average
potential and force. To control the validity of the approximation used, a computational method
is exactly derived in section 3, which is applicable to any random field. The theory is applied
in section 4 to a quantum well where the disorder is caused by modulation doping. Finally,
section 5 is devoted to conclusions.
2. Semiclassical non-perturbative approach
2.1. Path-integral formulation of the Landau-level DOS
We are dealing with a 2DEG subjected to a random field and a uniform magnetic field
perpendicular to the x–y plane of the 2DEG. The concepts of homogeneity and isotropy
are not applicable for an individual realization ofthestochastic potential. Instead, a stochastic
potential U is said to be macroscopically homogeneousand isotropic if U
(r) = U (αr +a) for
any orthogonal matrix α and any vector
a has the same statistical properties as U ,inparticular,
if all correlation functions of U
are the same as for U .Inwhatfollows, we will restrict
thediscussion to the case when the random field U (
r) is Gaussian, and it may therefore be
completely described by a binary correlation function, defined as
W (
r
1
− r
2
) =U(r
1
)U(r
2
), (1)
where the angular brackets ···stand for the averaging over all configurations of the random
field, which is assumed macroscopically homogeneous. As usual, the field is characterized by
its rms potential γ and correlation length ,definedas
γ
2
=U
2
=W (r)|
r=0
(2)
and
= γ/F, (3)
with the rms force given by
F
2
=(∇U )
2
=∇
1
∇
2
W (r
1
− r
2
)|
r
1
=r
2
. (4)
It is well known that the electronic DOS per unit area can be expressed in terms of a
Fourier transform of the Green function as
ρ(E) =
1
π¯h
+∞
−∞
dt exp(iEt/¯h)G(t), (5)
with the spin degeneracy included. Here G(t)=G(
r, r;t)denotes the diagonal part of the
averaged Green function,which describes the one-particle properties of a 2DEG in the presence
of random and magnetic fields. This is obviously independent of the spatial coordinate in the
x–y plane by virtue of the macroscopic homogeneity of the system.
Foraparticle with effective subband mass m and charge −e (where e is the absolute value
of the electron charge) in a magnetic field B,thelength and energy scales are given by
l
c
=
¯h/eB (6)
and
¯hω
c
= ¯h(eB/m). (7)
1308 SGlutsch et al
The quantities l
c
and ω
c
are known as the magnetic length and cyclotronfrequency,respectively.
These allow us to decide whether a parameter is small, intermediate or large.
Amagnetic field pointing along the z direction can be described in the symmetric gauge
by a vector potential in the x–y plane
A(r) = (−By/2, Bx/2).For a Gaussian random
field, the averaged Green function may then be written in terms of a Feynman path integral
as [22, 24, 30]
G(t)=
m
2πi¯ht
1
N
Dr(τ ) exp
i
¯h
t
0
dτ
m
2
˙
r
2
(τ ) +
mω
c
2
(x ˙y − y ˙x)
−
1
2¯h
2
t
0
dτ
t
0
dτ
W [r(τ ) − r(τ
)]
, (8)
in which
Dr(τ ) denotes the Feynman measure on the set of closed orbits in the plane of the
2DEG and
N is the normalization constant:
N =
Dr(τ ) exp
i
¯h
t
0
dτ
m
2
˙
r
2
(τ )
. (9)
Thus, with the help of a Feynman path integral the DOS of a 2DEG subjected to a
perpendicular magnetic field and a random field is represented in terms of the correlation
function W (
r),whose form is specified by the interaction of the 2DEG with disorder and,
hence, depends on the system geometry as well as the disorder origin [2, 17]. Equations (5)
and (8) set up a basis for discussion of the disorder effect on the Landau level DOS.
2.2. Averaged one-particle Green function
Hereafter, we assume that the random field is varying slowly on the average in space, which
crudely implies that
l
c
. (10)
The random field may then be dealt with within a semiclassical approach. The derivation
follows the same line as in the case treated in [34] for a disordered 2DEG in the absence of
amagnetic field. Under the condition (10), the dominant contribution to the path integral
entering equation (8) comes from classical orbits of the particle affected by both magnetic and
random fields. Furthermore, on these trajectories the correlation function W(
r) varies slowly.
Fororbits of relatively small size, we acquire a small quantity of the form
γ
2
t
2
2¯h
2
−
1
2¯h
2
t
0
dτ
t
0
dτ
W [r(τ ) − r(τ
)] 1. (11)
This enables the exponential in equation (8) to be expanded in its powers. As a result, we may
get an approximation for the averaged Green function:
G(t)=G
0
(t) exp
−
γ
2
t
2
2¯h
2
1+
γ
2
t
2
− J(t)
2¯h
2
, (12)
where G
0
(t) is the Green function in the presence of amagnetic field but in the absence of
disorder
G
0
(t) =
m
2πi¯ht
ω
c
t/2
sin ω
c
t/2
(13)
and J (t) is defined by
J (t) =
1
2πiG
0
(t)
1
N
Dr(τ ) exp
i
¯h
t
0
dτ
m
2
˙
r
2
(τ ) +
mω
c
2
(x ˙y − y ˙x)
×
t
0
dτ
t
0
dτ
W [r(τ ) − r(τ
)]. (14)
Density of states of a two-dimensional electron gas 1309
It has been demonstrated [37] that, for equation (13) to make sense at any time, one must
replace ω
c
→ ω
c
−iη with an infinitesimal η>0. It is worth noting that the first and second
terms inside the square brackets on the right-hand side of equation (12) refer to the classical
DOS and its quantum correction, respectively.
We have now to evaluate the path integral J (t).Thecorrelation function appearing in
equation (14) is to be replaced with its Fourier transform, defined by
W (
r) =
d
2
k
(2π)
2
exp(ikr )W (k). (15)
The path integral is then exactly performed, yielding
J (t) = t
2
d
2
k
(2π)
2
W (k)
1
0
dσ exp
−i
¯h
k
2
t
2m
sin(σ ω
c
t/2) sin[(1 − σ)ω
c
t/2]
(ω
c
t/2) sin(ω
c
t/2)
. (16)
Next, it is clearly observed from equations (5) and (12) that the main contribution to the
semiclassical DOS results from such a time region that
γ t/¯h 1, (17)
with γ the rms disorder potential. Further, the
k integral in equation (16) is apparently extended
primarily over such a wavevector region that
|
k| 1/, (18)
with the disorder correlation length. Upon combining inequalities (17) and (18), we are in
a position to estimate the upper limit of the variable of the exponential in equation (16):
¯h
k
2
t
2m
¯h
2
/2m
2
γ
. (19)
In accordance with the semiclassical nature of the random field, we may as usual adopt the
following inequality:
¯h
2
/2m
2
γ
1. (20)
Upon employing a Taylor series in powers of the small quantity ¯h
k
2
t/2m for the exponential
in equation (16) truncated after the first order, we are able to obtain for J (t) an approximate
expansion:
J (t) = γ
2
t
2
− i
¯hF
2
t
mω
2
c
1 −
ω
c
t
2
cot
ω
c
t
2
. (21)
Lastly, by inserting equation (21) back into (12), we may immediately arrive at a
simple expression fortheaveraged Green function describing the 2DEG moving in a high
perpendicular magnetic field and a random field of long-range correlation:
G(t)=G
0
(t) exp
−
γ
2
t
2
2¯h
2
1+i
¯hF
2
t
2mω
2
c
1 −
ω
c
t
2
cot
ω
c
t
2
. (22)
The rms potential and force of the random field figuring in equation (22) are now rewritten in
terms of the Fourier transform of the correlation function as
γ
2
=
d
2
k
(2π)
2
W (k) (23)
and
F
2
=
d
2
k
(2π)
2
k
2
W (k). (24)
1310 SGlutsch et al
2.3. Analytic solution for the Landau level DOS
Let us now return to the calculation of the Landau level DOS. Upon putting equation (22)
into (5) with the subsequent use of a spectral expansion of the disorder-free Green function:
G
0
(t) =
ω
c
2π
∞
n=0
exp(−iE
n
t), (25)
with E
n
= ¯hω
c
(n +1/2) as the Landau levels, we get a corresponding expansion for the DOS:
ρ(E) =
ω
c
2π
2
∞
n=0
+∞
−∞
dt exp
i
¯h
(E − E
n
)t −
γ
2
t
2
2¯h
2
1+i
¯hF
2
t
2mω
2
c
1 −
ω
c
t
2
cot
ω
c
t
2
.
(26)
Next, the cot(ω
c
t/2) in equation (26) is to be replaced with a Fourier series. The t integrals
appearing are then straightforward by means of [38]:
+∞
−∞
dxx
n
exp(−px
2
− qx) =
i
2
n
√
π
p
(n+1)/2
exp
q
2
4p
H
n
iq
2
√
p
, [Re p > 0],
(27)
with H
n
(x ) being a Hermite polynomial. Consequently, we are able to represent the DOS of
interest in terms of the following series:
ρ(E) = n
LL
∞
n=0
1
√
2πγ
exp
−
(E − E
n
)
2
2γ
2
1+
¯h
4mω
c
2
1 − 2
E − E
n
¯hω
c
−
(E − E
n
)
2
γ
2
+2
∞
k=1
exp
−
(k¯hω
c
)
2
2γ
2
+
k¯hω
c
(E − E
n
)
γ
2
1 −
(E − E
n
− k¯hω
c
)
2
γ
2
,
(28)
where n
LL
= 1/πl
2
c
= eB/π¯h is the degeneracy of a Landau level. The series thus obtained
can be rearranged. As a result, we may finally find an analytic solution for the Landau level
DOS of a 2DEG in the presence of a perpendicular magnetic field and a semiclassical random
field:
ρ(E) = n
LL
∞
n=0
1
√
2πγ
exp
−
(E − E
n
)
2
2γ
2
×
1+
1
2mω
2
c
2
E
n
− (E − E
n
) − E
n
(E − E
n
)
2
γ
2
. (29)
It should be stressed that the above expression is derived under the condition of smoothness of
therandom field, however, not of its weakness. The DOS in a form analogous to equation (29)
was supplied previously [31] but only for a Gaussian choice of the correlation function
W (
r) = γ
2
exp(−r
2
/L
2
) with r =|r|,bymeansofanexpansion of the Green function (8)
in the inverse correlation length L
−1
.
Equation (29) evidently indicates that our DOS describes, as expected, a series of
broadened Landau levels. Moreover, the semiclassical DOS is made from two contributions:
the purely classical component and itsquantum correction, correspondingto the firstandsecond
terms inside curly braces in equation (29), respectively. The former exhibits a symmetric
Gaussian DOS whose linewidth is determined merely by the disorder potential γ ,independent
of the correlation length and the applied magnetic field. In contrast, the latter results in an
asymmetric non-Gaussian DOS whose broadening depends not only on the potential γ and
correlation length (or, equivalently, force F)ofthe random field but the magnetic field B as
Density of states of a two-dimensional electron gas 1311
well. It is worth remarking that the role of the quantum correction and, hence, the deviation of
theLandau level DOS from the Gaussian form is inversely proportional to the effective mass,
the square of the correlation length and the magnetic strength, i.e. to 1/m
2
B
2
.Thisimplies
that the shorter the disorder correlation length and the weaker the magnetic field is, the more
non-Gaussian the DOS becomes. Moreover, the quantum correction may be significant for
a light particle. Interestingly, the contributionofeach Landau level to the integrated DOS is
equal to n
LL
both for the classical limit and the full semiclassical solution. Thus, the quantum
correction does not change the spectral weight of a Landau level. We will return to this point
in the next section.
It is interestingto note that a semiclassicalperturbativeexpansion of the one-particle Green
function (8) in the small ratio l
c
/ leads, with a Gaussian choice of the correlation function,
to an analytic solution to the Landau level DOS [28]. Thereby, the quantum correction is,
however, found to be independent of the correlationlength. Moreover, such an expansion is
proved not to be useful for the calculation of a two-particle Green function defining, e.g., the
conductivity of the 2DEGs. An analytic solution may be obtained for the DOS broadened by a
more general (Poisson’s) distribution of long-range scatterers with a Gaussian potential [29].
However, this is derived simply for the purely classical component of the lowest Landau level
DOS by an expansion of the partition function in which both the ratios l
c
/ and γ/¯hω
c
are
assumed small.
2.4. Conditions of applicability of the semiclassical non-perturbative approach
We now assess the validity of the Landau level DOS (29). In the earlier theories [35, 36] the
discussion of the quality of the adopted approximation has been focused mainly on the relation
in the lengths involved. Nevertheless, this is obviously decided by the scales not only in the
length, but in the strength of the magnetic and random fields, and the electron energy as well.
For a fixed energy range −∞ < E E
n
,theconditions are
2
2
l
2
c
n +
1
2
(30)
and
8γ
2
2
(¯hω
c
)
2
l
2
c
n +
1
2
. (31)
The above inequalities and the inequality (20) reveal that the DOS (29) becomes a better
approximation when increasing the correlation length and the rms potential γ of the random
field, and decreasing the Landau level index n.Thismeans that the disorder is smoother
and stronger, e.g., due to heavy doping and the electron energy is lower. The first inequality
replaces the condition (10). This requires that the correlation length has to be much larger
than the radius of the cyclotron orbit related to the nth Landau level, i.e. its spatial extension,
given by R
c
(E
n
) = l
c
√
2(n +1/2).Inparticular, this implies a high enough magnetic field.
The second inequality must be fulfilled in order to avoid a negative DOS between two Landau
levels, which implies a low enough magnetic field. Thus, both the conditions can be satisfied
simultaneously in a moderate range of magnetic fields. These also imply that, in the case of
relatively weak disorder, our analytic solution is useful for low Landau levels (see section 3.2).
Indeed, because the radius of a particle trajectory is reduced with decreasing the energy (or
n), at low energies it is of small size, so that the estimation (11) is well justified. In addition,
equation (29) goes over into a Gaussian DOS of the linewidth γ in the limit γ→∞,which
is the condition (31).
As already mentioned in section 1, for a disorder of arbitrary correlation there has been
giveninthe literature a version of the semiclassical theory of the Landau level DOS with the
help of a path-integral technique, provided in [36]. There, the disorder is, however, assumed
1312 SGlutsch et al
to be not only smooth, but weak as well. Then, a semiclassical perturbative path-integral
approach may be developed. The basic idea of the method is outlined as follows [36, 39]. The
smoothness condition (10) enables the Green function (8) to be expressed as a sum over all
classical orbits, in which each (orbit-dependent) term is the product of a classical amplitude
and a phase factor including the classical action. Furthermore, under the weakness condition,
given by [36]
ω
c
τ
tr
1, (32)
with τ
tr
the transport time, the disorder influence on classical orbits and their classical amplitude
is negligibly small. The dominant disorder effect on the Green function comes from the shifts
in phases due to the modification of the classical actions along the unperturbed trajectories.
The latter are to be chosen as saddle-point orbits in the absence of disorder, i.e. within the
stationary-phaseapproximation. As a result, the Landau levelDOS is obtained in the following
form [36]:
ρ(E) = n
LL
∞
n=0
1
√
2π(E)
exp
−
(E − E
n
)
2
2
2
(E)
. (33)
Here, the linewidth is fixed by
2
(E) =
d
2
k
(2π)
2
W (k)J
2
0
[kR
c
(E)], (34)
where k =|
k|, J
0
(x ) is the Bessel function of zero order and
R
c
(E) = l
c
2E /¯hω
c
(35)
is the radius of the cyclotron orbit as an increasing function of energy. The DOS expression
determined by equations (33)–(35) was already derived within perturbation theory with the use
of a diagrammatic technique [35]. It was indicated that this holds for high Landau levels and
obviously represents a system of Gaussian peaks with the broadeningdecreasing with energy
as E
−1/4
.
Next, we turn to the discussion of the condition for a random field to be weak. For
acorrelation function depending on the distance only, W (
r) = W (r),thetransport time
entering inequality (32) is specified by [36]
1
τ
tr
=−
1
m
2
ω
3
c
R
3
c
(E)
∞
0
dr
r
dW (r)
dr
. (36)
For simplicity, the integral in equation (36) is estimated for the case of a Gaussian-correlated
disorder (with the correlation length = L/2) to be
1
τ
tr
=
√
π
2
γ
2
m
2
ω
3
c
R
3
c
(E)
. (37)
Upon replacing R
c
(E) in equation (37) by equation (35) and then inserting the transport time
appearing into inequality (32), the latter becomes
4
√
π
¯hω
c
E
3/2
γ
2
¯h
2
/2m
2
1, (38)
which must also hold qualitatively for any weak disorder.
The inequality just obtained reveals that, although the condition for validity of the existing
theories [35, 36] about the length scale is the same as in our theory, the conditions about the
disorder strength and electronenergy scales are clearly seen in opposition to the ones from
which the formula(29) is derived. The DOS (33) becomes a better approximation when
Density of states of a two-dimensional electron gas 1313
increasing the correlation length ,butdecreasing the rms potential γ and elevating the
energy E.Theincrease of the magnetic field B also favours the approximation. According
to equation (34), the linewidth of Landau levels (E) tends to the rms potential γ in the limit
of ultra-long-range correlation →∞,independent of γ .Hence formula (33) fails for low
Landau levels in the case where the product γis large but is moderate. Thus, the earlier
DOS is accurate for the part of the energy spectrum related to high Landau levels, but does
not give the correct behaviour for low Landau levels, especially in the limit γ →∞and
fixed. This also is appropriate for light doping. In this sense our DOS, of which low Landau
levels and/or heavy doping are in favour, is a complement to the result obtained in [35, 36]. On
theother hand, the low-energy region is likely to be of more physical interest in comparison
to thehigh-energy one. Indeed, under a strong magnetic field and low temperature the DOS
values at energies less than the Fermi level play a key role, which is normally located within a
region of a few cyclotron energies, reckoned from the subband edge [10, 16]. In addition, it is
to be noted that, in the intermediate range of parameters, both the DOS formulae give nearly
identical results.
The energy regions in question suggest corresponding approximations. Indeed, according
to equation (35), the high Landau levels refer to classical orbits of large radius. In that case, one
may neglect their (relative) deformation caused by the random force and adopt the stationary-
phase approximation, keeping only saddle-point orbits in the classical-path sum for the Green
function [36, 39]. Moreover, it was pointed out [35] that the broadening of a Landau level is
then of purely classical origin, caused simply by its Gaussian fluctuations due to the random
potential, resulting in a Gaussian shape of the DOS, as indicated by equation (33). In contrast,
in the low-energy region the cyclotron orbits are of small size and their deformation may
become remarkable, in particular for a particle of small effective mass. Therefore, one must
take adequate account of the non-Gaussian quantum correction, which is closely connected
with therandom force and is of greaterimportance for a lightparticle,asshown in equation (29).
3. Computational approach
3.1. Reformulation for the Landau level DOS
To develop a computational approach to the DOS of interest, we start from its spectral
representation, defined by
ρ(E) = 2
λ
ϕ
λ
(r)δ(E
λ
− E)ϕ
∗
λ
(r)
, (39)
where E
λ
and ϕ
λ
describe the eigenstates of the Hamiltonian
ˆ
H for an electron moving in a
random potential and a magnetic field. In the asymmetric Landau gauge with a vector potential
for the magnetic field
A(r) = (0, Bx),theexplicit form of the Hamiltonian in real space is
ˆ
H =−
¯h
2
2m
∂
2
∂x
2
+
e
2
B
2
2m
x
2
+
eB
m
x
¯h
i
∂
∂y
−
¯h
2
2m
∂
2
∂y
2
+ U (x , y) =
ˆ
H
0
+ U (x , y), (40)
where
ˆ
H
0
is theHamiltonian with a magnetic field but without a random potential. It is
convenient both for the analytical and computational treatments to employ Born–von K
´
arm
´
an
cyclic boundary conditions for the y direction with normalization length L
y
.Hereafter, in all
equations the limit L
y
→∞is implicitly assumed.
Equation (39) requires the numerical solution of an eigenvalue problem and the statistical
average over many realizations of the stochastic potential. This is not feasible, as the dimension
of the matrix is typically of the order of a few million. Therefore, we derive a formulation,
1314 SGlutsch et al
which is self-averaging, so that ρ(E) can be determined from one realization of the stochastic
potential U.Furthermore, we reformulate the eigenvalue problem as an initial-value problem
and use a discretization of the differential operator in real space. Then the time-dependent
Schr
¨
odingerequation is solved numerically, taking fully into account the sparsity of the matrix.
The Green function appearing in equation(5),which followsfrom equation (39),obviously
takes the following form [37]:
G(
r, r, t) = (r|e
−i
ˆ
Ht/¯h
|r), (41)
where the round brackets (|···|) stand for theexpectation value of an operator.
In principle, G(t)=G(
r, r, t) can be evaluated by computationally solving
equation (41) and averaging over thousands of realizations of the disorder potential U(
r).
The task can be simplified by exploiting the self-averaging properties of the random field [40].
For this purpose, we replace the ensemble average by an average over the y coordinate and
employ the conservation of trace:
G(t)=
1
L
y
+L
y
/2
−L
y
/2
dy(x = 0, y|e
−i
ˆ
Ht/¯h
|x = 0, y)
=
1
2π
+∞
−∞
dk
y
(x = 0, k
y
|e
−i
ˆ
Ht/¯h
|x = 0, k
y
). (42)
The wavefunction which belongs to the quantum number k
y
is exp(ik
y
y)/
L
y
.With the help
of the identity
e
−ik
y
y
ˆ
H
0
(r, −i¯h∇)e
+ik
y
y
···=
ˆ
H
0
(r + (¯hk
y
/eB)e
x
, −i¯h∇) ···, (43)
and the translational invariance of U (see section 2.1), the case of k
y
= 0 can be reduced back
to k
y
= 0. At the same time, the x coordinate is shifted by ¯hk
y
/eB and the averaged Green
function becomes
G(t)=
eB
2π¯h
+∞
−∞
dx(x , k
y
= 0|e
−i
ˆ
Ht/¯h
|x, k
y
= 0). (44)
The integration over x can be replaced by the trace over any complete set of eigenstates. Here,
we use the unperturbed eigenfunctions for k
y
= 0, known as
ϕ
n
(x ) =
1
2
n
√
πn!l
c
exp
−
x
2
4l
2
c
H
n
x
l
c
, (45)
with n = 0, 1, 2, ....Thenthe final expression for the averaged Green function is
G(t)=
eB
2π¯h
∞
n=0
(n, k
y
= 0|e
−i
ˆ
Ht/¯h
|n, k
y
= 0) (46)
and the matrix elements of the Hamiltonian take the form
ˆ
H
nn
= ¯hω
c
n +
1
2
δ
nn
− i¯hω
c
X
nn
d
dy
−
¯h
2
2m
δ
nn
d
2
dy
2
+ U
nn
(y), (47)
in which
X
nn
=
1
2
l
c
√
n + n
+1, for |n − n
|=1,
0, otherwise
(48)
and
U
nn
(y) =
+∞
−∞
dx ϕ
∗
n
(x )U (x , y)ϕ
n
(x ). (49)
Density of states of a two-dimensional electron gas 1315
For U = 0, the solution for the disorder-free Green function G
0
(t) (25) is reproduced,
whichleads to the DOS in the absence of a random field:
ρ
0
(E) =
eB
π¯h
∞
n=0
δ
E − ¯hω
c
n +
1
2
. (50)
In the limit B → 0, the latter expression changes over into a step function:
ρ
0
(E, B = 0) =
m
π¯h
2
θ(E), (51)
known as the DOS of a 2DEG in the absence of both random and magnetic fields.
By virtue of the Fourier transform (5), the integrated DOS including the spin degeneracy
is equal to 2G(t = 0),anditfollows clearly from equation (46) that the spectral weight of
each Landau level is equal to n
LL
,independently of the random field.
The Gaussian disorder potential can be generated as a Fourier series with random
phases [33, 41]:
U(
r) =
1
L
x
L
y
k
e
ik·r
C(k)
W (k), (52)
where the prime over the summation means that the point
k = 0istobeexcluded and
C(
k) = e
i(k)
(53)
with therandom phases (
k) distributed independentlyanduniformly in the interval[−π, +π).
The reality of U(
r) requires the constraint (k) =−(−k).Thepotential is periodic in
x and y directions with periods L
x
and L
y
,respectively. In the limit L
x
, L
y
→∞it is
macroscopically homogeneous and isotropic. The correlation function (1) follows from the
relation C(
k
1
)C(k
2
)=δ
k
1
,−k
2
.
The expression (46) is evaluated computationally by solving an initial-value problem and
the DOS (5) is calculated by means of a fast Fourier transform. The method is described
in [41].
3.2. Examples
Forthe numerical calculations, we use dimensionless quantities, defined by ¯h = l
c
= ω
c
= 1.
In addition, the DOS is normalized by ρ
∗
= m/π¯h
2
,whichisthe DOS of a two-dimensional
gas of free electrons (51). It is observed that in the classical limit (30) the coupling of different
Landau levels is small so that only a finite number |n − n
| n
off
of off-diagonal elements
U
nn
has to be taken into account. Furthermore, the sum in equation (46) can be restricted
to a finite number of Landau levels n
max
,whichneeds to be only somewhat larger than the
upper limit of the argument E.Forthecomputational solution, we used n
off
= 2, n
max
= 15
for an energy range E 10. The normalization length L
y
has to be much larger than the
correlation length ,inordertoensure self-averaging. For the present parameters a value of
L
y
= 32 000 was sufficient. We checked that the results were independent of L
y
and that
different realizations of the potential, given by different sets of random numbers, lead to the
same result for the DOS. For technical reasons, the stochastic potential (52) is also periodic
in the x direction. Here, it is sufficient to choosethe normalization length to be somewhat
larger than the extension of the highest Landau level in the x direction, as given by the distance
between the classical points of return 2
√
2n
max
−1. In the calculation, we used L
x
= 16. The
differential operators in equation (47) were discretized using second-order finite differences
with a mesh size y = 1/4. With this set of parameters we found full convergence and we
shall hereafter refer to the computational solution as the exact DOS.
1316 SGlutsch et al
3
2
1
0
–1
–20 2 4 6 810
ρ (E)
γ = 0.4
3
2
1
0
–1
–20 2 4 6 810
ρ (E)
γ = 0.2
E
E
Figure 1. Landau levelDOS ρ(E) versus energy E in dimensionless units for a Gaussian correlation
function W (
r) = γ
2
exp(−r
2
/L
2
) with the disorder parameters L = 6, γ = 0.4(upper part) and
0.2 (lower part). The full and dotted curves refer to the computational and analytic solutions,
respectively.
As a first example, we have evaluated the DOS in the case of a Gaussian correlation
function W(
r) = γ
2
exp(−r
2
/L
2
) for L = 6anddifferent values γ = 0.4and0.2. The same
function was addressed in [32]. Let us first check the smoothness criteria (20), (30) and (31).
The correlation length is equal to = L/2. For γ = 0.4, we have γ
2
= 3.6, 2
2
= 18
and 8γ
2
2
= 11.52. For γ = 0.2, we obtain γ
2
= 1.8, 2
2
= 18 and 8γ
2
2
= 2.88.
From these estimates we expect that, with γ = 0.4, the analytical formula (29) is a good
approximation for E 10. With γ = 0.2, the first and third inequalities are hardly fulfilled
and we expect negative values for the DOS after a few Landau levels.
The result of the numerical calculations is displayed in figure 1. The computational result
(full curve) is compared with theanalytic one (dotted curve). Let us first examine the general
properties of the exact solution. For the first Landau level, the lineshape is virtually the same
as for the classical result (not shown). With increasing energy, the peaks become increasingly
narrow, but the shape of the curve is not distinctively different from the Gaussian form, which
is, as mentioned in section 2.3, due to the large correlation length in use. In a simplified
picture, this line narrowing could be interpreted as an averaging of the random potential with
the probability density of the Landau level |ϕ
n
(x )|
2
.Thenthe linewidth of the nth Landau
levelshould be equal to U
nn
.Itturns out that the linewidth is smaller than this value, which
Density of states of a two-dimensional electron gas 1317
1.50
1.25
1.00
0.75
0.50
02
4
6810
ρ (E)
E
Figure 2. Computational solution for the Landau level DOS ρ(E) versus energy E in dimensionless
units for different correlation functions: exponential W (
r) = γ
2
exp(−r/L) (full curve) and
Gaussian W (
r) = γ
2
exp(−r
2
/L
2
) (dotted curve). The disorder parameters in both cases are
L = 6andγ = 0.4.
means that motional narrowing due to quantum-mechanicalmotion in the y direction plays an
important role. Inthewhole energy range the minima are located near E = n and the maxima
near E = n +1/2. The height of the maxima is about a linear function of the energy. The
same is true for the minima, as long as they do not come close to zero. We will see later that
the linear scaling of the maxima and minima is not a universal feature.
Next, we check the quality of the analytical solution (29). With γ = 0.4, the approximation
is reasonable in the whole energy range. Negative values for the DOS will occur only at much
larger energies (E ≈ 35). For the first three (n = 2) Landau levels, the analytical and
computational solutions are virtually indistinguishable. A small quantitative deviation from
the exact result is observed for n 3. With γ = 0.2, the approximation for the first three
Landau levels is also very good. Also, the overall relative error is small. However, starting
with n 3, the analytical DOS becomes negative between two Landau levels (at E = n).
Despite a large time consumption, a major advantage of the computational approach
is that its validity is independent of a specific choice of the correlation function. As an
illustrating example, we have calculated the DOS for an exponential correlation function
W (
r) = γ
2
exp(−r/L).Thenthe relevant average force F is, according to equation (24),
divergent so that the semiclassical method developed in the preceding section fails to be valid,
whereas the computational method works quite well. Further, since W (
r) is non-analytic at
r = 0, analytic solutions based upon a Taylor expansion at r = 0derived, e.g., in [32] cannot
be used.
The computational DOS for an exponential correlation function with L = 6andγ = 0.4
is depicted by a full curve in figure 2. For comparison, the corresponding solution for a
Gaussian correlation function with the same parameters is indicated by a dotted curve. In both
cases, the qualitative behaviour of the DOS is the same regarding the curve narrowing with
increasing energy and the positions of the maxima and minima. A difference is observed in
the behaviour of the peak heights as a function of energy. For the full curve, the peak heights
scale approximately like
√
E,incontrast to the linear scaling for the dotted curve.
1318 SGlutsch et al
4. Quantum well
4.1. Correlation function
We shall now apply the forgoing theory to evaluate the Landau level DOS in a quantum well
where the motion oftheelectrons are confined in the z direction by two infinite potential
barriers at its boundaries, 0 z a,with a the width of the well. The disorder is normally
caused by impurity doping, surfaceroughnessand alloying. As indicated above, the theoretical
analysis of the disorder effect on the Landau levels is simply reduced to finding the correlation
function for the 2DEG to be treated.
Fluctuations in the density of ionized impurities give rise to a random field. We assume
amodulation doping where the impurities are implanted in the system at a distance z
i
from a
boundary of the quantum well. The correlation function of the random field created by a 2D
sheet of impurities was derived in [17] to be
W (
k) =
Ze
2
2
L
0
2
n
i
k
2
2
(k)
F
2
ei
(k). (54)
with k =|
k|.Heren
i
denotes the 2D impurity density and Z is the impurity charge. The
L
is
the dielectric constant of the background lattice and (k) is the dielectric function allowing for
the screening of disorder interaction bythe2DEG. The form factor F
ei
(k) takesinto account
afinite extension of the electron state along the z direction, so that it depends on the geometry
of the system, viz. the well width a and the position of the impurity sheet z
i
,asfollows:
F
ei
(k) =
4π
2
ka(4π
2
+ k
2
a
2
)
e
kz
i
(1 − e
−ka
), for z
i
0,
2 −e
−kz
i
− e
−k(a−z
i
)
+
k
2
a
2
2π
2
sin
2
π z
i
a
, for 0 z
i
a.
(55)
Within the Thomas–Fermi approximation, the dielectric function is supplied by
(k) = 1+
k
s
k
F
ee
(k), (56)
where k
s
is the screening wavevector (inverse of the screening length) and F
ee
(k) is the
form factor for the electron–electron interaction potential in the 2DEG. The electron–electron
interaction is to be modified by the z extension of the electron state, so that
F
ee
(k) =
1
4π
2
+ k
2
a
2
3ka +
8π
2
ka
−
32π
4
k
2
a
2
1 − e
−ka
4π
2
+ k
2
a
2
. (57)
Aparticular problem is the calculation of the screening wavevector in the presence of both
disorder and magnetic field. Several authors proposed a formula of the kind [2]
k
s
=
e
2
2ε
L
ε
0
ρ(E
F
). (58)
For a given DOS, the FermienergyE
F
implicitly depends on the density of electrons. Thus,
the DOS, the Fermi level and the screening wavevector have to be determined self-consistently.
Unfortunately,this procedure has a fixpoint,characterizedby ρ(E
F
) =∞, k
s
=∞, ε(k) =∞,
U(
r) ≡ 0, and ρ(E) = ρ
0
(E),asgiven by equation (50). As a consequence, the random field
created by the charged impurities will always be totally screened. To avoid this spurious
solution within the self-consistent Thomas–Fermi screening, one has to use an improved
formula for the dielectric function, which has not been available so far. For simplicity, we
Density of states of a two-dimensional electron gas 1319
1
d
c
b
a
0.1
ρ (E = 1)
B (T)
0.01
2 34567
Figure 3. Analytical solution for the Landau level DOS at the cyclotron energy ρ(E = 1) versus
magnetic field B for a quantum well of width a = 140 Å, doped with Z = 1, z
i
=−200 Å,
and various impurity densities n
i
= 0.8(a),1.2(b), 1.4 (c), and 1.6 × 10
12
cm
−2
(d). The full,
broken and dotted curves refer to our calculation, the prediction of [32] (figure 3, diamonds) and
the measurement of [4], respectively.
replace ρ(E
F
) by the DOS without both random and magnetic fields, ρ
0
(E, B = 0), according
to equation (51). Then the screening wavevector is equal to
k
s
=
me
2
2πε
L
ε
0
¯h
2
. (59)
The principal results are rather insensitive to the special choice of k
s
.
4.2. Numerical results and comparison with experiment
The numerical results for the Landau level DOS are to be presented in connection with some
available experimental findings. These are specified for a modulation-doped quantum well
made from GaAs with the effective mass m = 0.067 m
e
.TheDOS is measured in units of
ρ
∗
= 2.8 ×10
10
meV
−1
cm
−2
and the energy in units of ¯hω
c
.
First, we have calculated the DOS at the cyclotron energy (E = 1)—between the lowest
and first excited Landau levels—as a function of the applied magnetic field. The geometrical
parameters are taken from [4], sample 1 for magnetization measurement. The well width is
a = 140 Å. The layer doped with the impurity charge Z = 1isassumed to be located in the
middle of the barrier so that z
i
=−200 Å. Since the impurity density n
i
was not indicated,
we perform the calculation with various values n
i
= 0.8, 1.2, 1.4 and 1.6 × 10
12
cm
−2
.The
result thus obtained for the DOS ρ(E = 1) is displayed in dimensionless units for a range of
magnetic fields from B = 2–7 T by full curves in figure 3. There, the theoretical prediction
of [32] (brokencurves) and the data of [4] (dotted curve), which were fitted by a Gaussian DOS
with a broadening γ(meV) =
√
B (T),arealsoplotted for comparison. At the values of B
used, our results are clearly found in quantitative agreement with the experimental data, both
in the order of magnitude of the DOS and its magnetic-field behaviour as well. In particular,
thefitisvery good at the doping level n
i
= 1.4 ×10
12
cm
−2
.Indistinction, the exact solution
of the simple model based on a quadratic correlation function derived in [32] gives at a large
magnetic field a small DOS between the Landau levels. At B = 7T,thepredicted DOS is
1320 SGlutsch et al
5
c
b
a
4
3
2
1
0
0.0
2.0
0.5 1.51.0
ρ (E)
E
d
Figure 4. Analytical solution for the Landau level DOS ρ(E) versus energy E in dimensionless
units for a quantum well of width a = 75 Å, doped with Z = 1, z
i
=−200 Å, under a doping
level n
i
= 5 ×10
11
cm
−2
and different magnetic fields B = 1(a),3(b), 7 (c) and 10 T (d).
8
a
b
c
d
6
4
2
0
0.0
2.0
0.5 1.51.0
ρ (E)
E
Figure 5. Analytical solution for the Landau level DOS ρ(E) versus energy E in dimensionless
units for a quantum well of width a = 75 Å, doped with Z = 1, z
i
=−200 Å, under a magnetic
field B = 7Tand various impurity densities n
i
= 0.1(a),0.5(b), 1 (c) and 1.5 × 10
12
cm
−2
(d).
The full and broken lines refer to our calculation and the measurement of [8], respectively.
smaller than the measured one by up to more than one order of magnitude, even with a best
choice of the fitting parameters. Moreover, an inspection of the full curves in figure 3 reveals
that the Landau level DOS at the cyclotron energy exhibits a rapid decrease when increasing B
(about square exponential) and lowering n
i
(about linear exponential), which is expected from
the semiclassical solution (29) and the linear n
i
dependence of the mean squared potential γ
2
.
Next, we evaluatedtheLandau levelDOSof a sample studied in [8] by magnetocapacitance
measurement. The parameters are the same as in figure 3, except for the well width a = 75 Å.
Figures 4 and 5 depict the DOS as a function of energy ρ(E).Infigure 4 this is plotted under
a doping level n
i
= 5 ×10
11
cm
−2
and different magnetic fields B = 1, 3, 7 and 10 T, whereas
Density of states of a two-dimensional electron gas 1321
in figure 5 this is under a value of B = 7Tand various impurity densities n
i
= 0.1, 0.5, 1.0
and 1.5 × 10
12
cm
−2
.There,the data of [8] at B = 7T(broken curve) is also reproduced for
comparison. Figure 4 illustrates an apparent deviation of the DOS from the Gaussian form
at lower magnetic fields. With a reduction of B the Landau levels become more asymmetric
and their overlap becomes larger, and at B = 1Tthelineshape is almost flat. The apparent
non-Gaussian feature of the broadening is due to the significant quantum correction associated
with asmall effective mass of the material under consideration, as quoted in section 2.3. The
structureless behaviour of the Landau level DOS at low magnetic fields was already seen for
the case of weakly disordered 2DEGs, basedontheSCBAtheorywith the use of nonlinear
self-consistent screening [16, 42]. It is observed from figure 5 that the lineshape calculated with
n
i
= 1 ×10
12
cm
−2
is in reasonable agreement with the measured one. It has to be mentioned
that our theory may provide a remarkable DOS between the Landau levels, whereas the authors
of [8] had to invoke a constant background DOS of 28% of ρ
∗
to fit their data.
5. Conclusions
In summary, in this paper we have proposed two schemes for calculating the Landau level DOS
of a 2DEG subjected to a uniform perpendicular magnetic field and a random field of arbitrary
correlation. Both of the methods are able to work with the realistic correlation function of
thelatter. This allows us to take complete account of various details of the actual geometry of
a2DEG, such as its extension into the bulk and its 2D screening of the disorder interaction.
Also, this allows us to take the actual origin of the disorder and to evaluate the disorder effect
on the lineshape due, for example, to impurity doping, surface roughness and alloying. Thus,
our theory offers a microscopic description of the Landau level DOS of actual quasi-2DEGs.
It is to be stressed that all existing theories of the Landau level DOS are approximation
schemes. Almost all of them were based on simple models of the correlation function with
some fitting parameters, and developeddifferent methods for exact or approximate calculation.
However, the theoretical prediction may not be in agreement with experiment, even with an
exact solution, as in the case of a quadratic correlation function [32]. In contrast, our theory
starts from the realistic correlation function and turns out to be able to give a quantitative
explanation of experimental data with no fitting parameters.
The analytic theory is established within a semiclassical non-perturbative path-integral
approach to the random field and provides a simple closed solution (29) for the Landau level
DOS, whereas the computational approach is exactly derived, whose validity is independent of
aspecific choice of the correlation function. As a consequence, the former is applicable only
to asmooth random field, whereas the latter works quite well with any random field. Under
thecriteria (20), (30) and (31), the analytical solution has been found to fit the computational
solution very well.
The semiclassical formula (29) for the Landau level DOS is composed of the classical
component and its quantum correction. The former is responsible for a symmetric Gaussian
DOS and is associated merely with fluctuations in the disorder potential, whereas the latter
is for an asymmetric non-Gaussian DOS and is connected not only with fluctuations both in
the potential and force of the random field, but with the applied magnetic field as well. This
implies that there is not necessarily a universal form for the Landau level DOS, which can go
over from a Gaussian form to another one, depending on thedisorder correlation length and
the magnetic field. Suchadependence of the DOS behaviour on the correlation length has
already been shown in [43] for disordered 2DEGs in the absence of a magnetic field. Under
adisorder of rather short correlation length and rather low magnetic field, the deviation from
theGaussian form is found to be significant, especially for a particle of small effective mass,
as in GaAs.
1322 SGlutsch et al
Our solution (29) is useful for low Landau levels, whereas the solution (33) due to previous
authors [35, 36] is accurate for high Landau levels. It is worth noting that the classical
DOS (33) includes the disorder effect associated, via the linewidth (34), with the random
potential only, whereas the semiclassical DOS (29) allows for, via the rms potential and force,
theinfluence of both the potential and its spatial derivative of the first order. This suggests
that our semiclassical non-perturbative path-integral approach is to be regarded as a first step
towards a better approximation,especially for a non-smoothdisorder, where one might develop
a quantum theory of the broadening of Landau levels with the aid of an expansioninvolving the
spatial derivatives of all orders of the disorder potential. This idea was, in fact, proved in [43]
to be successful for disordered low-dimensional electron systems in the absence of magnetic
field.Inthe case of a non-smooth random potential its short-range fluctuations make a key
contribution to the energy spectrum. This requires a quantum description of electron states
and may likely give rise to an apparently non-Gaussian Landau level DOS.
Another merit of our theory is that it might be applied on an equal footing to calculation
of the disorder effect on a two-particle Green function defining, for example, the conductivity
of 2DEGs. In contrast, it was shown [36, 39] that the semiclassical perturbative path-integral
approach with the use of saddle-point orbits turns out to be unable to produce the relevant
disorder effect, so that one has to go beyond the stationary-phase approximation, taking
explicitly into account the deformation of classical orbits.
Acknowledgments
This work was done during the stay of one of the authors (DNQ) at the Institut f
¨
ur
Festk
¨
orpertheorie und Theoretische Optik,Friedrich-Schiller-Universit
¨
at Jena,Germany under
financial support by the Deutscher Akademischer Austauschdienst.
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