PHYSICAL REVIEW B 71, 045329 共2005兲
Microwave photoconductivity of two-dimensional electron systems with unidirectional
periodic modulation
Jürgen Dietel
Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
Leonid I. Glazman
W.I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55454, USA
Frank W. J. Hekking
Laboratoire de Physique et Modélisation des Milieux Condensés, CNRS Université Joseph Fourier, BP 166,
38042 Grenoble Cedex 9, France
Felix von Oppen
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
and Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
共Received 13 July 2004; revised manuscript received 22 October 2004; published 24 January 2005兲
Motivated by the recently discovered microwave-induced “zero-resistance” states in two-dimensional electron systems, we study the microwave photoconductivity of a two-dimensional electron gas 共2DEG兲 subject to
a unidirectional static periodic potential. The combination of this potential, the classically strong magnetic
field, and the microwave radiation may result in an anisotropic negative conductivity of the 2DEG. Similar to
the case of a smooth random potential, two mechanisms contribute to the negative photoconductivity. The
displacement mechanism arises from electron transitions due to disorder-assisted microwave absorption and
emission. The distribution-function mechanism arises from microwave-induced changes in the electron distribution. However, the replacement of a smooth random potential by the unidirectional one, leads to different
relative strengths of the two contributions to the photoconductivity. The distribution function mechanism
dominates the photoconductivity in the direction of the static potential modulation, while both mechanisms
contribute equally strongly to the photoconductivity in the perpendicular direction. Moreover, the functional
dependence of the negative photoconductivity on the microwave frequency is different for the two mechanisms, which may help to distinguish between them. In another marked difference from the case of smooth
disorder, the unidirectionality of the static potential simplifies greatly the evaluation of the photoconductivities,
which follow directly from Fermi’s golden rule.
DOI: 10.1103/PhysRevB.71.045329
PACS number共s兲: 73.40.⫺c, 73.50.Pz, 73.43.Qt, 73.50.Fq
I. INTRODUCTION
Recent experiments1–5 on a two-dimensional electron gas
共2DEG兲 in weak magnetic fields under microwave irradiation
have led to the unexpected discovery of regions in magnetic
field where the longitudinal resistance is very close to zero.
Unlike for quantized Hall states, the Hall resistance remains
essentially classical and nonquantized for these novel “zeroresistance states.” These states occur near magnetic fields
where, up to an additive constant, the microwave frequency
␻ is an integer multiple of the cyclotron frequency ␻c.
This discovery initiated a flurry of theoretical activity
from which the following basic picture emerges. It has been
argued6,7 that under microwave irradiation, the microscopic
diagonal conductivity can become negative. This would lead
to a 共macroscopic兲 instability towards a current carrying
state. Macroscopic resistance measurements on this state
show zero resistance because current can be made to flow
through the sample by a rearrangement of large current domains. Different microscopic mechanisms for a negative
contribution to the microwave-induced photoconductivity
have been proposed. One mechanism relies on disorder1098-0121/2005/71共4兲/045329共15兲/$23.00
assisted absorption and emission of microwaves8–11 共see also
Ref. 12兲. Depending on the detuning ⌬␻ = ␻c − ␻, the displacement in real space associated with these processes is
preferentially in or against the direction of the applied dc
electric field. In an alternative mechanism, microwave absorption leads to a change in the electron distribution function, which can result in a negative photoconductivity.4,13,14
Detailed calculations within the self-consistent Born approximation suggest that the latter mechanism is larger by a
factor ␶in / ␶s*, where ␶in is the inelastic relaxation time and ␶s*
denotes the single-particle elastic scattering time in a magnetic field.
In the present paper, we study the microwave-induced
photoconductivity within a model in which the 2DEG is subjected to a unidirectional and static periodic potential.15 Our
motivation for doing so is twofold. First, the study of periodically modulated 2DEGs in a perpendicular magnetic field
has led to the discovery of a number of interesting effects
such as transport anisotropies16 and commensurability effects
such as the Weiss oscillations of the conductivity.17–20 Thus,
investigating the effects of a static periodic potential 共which
is not present in the experiments performed to date兲 in the
045329-1
©2005 The American Physical Society
PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
regime of microwave-induced zero-resistance states may
shed light on the underlying physics. In addition, the periodic
potential lifts the Landau level 共LL兲 degeneracy. This allows
one to exploit the familiar relation between momentum
transfer and distance in real space in high magnetic fields to
compute the current by applying Fermi’s golden rule. In this
way, one finds in the absence of microwaves that scattering
from the disorder potential U leads to a current
jx =
distribution-function mechanism is worked out in Sec. V
with the main results in Eqs. 共66兲 and 共68兲 along the modulation direction and Eqs. 共70兲 and 共72兲 across the modulation
direction. Section V also contains a discussion of the Weiss
oscillations of the photocurrent. The polarization dependence
is considered in Sec. VI. We summarize in Sec. VII. Some
technical details are given in a number of appendices. In the
remainder of this paper, we set ប = 1.
␲e
兺 兺 共k⬘ − k兲lB2 兩具n⬘k⬘兩U兩nk典兩2关f nk0 − f 0n⬘k⬘兴
LxLy nn kk
⬘
II. THE MODEL
⬘
⫻␦共⑀nk − ⑀n⬘k⬘兲
A. Basics
共1兲
0
for an applied dc electric field in the x direction.22 Here, f nk
is the Fermi-Dirac distribution function of the Landau level
states 兩nk典 which remain approximate eigenstates even in the
presence of the periodic potential 共n is the LL index兲. The ␦
function involves the energies ⑀nk of these states including
the effect of both periodic potential and dc electric field.
It is evident from Eq. 共1兲 that the microwaves will affect
the current in two ways. 共i兲 The joint effect of disorder and
microwaves can give additional contributions to the transition matrix elements. This is the origin of the displacement
photocurrent which relies on the displacements in real space
associated with disorder-assisted absorption and emission of
microwaves. In more conventional terms, this contribution
can be associated with the effect of the microwaves on the
collision integral in a kinetic equation. 共ii兲 The microwaves
will also result in a redistribution of electrons, changing the
electron distribution function f nk away from its equilibrium
0
. This distribution-function contribution to the phoform f nk
tocurrent will be important if inelastic relaxation is sufficiently slow. Our model allows us to compute the various
contributions to the photocurrent straight-forwardly within
Fermi’s golden rule.
For the parallel photocurrent 共i.e., parallel to the wave
vector of the static periodic modulation兲 we find that the
distribution-function mechanism gives a larger contribution
than the displacement mechanism, by a factor ␶in / ␶s*. In addition, we find in this case that our results, with suitable
identifications, are parametrically consistent with earlier results for disorder broadened Landau levels in the selfconsistent Born approximation.11,14 By contrast, we find a
strong enhancement of the displacement mechanism for the
perpendicular photocurrent so that in this case, both contributions are of the same order.
For readers not interested in the details of the derivation,
we include a guide to our main results. In Sec. II, we introduce the model and collect the relevant background material.
In Sec. III, we compute the dark conductivity. The condcuctivity along the modulation direction is given in Eqs. 共30兲
and 共31兲. The conductivity across the modulation direction is
presented in Eqs. 共36兲 and 共37兲. In both cases, an interpretation of the result is given in the following pararaph. The
displacement mechanism for the photocurrent is discussed in
Sec. IV. Our main results are in Eqs. 共51兲 and 共52兲 along the
modulation direction and in Eqs. 共57兲 and 共58兲 across the
modulation direction. Estimates interpreting these results are
again given in the paragraphs following the equations. The
In this section, we specify our model and review some
relevant background material. We consider a twodimensional electron gas 共2DEG兲 subject to a perpendicular
magnetic field B and a unidirectional static periodic potential
共2兲
V共r兲 = Ṽ cos共Qx兲
with period a = 2␲ / Q. The periodic potential which lifts the
Landau level degeneracy, is assumed to be stronger than the
residual disorder potential U共r兲. The disorder potential is
characterized by zero average and variance
具U共r兲U共r⬘兲典 = W共r − r⬘兲.
共3兲
For white-noise disorder, W共r − r⬘兲 = 共1 / 2␲␯␶兲␦共r − r⬘兲 with
Fourier transform W̃共q兲 = 1 / 2␲␯␶. Here, ␯ denotes the density of states at the Fermi energy in zero magnetic field and ␶
is the zero-field elastic scattering time. For smooth disorder
potentials, the correlator W共r兲 falls off isotropically on the
scale of the correlation length ␰ of the disorder potential
共␰ Ⰷ ␭F for smooth disorder; ␭F denotes the zero-field Fermi
wavelength兲. We also note that the impurity average of the
disorder matrix element between oscillator states 兩nk典 for
electrons in a magnetic field in the Landau gauge is
兩具nk⬘兩U兩nk典兩2 =
冕
冋 冉 冊册
q2lB2
2
d 2q
−q2lB
/2
␦
e
L
q ,k −k
n
2
共2␲兲2 y ⬘
2
W̃共q兲.
共4兲
Here, n denotes the LL quantum number and k the momentum in the y direction. Ln共x兲 denotes the Laguerre polynomial and lB = 共ប / eB兲1/2 the magnetic length.
In this paper, we focus on the regime of high Landau
levels so that ␭F Ⰶ lB Ⰶ Rc. 共Here, Rc denotes the cyclotron
radius.兲 We assume that the period a of the modulation satisfies the condition
␭F Ⰶ a Ⰶ Rc .
共5兲
This is essentially a technical condition, which simplifies
some of the calculations. For smooth disorder, we assume, in
addition, that the correlation length ␰ of the disorder potential satisfies the inequality
␭F Ⰶ ␰ Ⰶ lB2 /a.
共6兲
Here, the first inequality reflects the fact that the disorder is
smooth, while the second inequality ensures that the typical
jump in real space of length lB2 / ␰ associated with a disorder
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MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
scattering event is large compared to the period a of the
periodic modulation.
The 2DEG is irradiated by microwaves described by the
electric potential
e
␾共r,t兲 = − r共E*ei␻t + Ee−i␻t兲 = ␾+e−i␻t + ␾−ei␻t ,
2
共7兲
where ␾+ = 关␾−兴* = −eE · r / 2. The complex vector E contains
both strength and polarization of the microwaves.
In the absence of disorder and microwaves, and for sufficiently weak periodic potential, the single-particle spectrum
of the electrons can be obtained by treating the periodic potential perturbatively. Starting with the oscillator states 兩nk典,
one obtains
冉 冊
0
⑀nk
⯝ ␻c n +
1
+ Vn cos共QklB2 兲.
2
共8兲
The amplitude Vn is given by
Vn = Ṽe−Q
2l2 /4
B
Ln共Q2lB2 /2兲.
共9兲
In the limit of high Landau levels, Vn can be approximated as
Vn ⯝ ṼJ0共QRc兲 and thus exhibits slow oscillations with period kFa Ⰷ 1 as a function of LL index n. 共The LL index n
enters via the cyclotron radius.兲 This also implies oscillations
of Vn as function of the magnetic field. It is these oscillations
of Vn which are responsible for the Weiss oscillations18 of the
conductivity.
If in addition, a dc electric field Edc is applied in the x
direction, the eigenenergies take the form
⑀nk ⯝
0
⑀nk
−
␯*共⑀兲 = ␯*˜␯*共⑀兲
共11兲
with the density of states at the band center
1
冉 冊 冉 冊
⳵ f nk
⳵ f nk
=
⳵t
⳵t
1
共12兲
2␲lB2 ␲Vn
冉 冊
⳵ f nk
⳵t
˜␯*共⑀兲 =
冑1 − 关共⑀ − En兲/Vn兴
2
.
B. Kinetic equation
We now turn to setting up the kinetic equation for the
nonequilibrium electronic distribution function f nk which describes the occupation of the LL oscillator eigenstates 兩nk典.21
These occupations change due to disorder scattering,
disorder-assisted microwave absorption and emission, as
well as inelastic relaxation which we include within the
−
mw
f nk − f
␶in
0
nk
共14兲
.
=
dis
兺 2␲兩具n⬘k⬘兩U兩nk典兩2关f n⬘k⬘ − f nk兴␦共⑀nk − ⑀n⬘k⬘兲.
n⬘k⬘
The collision integral for disorder-assisted microwave absorption and emission is
冉 冊
⳵ f nk
⳵t
=
mw
兺 ␴兺=± 2␲兩具n⬘k⬘兩T␴兩nk典兩2关f n⬘k⬘ − f nk兴
n⬘k⬘
⫻␦共⑀nk − ⑀n⬘k⬘ + ␴␻兲.
共16兲
The precise nature of the operator T␴ will be given in Eq.
共42兲 below. Note that these collision integrals involve the
electron energies including the effects of the dc electric field.
If the dc electric field points along the y direction, we can
no longer include it in the eigenenergies. Instead, it enters the
kinetic equation through an additional term describing the
associated drift in the x direction,
冉 冊 冉 冊
⳵ f nk
⳵ f nk
⳵ f nk
= − eEdc
+
⳵t
⳵k
⳵t
共13兲
Here, n and ⑀ satisfy 兩⑀ − En兩 ⬍ Vn 关with the LL energy
En = ␻c共n + 1 / 2兲兴. Note that the DOS can also be expressed as
␯* ⬃ ␯共␻c / Vn兲, reflecting the increased density of states due
to the Landau quantization.
dis
⳵ f nk
⳵t
共15兲
and the normalized density of states
1
+
In principle, there should also be a term which describes the
drift in the y direction induced by the dc electric field. However, this term has no consequences when considering distribution functions which are independent of y. In the last term
0
on the right-hand side, f nk
denotes the equilibrium FermiDirac distribution and ␶in denotes a phenomenological inelastic relaxation rate. The collision integral for disorder scattering is explicitly given by
共10兲
eEdcklB2 .
It is useful to define the density of states 共DOS兲 of a periodic
potential broadenend LL by
␯* =
relaxation-time approximation. Note that, in principle, this
distribution function also depends on the spatial coordinate
y. However, it will be sufficient throughout this work to consider distribution functions which are uniform in the y direction. 共The dependence on x, on the other hand, is included, as
the momentum k also plays the role of a position in the x
direction.兲
If the dc electric field points in the x direction, the kinetic
equation takes the form
+
dis
⳵ f nk
⳵t
−
mw
f nk − f
␶in
0
nk
.
共17兲
The collision integrals are given by the expressions in Eqs.
共15兲 and 共16兲 with the energies in the ␦ functions taken in the
absence of the dc electric field.
We close this section with a calculation of the elastic scattering rate 1 / ␶* in high magnetic fields. The motivation for
doing this is twofold. First, ␶* is a natural parameter in terms
of which to write our final results for the conductivity. On a
more technical note, computing ␶* gives us the opportunity
to introduce a convenient way of dealing with integrals involving Laguerre polynomials, which will be used repeatedly
throughout this paper. From the collision integral for elastic
disorder scattering, we obtain
045329-3
PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
1
=
␶ 共⑀兲 n
兺 2␲兩具n⬘k⬘兩U兩nk典兩2␦共⑀ − ⑀n⬘k⬘兲
*
共18兲
⬘k⬘
with ⑀ = ⑀nk. Noting that n = n⬘ and inserting the expression
共4兲 for the matrix element, we obtain
1
= 2␲
*
␶ 共⑀兲
冕
冋 冉 冊册
q2lB2
d2q −q2l2 /2
B
e
L
n
2
共2␲兲2
2
W̃共q兲␦共⑀ − ⑀nk+qy兲.
This current can be expressed by counting the number of
disorder scattering events that take an electron from a state k,
localized in the x direction at klB2 to the left of an imaginary
line x0 parallel to the y axis, to a state k⬘, localized at k⬘lB2 to
the right of this imaginary line, and vice versa. Due to current conservation, the current is independent of the particular
choice of x0 and it turns out to be useful to average over all
possible x0. This results in the expression
共19兲
The Laguerre-polynomial factor arising from the matrix elements of the disorder potential decays as a function of qlB2 on
the scale of the cyclotron radius Rc, in addition to fast oscillations on the scale of the zero-field Fermi wavelength ␭F.
On the other hand, the argument of the ␦-function changes
with klB2 on the scale of the period a of the periodic potential.
Thus, for white-noise disorder and in the limit ␭F Ⰶ a Ⰶ Rc,
we can average the ␦ function separately over qy. Using the
identity
具␦共⑀ −
0
⑀nk
兲典
⬘ k⬘
=
2␲lB2 ␯*共⑀兲
冋 冉 冊册
q2lB2
d2q −q2l2 /2
B
Ln
2e
2
共2␲兲
2
=
1
,
2␲lB2
jx =
⬘
0
⬘
共24兲
kk⬘
共25兲
jx =
␲e
兺
LxLy nk
0
nk
冕
−f
冋 冉 冊册
q2lB2
2
d 2q
2 −q2lB
/2
q
l
e
L
y
n
B
2
共2␲兲2
0
0
nk+qy兴␦共⑀nk
2
W̃共q兲
0
− ⑀nk+q
+ eEdcqylB2 兲.
y
共26兲
Expanding to linear order in the dc electric field, one obtains
for the conductivity
␴xx =
冉 冊
0
⳵ f nk
␲e2
−
兺 ⳵⑀0
LxLy nk
nk
冕
冋 冉 冊册
q2lB2
2
d 2q
2 2 −q2lB
/2
共q
l
兲
e
L
y
n
B
2
共2␲兲2
2
W̃共q兲
0
0
− ⑀nk+q
兲.
⫻␦共⑀nk
共27兲
y
For white-noise disorder and ␭F Ⰶ a Ⰶ Rc, the Laguerrepolynomial integral can be computed in analogy with the
evaluation of Eq. 共19兲 above. Using
冕
冋 冉 冊册
q2lB2
2
d 2q
2 2 −q2lB
/2
共q
l
兲
e
L
y
n
B
2
共2␲兲2
this yields
A. Conductivity ␴xx along the modulation direction
In this section, we compute the dark conductivity, i.e., the
conductivity in the absence of microwaves. We start with the
situation in which the dc electric field is applied in the x
direction, i.e., parallel to the wave vector of the static periodic modulation. We assume that the dc electric field is sufficiently weak so that heating effects can be ignored. In this
case, the distribution function remains in equilibrium
0
, and the system responds to the dc electric field with
f nk = f nk
a current in the x direction.
2␲兩具n⬘k⬘兩U兩nk典兩2
Inserting the explicit expression 共4兲 for the disorder-averaged
matrix element and performing the sum over k⬘, one obtains
⫻
III. DARK CONDUCTIVITY
兺
2
2
k⬍x0/lB
k⬘⬎x0/lB
兺 兺 共k⬘ − k兲lB2 兩具n⬘k⬘兩U兩nk典兩2关f nk0 − f n0⬘k⬘兴
共23兲
This result reflects the increased density of final states in the
limit of well-separated Landau levels.
For smooth disorder, we need to distinguish between the
single-particle scattering time and the transport scattering
time. Their zero-field values ␶s and ␶tr are related to the finite
field values ␶s*共⑀兲 and ␶tr*共⑀兲 in analogy to Eq. 共22兲, i.e.,
␶s*共⑀兲 = ␶s␯ / ␯*共⑀兲 and ␶tr*共⑀兲 = ␶tr␯ / ␯*共⑀兲. Some details of the
calculation are given in Appendix A.
兺 兺
− f n⬘k⬘兴␦共⑀nk − ⑀n⬘k⬘兲
0
nk
␲e
LxLy nn
⫻关f
In the following, we will also use the notation ␶* = ␶*共⑀
= En兲, i.e.,
␲Vn
␶* = ␶
.
␻c
−Lx/2
dx0
Lx nn
⫻␦共⑀nk − ⑀n⬘k⬘兲.
共21兲
共22兲
Lx/2
for the current in the x direction. Performing the integral over
x0 gives
we find the result
1 ␯ *共 ⑀ 兲
1
=
.
*
␶ 共⑀兲 ␶ ␯
冕
e
Ly
⫻关f
共20兲
and performing the remaining integral over the Laguerre
polynomial
冕
jx =
␴xx =
e 2N 1
LxLy 2␲␯␶
兺
nk
冉 冊
−
⳵f
0
nk
0
⳵⑀nk
2
N
,
␲
共28兲
2␲lB2 ␯*共⑀0k 兲.
共29兲
=
Expressing the sum by an integral involving the density of
states, we can cast this result in the final form
␴xx =
in terms of
045329-4
冕 冉
d⑀ −
冊
⳵ f共⑀兲
␴xx共⑀兲
⳵⑀
共30兲
PHYSICAL REVIEW B 71, 045329 共2005兲
MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
␴xx共⑀兲 = e2
冉 冊
R2c
2␶tr*共⑀兲
␯*共⑀兲.
共31兲
This equation is written such that it includes both types of
disorder. For white-noise disorder ␶tr* = ␶s* = ␶*, while for
smooth disorder ␶tr* ⫽ ␶s*. The derivation of the result for
smooth disorder is sketched in Appendix A.
This result for the dark conductivity can be interpreted as
follows. The bare rate for disorder scattering is 1 / ␶s*, where
each scattering event is associated with a momentum transfer
1 / ␰. This momentum transfer translates into a jump of magnitude lB2 / ␰ in real space so that the electron diffuses in the x
direction with a diffusion constant Dxx ⬃ 共lB2 / ␰兲2 / ␶s*. Alternatively, this diffusion constant can be written in terms of the
transport time as Dxx = R2c / 2␶tr* 关using that ␶tr* / ␶s* ⬃ 共kF␰兲2兴.
By the Einstein relation, this diffusion constant translates
into the conductivity given in Eq. 共31兲. The conductivity 共31兲
can also be expressed in terms of the zero-B conductivity
␴xx共B = 0兲 as ␴xx = ␴xx共B = 0兲 / 共␻c␶tr*兲2. We also note that
␴xx ⬃ 1 / V2n so that the oscillations of V with magnetic field B
关see Eq. 共9兲 above兴 lead to Weiss oscillations of the conductivity, in agreement with previous results.18
The energy integral in Eq. 共30兲 is formally logarithmically
divergent due to the square-root singularity of the density of
states ␯*共⑀兲 at the band edge. This singularity is cut off by
smearing of the band edge by disorder or by the applied dc
electric field, when the latter is kept beyond linear order.
B. Conductivity ␴yy perpendicular to the modulation
direction
An applied dc electric field in the y direction leads to a
nonequilibrium distribution function f nk due to the drift term
in the kinetic equation 共17兲. In the absence of microwaves,
linearizing the stationary kinetic equation in the applied dc
electric field yields
eEdc
0
⳵ f nk
= 2␲
⳵k
冕
冋 冉 冊册
q2lB2
d2q −q2l2 /2
B
e
L
n
2
共2␲兲2
2
0
0
⫻关␦ f nk+qy − ␦ f nk兴␦共⑀nk
− ⑀nk+q
兲.
y
W̃共q兲
共32兲
0
denotes the deviation from the equilibHere, ␦ f nk = f nk − f nk
rium distribution function, and we have neglected inelastic
processes relative to elastic disorder scattering. Due to the
periodicity in the x direction, ␦ f nk = ␦ f nk+a/l2 . Moreover, if k
B
0
and k + qy are two momenta with the same energy ⑀nk
, but
0
opposite signs of the derivative ⳵⑀nk / ⳵k, then ␦ f nk = −␦ f nk+qy.
Using that for white-noise disorder and ␭F Ⰶ a Ⰶ Rc, we can
split the q integration as in the evaluation of Eq. 共19兲 and
obtain
␦ f nk = − eEdc␶
*
0
0 ⳵ f nk
.
共⑀nk
兲
⳵k
兺
nk
0
⳵⑀nk
␦ f nk .
⳵k
共34兲
Inserting the expression for the distribution function gives
for the conductivity
␴yy = −
e2
L xL y
兺
nk
冉 冊
0
⳵⑀nk
⳵k
2
0
␶*共⑀nk
兲
⳵f
0
nk
0 .
⳵⑀nk
共35兲
Expressing the sum over nk as an energy integral involving
the DOS, we obtain
␴yy =
冕 冉
d⑀ −
冊
⳵ f 0共 ⑀ 兲
␴yy共⑀兲
⳵⑀
共36兲
with
␴yy共⑀兲 = e2共关vy共⑀兲兴2␶s*共⑀兲兲␯*共⑀兲.
共37兲
As above for ␴xx, this result is written such that it includes
both the case of white-noise and of smooth disorder. The
derivation for the case of smooth disorder is sketched in
Appendix A. We have defined the drift velocity
兩vy共⑀兲兩 =
冏 冏
0
⳵⑀nk
1
=
⳵k
␲ a ␯ *共 ⑀ 兲
共38兲
in the y direction, induced by the periodic modulation.
The result 共37兲 can be interpreted as follows. With respect
to the motion in the y direction, a partially filled LL consists
effectively of a set of two “internal edge channels” parallel to
the y axis per period a. Neighboring channels flow in opposite directions so that disorder scattering randomizes the direction of the motion in the y direction after time ␶s*. The
factor Dyy = v2y ␶s* can thus be interpreted as the diffusion constant of the resulting diffusion process. We note that unlike
␴xx共⑀兲, the conductivity ␴yy共⑀兲 remains finite at the band
edge. The anisotropy ␴yy / ␴xx of the dark conductivity is thus
of order 共vy␶s* / Rc兲2共kF␰兲2, where both factors are larger than
unity. The dark conductivity ␴yy depends on the modulationinduced LL broadening as ␴xx ⬃ V2n, so that the Weiss oscillations in ␴yy are phase shifted by ␲ relative to the oscillations in ␴xx, in agreement with standard results.18
The dc electric field also leads to heating of the electron
*
where this becomes
system. The characteristic field Edc
relevant, can be estimated as follows. The dc electric
field causes a drift in the x direction with drift velocity
共Edc / B兲, changing the potential energy of the electron by
共V / a兲共Edc / B兲␶s*. This gives rise to a diffusion constant in
energy of D⑀ ⬃ 共V / a兲2共Edc / B兲2␶s*. Heating can be neglected
as long as the typical energy change 共D⑀␶in兲1/2 is small compared to V. This gives the condition
*
=
Edc Ⰶ Edc
共33兲
In terms of the distribution function, the current in the y
direction is given by
1
L xL y
jy = e
Ba
2␲冑␶in␶s*
共39兲
for the dc electric field. It is only for these electric fields that
the result 共37兲 is valid. A more formal derivation of this
result is given in Appendix B.
045329-5
PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
*
For larger dc electric fields Edc Ⰷ Edc
, the effect of heating
needs to be taken into account. Following the arguments
given in Appendix B, one expects that the conductivity is
suppressed by heating effects and behaves in magnitude as
冉 冊
E*
␴yy ⬃ dc
Edc
兩具n ± 1k⬘兩T±兩nk典兩2 ⯝
共40兲
The reason for this suppression is that heating reduces the k
dependence of the distribution function.
IV. DISPLACEMENT PHOTOCURRENT
A. t-matrix elements
The microwaves lead to additional contributions to the
transition matrix element between LL oscillator states which
enters into the current expression 共1兲. Direct microwave absorption or emission does not contribute to the current, because the microwaves do not transfer momentum to the electrons so that such processes are not associated with
displacements in real space. In addition, such processes occur only for ␻ = ␻c. On the other hand, disorder-assisted microwave absorption and emission is associated with displacements in real space of the order of Rc 共lB2 / ␰ for smooth
disorder兲. This process is allowed for microwave frequencies
away from ␻c. In this section, we compute the contribution
of this displacement mechanism to the photoconductivity
within our model.
The transition rate between LL oscillator states involves
the t matrix
T = 共U + ␾兲 + 共U + ␾兲G0共U + ␾兲 + ¯ ,
共41兲
具n ± 1k⬘兩T±兩nk典 = ±
冉 冊
eERc
关具n ± 1k⬘兩U兩n ± 1k典
4⌬␻
− 具nk⬘兩U兩nk典兴,
共43兲
G0,nk⬘共⑀nk兲 =
=
jphotoI
x
2␲e
L xL y
⫻关f
W̃共q兲.
兺n 兺 共k⬘ − k兲lB2 兩具n + 1k⬘兩T+兩nk典兩2
kk⬘
0
nk
−f
0
n+1k⬘兴␦共⑀nk
− ⑀n+1k⬘ + ␻兲.
共46兲
Here, we assume that the microwaves frequency ␻ ⬎ 0 is
such that it couples neighboring LLs.
In order not to complicate the calculations unnecessarily,
we will consider temperatures T Ⰷ V. In this regime, the temperature smearing is over an energy range large compared to
the LL width and the distribution function depends only on
the LL index n, but not on the momentum k. Inserting the
explicit expression 共45兲 for the t-matrix element, we obtain
1
1
= ⫿
,
⑀nk ± ␻ − ⑀n±1k
⌬␻
冉 冊兺
2␲lB2 e eERc
LxLy 4⌬␻
⫻
共44兲
Using the disorder matrix elements and neglecting corrections of order 1 / n, one finds
2
In this section, we compute the displacement contribution
to the photocurrent in the modulation direction. The current
in the x direction can now be computed in terms of Fermi’s
golden rule in the same manner as for the dark current in
Sec. III. In this way, one obtains
I
=
jphoto
x
1
1
= ±
.
⑀nk − ⑀nk⬘
⌬␻
q2lB2
q2lB2
− Ln
2
2
B. Displacement photocurrent jx along the modulation
direction
where we used
G0,n±1k共⑀nk ± ␻兲 =
2
d 2q
−q2lB
/2
2 ␦qy,k⬘−ke
共2␲兲
Thus in this approximation, the matrix elements are identical
for absorption and emission and depend only on the absolute
value of k − k⬘. It is worthwhile to point out that the difference between the two Laguerre polynomials reflects the fact
that disorder-assisted microwave absorption involves a coherent sum of two processes: In one process, a microwave
photon is first absorbed, resulting in a transition from the nth
to the 共n + 1兲th LL, with disorder subsequently inducing a
transition between states in the 共n + 1兲th LL. In the second
process, disorder first leads to a transition between states in
the nth LL with a subsequent absorption of a microwave
photon. The divergence of the matrix element for ⌬␻ → 0 is
an artefact of low-order perturbation theory in the disorder
potential U. In a more accurate treatment, this divergence
would be removed by disorder broadening.
共42兲
Note that T+ and T− contribute incoherently. Assuming that ␻
couples only neighboring LLs and that the microwaves are
linearly polarized in the x direction, the corresponding matrix
elements between LL oscillator states are
2
共45兲
where G0 denotes the retarded Green function of the unperturbed system. The dark conductivity, computed in the previous section, follows in the approximation T ⯝ U. Disorderassisted microwave absorption T+ and emission T− is given
by
T± = 关UG0␾± + ␾±G0U兴.
eERc
4⌬␻
⫻ Ln+1
2
e2Dyy␯* .
冉 冊冕
冋 冉 冊 冉 冊册
兺k
− Ln
冕
045329-6
n
关f n − f n+1兴
冋 冉 冊
q2lB2
2
d 2q
−q2lB
/2
q
e
L
y
n+1
2
共2␲兲2
冉 冊册
q2lB2
2
The k summation gives
2
2
W̃共q兲␦共⑀n+1k − ⑀nk+qy − ␻兲. 共47兲
PHYSICAL REVIEW B 71, 045329 共2005兲
MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
兺k ␦共⑀n+1k − ⑀nk+q
=
L xL y
2␲lB2 2␲Vn
y
− ␻兲
冋
1
˜
Qq
⌬␻
y
sin2
−
2
2Vn
冉 冊册
共48兲
2 1/2 ,
˜ = ⌬␻ − eEdcqylB2 . Thus, we obtain for
where we introduced ⌬␻
the current
I
=
jphoto
x
冉 冊兺
e eERc
2␲lB2 4⌬␻
⫻
冕
− Ln
2
关f n − f n+1兴
n
冋 冉 冊
q2lB2
2
d 2q
2 −q2lB
/2
q
l
e
L
y
n+1
2
共2␲兲2 B
冉 冊册
q2lB2
2
2
冋
W̃共q兲
冉 冊册
˜
Qqy
⌬␻
−
Vn sin
2
2Vn
2
FIG. 1. The functions A1共⌬␻ / 2VN兲 共full line兲 and
B1共⌬␻ / 2VN兲 / ln共VN / ⌬兲 共dashed line兲, describing the dependence of
the parallel photoconductivity on the microwave frequency, see
Eqs. 共52兲 and 共68兲.
2 1/2 ,
共49兲
where the integral is only over the region where the square
root in the denominator is real. It is useful to interpret the
various factors in this expression. It consists of a charge
density per LL e / 2␲lB2 , and a rate 共per LL兲 for jumps in the
x direction with lengths between qylB2 and 共qy + dqy兲lB2 , multiplied by the jump lengths qylB2 Finally, the expression is
integrated over all jump lengths qy and summed over all LLs.
The sum over LLs n is trivial and for white-noise
disorder, the integral over q can again be decoupled for
␭F Ⰶ a Ⰶ Rc. Expanding to linear order in the dc electric field
and using the integral
冕
冋 冉 冊 冉 冊册
q2lB2
q2lB2
2
d 2q
2 2 −q2lB
/2
共q
l
兲
e
L
−
L
y B
n+1
n
2
2
共2␲兲2
2
=
3n
,
␲
共50兲
we obtain the linear-response conductivity
photo I
␴ xx
= 关e2Dxx␯*兴
冉 冊
␶s* eERc 2
A1共⌬␻/2VN兲
␶tr* 4⌬␻
共51兲
K共冑1 − 共⌬␻/2VN兲2兲
冦冉
− ln共兩⌬␻兩/8VN兲,
冊
⯝ ␲
␣
1+
,
4
4
兩⌬␻/2VN兩 Ⰶ 1,
␣=1−
冉 冊
⌬␻
2VN
2
Ⰶ 1.
冧
.
共53兲
This implies that A1共⌬␻ / 2VN兲 remains finite for 兩⌬␻ / 2VN兩
→ 1 共see Fig. 2兲 and is proportional to 1 / ⌬␻ for small
兩⌬␻ / 2VN兩. The sign of the displacement photocurrent is
given by sgn共␻c − ␻兲, similar to previous work on disorderbroadenend LLs.8,11
The magnitude of the displacement contribution 共51兲 to
the photoconductivity can be understood as follows. The
bare rate of disorder-induced microwave absorption is
共1 / ␶s*兲共eERc / ⌬␻兲2, where the second factor is the dipole
coupling of the microwave field for Landau states divided by
the relevant energy denominator of the intermediate state.
Each of these scattering events is associated with a jump in
real space of the order of lB2 / ␰, resulting in an effective diffusion constant Dxx共eERc / ⌬␻兲2. An additional factor 共␶s* / ␶tr*兲
arises because of the partially destructive interference of the
with the function 共see Fig. 1兲
A1共x兲 = −
3 ⳵
K共冑1 − x2兲,
␲2 ⳵x
共52兲
expressed in terms of the complete elliptic function K. Note
that the first factor in Eq. 共51兲 is just the dark conductivity
␴xx 关see Eq. 共31兲兴 and that the result includes the case of
smooth disorder. 共Strictly speaking, the result for smooth disorder is valid only up to a numerical prefactor that depends
on the precise nature of the smooth disorder potential, see
Appendix A for details.兲 The behavior of this displacement
photocurrent for ⌬␻ Ⰶ 2VN and ⌬␻ ⬃ 2VN follows from the
asymptotic expressions
FIG. 2. Illustration of disorder-assisted microwave absorption
for ⌬␻ ⯝ 2VN. In the x direction, the Landau levels are modulated
by the periodic potential and tilted by the dc field. The square-root
singularity of the Landau level DOS 共13兲 at the band edge leads to
the singular behavior of the photocurrent for these detunings ⌬␻,
see Fig. 1.
045329-7
PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
two contributions which were discussed below Eq. 共45兲. This
interference leads to the difference of Laguerre polynomials
in Eq. 共49兲 which introduces an additional factor 共q / kF兲2 into
the integral. This factor is of order 1 / 共kF␰兲2 ⬃ 共␶s* / ␶tr*兲.
C. Displacement photocurrent jy perpendicular to the
modulation direction
The current in the y direction can also be computed in a
semiclassical approach. If a dc electric field is applied in the
y direction, the equipotential lines of energy E are “meander”
lines defined by E = Vn cos共Qx兲 − eEdcy. This gives
y=
1
关Vn cos共Qx兲 − E兴
eEdc
共54兲
with an average y value ȳ = −E / eEdc. Quantum mechanically,
we can think of these equipotential lines as states. This
relation as well as the calculation sketched in this section
is worked out more formally in Appendix C. Scattering
between such meander states that differ in energy by
⌬␻ = ␻c − ␻ 共ignoring the LL energy兲 involves jumping a distance ⌬␻ / eEdc in the y direction. Thus, we can again compute the current by Fermi’s golden rule. It is important to
observe that the direction of the jumps is fixed by the sign of
the energy difference. Below, we will comment on the limits
of validity of this approach.
The current expression involves the rate of jumps. If Edc
is sufficiently weak, the amplitude of the meander line is
very large compared to the scale Rc over which jumps occur.
On the scale of the jump, the meander lines are therefore
essentially indistinguishable from the equipotential lines in
the absence of the dc field. This allows us to employ the rate
of jumps which we obtained from the calculation of the current in the x direction. 关We have to set Edc = 0 in the formulas
obtained there, see Eq. 共49兲.兴 In this way, we obtain for the
displacement photocurrent in the y direction
I
=
jphoto
y
冉 冊兺
e eERc
2␲lB2 4⌬␻
⫻
⌬␻
eEdc
− Ln
冕
关f n − f n+1兴
n
冋 冉 冊
q2lB2
d2q −q2l2 /2
B
e
L
n+1
2
共2␲兲2
冉 冊册
q2lB2
2
2
2
W̃共q兲
Qqy
⌬␻
−
Vn sin2
2
2Vn
冋
冉 冊册
2 1/2 .
共55兲
As mentioned above, we derive this expression more formally in Appendix C. Computing the integral for white-noise
disorder, we obtain the result
I
=
jphoto
y
8⌬␻
共2␲兲
3
VNlB4 Edc
冉 冊
eERc
4⌬␻
2
1
K共冑1 − 共⌬␻/2VN兲2兲,
2␲␯␶
共56兲
valid for ␭F Ⰶ a Ⰶ Rc and T Ⰷ VN. Note that the current is not
linear in the applied dc electric field but rather diverges as as
1 / Edc. This anomalous behavior is associated with the fact
FIG. 3. The functions A2共⌬␻ / 2VN兲 共full line兲 and B2共⌬␻ / 2VN兲
共dashed line兲 describing the dependence of the perpendicular photoconductivity on the microwave frequency, see Eqs. 共58兲 and 共71兲.
that the length of the jumps diverges with decreasing dc electric field and that the direction of all jumps is the same, fixed
by the sign of ⌬␻. Rewriting the result in terms of a conductivity, we obtain
I
␴photo
= 关e2Dyy␯*兴
yy
冉
aB/␲冑␶s*␶tr*
Edc
冊冉 冊
2
eERc
4⌬␻
2
A2共⌬␻/2VN兲.
共57兲
Note that the first factor is just the dark conductivity ␴yy. The
derivation of the result for smooth disorder is sketched in
Appendix A. The function A2共x兲 is defined by
A2共x兲 = 2xK共冑1 − x2兲
共58兲
and plotted in Fig. 3. The sign of the photocurrent is given by
photoI
.
sgn共␻c − ␻兲, as in the case of ␴xx
photo
The magnitude of ␴ yy can be understood as follows.
Since all jumps are in the same direction, we estimate the
current density directly. Effectively one LL contributes
so that the relevant density of electrons is 1 / 2␲lB2 . The
step length is ⌬␻ / eEdc and the rate of jumps is given
by 共1 / ␶s*兲共eERc / ⌬␻兲2共␶s* / ␶tr*兲 where the first two factors
are the bare rate and the last factor again reflects the partially
destructive interference between the two contributions
to disorder-assisted microwave absorption. Thus, we
⬃ 共e / 2␲lB2 兲共⌬␻ / eEdc兲共1 / ␶s*兲
find a current of order jphotoI
y
2 * *
⫻共eERc / ⌬␻兲 共␶s / ␶tr兲, in agreement with Eq. 共57兲.
The limits of validity of this result are most naturally
discussed in terms of a semiclassical picture. Semiclassically,
the individual scattering events such as disorder-assisted microwave absorption leave the y coordinate of the electron
essentially unchanged 共to an accuracy of Rc兲. The full jump
by ⌬␻ / eEdc is realized only if the electron remains in the
meander state it scattered into for sufficiently long times to
explore its entire y range. Under the condition ␶s* Ⰶ ␶in, the
electron will diffuse on the meander line before equilibrating
by inelastic processes. The typical diffusion distance 冑Dyy␶in
in the y direction should be larger than the amplitude
045329-8
PHYSICAL REVIEW B 71, 045329 共2005兲
MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
VN / eEdc of the meander line. Thus, we find that the condition
for the validity of the expression 共57兲 is
0
␦ f nk = f nk − f nk
= ␶in 兺
兺 2␲兩具n⬘k⬘兩T␴兩nk典兩2关f n0⬘k⬘ − f nk0 兴
n⬘k⬘ ␴=±
0
− ⑀n⬘k⬘ + ␴␻兲.
⫻␦共⑀nk
0
Edc Ⰷ
*
Edc
=
aB
2␲冑␶*␶in
共59兲
.
Note that this is just the opposite of the range of validity of
the dark conductivity ␴yy computed in Eq. 共37兲.
For smaller dc electric fields, the jumps are no longer all
in the same direction and we can estimate the displacement
photoconductivity as follows. The disorder-induced microwave absorption excites the electrons to a meander 共equipotential兲 line which is shifted in the y direction by ⌬␻ / eEdc
relative to the initial state. For definiteness, assume that this
shift is in the positive y direction. Since quasiclassically, the
jump itself leaves the y coordinate of the electrons unchanged 共to an accuracy of Rc兲, the electrons will initially
populate only those parts of the excited meander line, which
are at least a distance ⌬␻ / eEdc from its top 共in the y direction兲. After the excitation, the electrons begin to diffuse on
the equipotential line due to disorder scattering, typically a
distance 冑Dyy␶in before they relax back. Thus, after time ␶in,
the population of the excited meander line will extend further
in the positive y direction by a distance 冑Dyy␶in, and the
average
positive
drift
per
electron
is
关冑Dyy␶in / 共VN / eEdc兲兴冑Dyy␶in. These arguments lead to a displacement photoconductivity of
I
␴ photo
⬃ 关e2Dyy␯*兴
yy
冉 冊冉 冊
eERc
⌬␻
2
␶in
␶tr*
As before, we restrict attention to temperatures T Ⰷ V so that
0
⯝ f 0n, independent of k. Inserting the explicit expression
f nk
共45兲 for the t-matrix element for white-noise disorder and
using the decoupling of the q integration for ␭F Ⰶ a Ⰶ Rc, we
obtain for the change in the distribution function
␦ f nk = 2
result matches with Eq.
note that even the linearresponse displacement photoconductivity involves the inelastic relaxation time ␶in.
V. THE EFFECT OF A NONEQUILIBRIUM ELECTRON
DISTRIBUTION ON THE PHOTOCURRENT
A. Distribution function
For nonzero inelastic relaxation time ␶in, the microwave
irradiation changes the electron distribution function, away
from the equilibrium Fermi-Dirac distribution. In this section, we consider the contribution to the photocurrent arising
from this change in f nk.
The microwave-induced change in the distribution function can be computed from the stationary kinetic equation in
the absence of the dc electric field. Elastic disorder scattering
contributes only when states of the same energy have different occupations. Since this is not the case for Edc = 0, we can
ignore elastic disorder scattering when computing the
microwave-induced change in f nk. Thus, the kinetic equation
reduces to a balance between the microwave-induced collision integral and inelastic relaxation which yields to linear
order in the microwave intensity
冉 冊兺
eERc
4⌬␻
␶in
− ␴⌬␻兲
2
␴=±
关f
0
n+␴
− f 0n兴
0
␶tr*共⑀nk
0
− ␴⌬␻兩兲.
⫻␪共V − 兩⑀nk
共62兲
0
⑀nk
.
Note that k enters this expression only through
Strictly
0
− ␴⌬␻ apspeaking, this expression breaks down when ⑀nk
proaches the band edge. In this limit, it is no longer sufficient
to treat the microwave field to linear order in the intensity.
Effectively, this divergence is cut off when ␦ f nk becomes of
order unity, i.e., for distances ⌬⑀ from the band edge satisfying
⌬⑀ Ⰶ V
冋冉 冊 冉 冊册
2
eERc
⌬␻
␶in
␶tr*
2
共63兲
.
Here, we used that the DOS has a square-root divergence at
the band edge. We also note that our linear approximation in
the microwave intensity breaks down completely beyond microwave intensities given by 共eERc / ⌬␻兲2共␶in / ␶tr*兲 ⬃ 1.
In the estimate
␦ f nk ⬃ 共1/␶s*兲共eERc/⌬␻兲2共␶s*/␶tr*兲␶in
共60兲
*
. Note that this
valid for Edc Ⰶ Edc
*
共57兲 for Edc = Edc. It is interesting to
共61兲
共64兲
for the magnitude of ␦ f nk, the first three factors are the rate
for disorder-assisted microwave absorption and emission.
The last factor ␶in represents the time interval during which
electrons are excited. The expression 共62兲 will form the basis
of our calculation of the distribution-function mechanism for
the photocurrent to which we now turn.
B. Distribution-function contribution to the photoconductivity
along the modulation direction
Going through the same steps as in the derivation of the
dark current in the x direction, one finds that Eq. 共29兲 for ␴xx
remains valid even for a nonequilibrium distribution func0
only. Thus, we
tion, as long as it depends on k through ⑀nk
find for the distribution-function contribution to the photoconductivity
photo II
␴ xx
=
1
e 2N
2␲lB2
L xL y
2␲␯␶
兺冉
nk
−
0
⳵␦ f共⑀nk
兲
0
⳵⑀nk
冊
␯*共⑀0k 兲. 共65兲
Inserting ␦ f nk from Eq. 共62兲, performing the sum over the LL
index n, and rewriting the sum over k as an energy integral
involving the density of states, we obtain our result
photo II
␴ xx
=4
冉 冊冉 冊
␶in
␶tr*
eERc
4⌬␻
2
关e2Dxx␯*兴B1共⌬␻/2VN兲 共66兲
for the distribution-function contribution to the photoconductivity. The function
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PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
B1共⌬␻/2VN兲 = −
⳵
⳵⌬␻
冕
VN−兩⌬␻兩
冋 冉
B2共x兲 = 8兩x兩 arcsin共1 − 2兩x兩兲 +
d⑀˜␯*共⑀ + 兩⌬␻兩兲关˜␯*共⑀兲兴2 .
−VN
册
冊
␲
− 8冑兩x兩 − 兩x兩2 sgn x.
2
共72兲
共67兲
is of order unity for ⌬␻ ⬃ VN. This result shows that for the
parallel photocurrent, the distribution-function contribution
is larger than the displacement contribution 共51兲 by a large
parameter ␶in / ␶s*. This result is consistent with earlier results
for disorder-broadened Landau levels.11,14
The integral 共67兲 has a logarithmic singularity at the
lower limit, which has the same origin as the divergence of
the dark conductivity ␴xx in Eq. 共31兲. Thus, the singularity is
cut off in the same way as for the dark conductivity. 关For
small ⌬␻, one may seemingly have a more serious singularity. However, we should remember that the DOS arising
from ␦ f never really becomes singular, see the discussion
above around Eq. 共63兲.兴 Since the logarithmic singularity
dominates the integral 共67兲, we can replace ⑀ by the lower
limit in the DOS ˜␯*共⑀ + 兩⌬␻兩兲. In this way, we obtain the
result 共see Fig. 1兲
B1共x兲 =
1 1 − 2兩x兩
VN
sgn x.
2 3/2 ln
16 共兩x兩 − 兩x兩 兲
⌬
共68兲
Here, ⌬ denotes an effective broadening in energy of the
band edge, either due to disorder or a finite dc electric
field, which cuts off the logarithmic singularity. Interestingly,
this shows that the sign of the photocurrent is given by
sgn共␻c − ␻兲 for 兩⌬␻兩 ⬍ VN only. For 兩⌬␻兩 = VN, we find additional sign changes which are associated with the singular
nature of the DOS at the Landau level edge, see Eq. 共13兲.
C. Distribution function contribution to the photocurrent
perpendicular to the modulation direction
To compute the distribution-function contribution to the
photoconductivity ␴yy, we note that Eq. 共35兲 remains valid
for nonequilibrium distribution functions which depend on k
0
, only. Thus, our starting point is
through ⑀nk
II
␴ photo
=−
yy
1
e2
共2␲␯␶兲
2␲
L xL y
兺
nk
冉 冊
0
⳵⑀nk
⳵k
2
1
␯
*
0
共⑀nk
兲
⳵␦ f nk
.
⳵⑀nk
共69兲
Inserting ␦ f nk from Eq. 共62兲 and rewriting the sum over nk as
an integral, we obtain the final result
␴
photo II
=
yy
冉 冊冉 冊
␶in
4 *
␶tr
eERc
4⌬␻
⳵
⳵⌬␻
冕
B2共x兲 ⯝
− 8冑兩x兩sgn x,
4冑1 − 兩x兩sgn x,
兩x兩 Ⰶ 1,
1 − 兩x兩 Ⰶ 1.
冎
共73兲
Interestingly, B2共x兲 has different signs in these two limits,
implying that the function B2共x兲 must have a node between
the arguments x = 0 and x = 1. This node is approximately at
兩⌬␻兩 ⬇ 2VN / ␲ 2 ⬇ 0.2V. As a result, the distribution-function
II
has the same
contribution to the photoconductivity ␴ photo
yy
photo I
only in the range 兩⌬␻兩 ⲏ 0.2VN. Again, we
sign as ␴ yy
associate this behavior with the anomalous behavior of the
DOS.
The magnitude is, apart from a function of ⌬␻ / VN, of
order ␴yy␦ f. Remarkably, in this case the magnitude is of the
same order as the displacement contribution in Eq. 共60兲. The
reason for this is that the displacement mechanism exhibits a
singular, non-Ohmic dependence on the dc electric field
which is cut off at small fields by inelastic processes only. A
more detailed analysis beyond this order of magnitude comparison would require an accurate calculation of the displacement contribution to the photocurrent in the linear-response
regime. Such a calculation is beyond the scope of this paper
and the result may in any case be sensitive to details of the
model for inelastic relaxation.
We remark that our approach can be extended to
higher harmonics ␻ ⬃ n␻c 共with n ⬎ 1兲 of the cyclotron resonance. In this case, one needs to include disorder matrix
elements coupling different Landau levels. We find that
parametrically, all photoconductivities considered here are
suppressed with the harmonic index n in the same manner.
Indeed, on a naive level one expects that higher harmonics
are suppressed relative to the cyclotron resonance by
共⌬␻ / n␻c兲2n ⬃ 共⌬␻ / ␻c兲2共1 / n兲. Note that one factor of n
arises from the fact that n Landau levels contribute for the
nth harmonic. However, it turns out that there are cancellations in the matrix elements, leading to an actual suppression
by 共⌬␻ / ␻c兲2n−3. We note that in particular, the displacement
and distribution-function contributions to the photoconductivity perpendicular to the modulation direction remain of the
same order even in this situation. Detailed results on higher
harmonics will be presented elsewhere.23
D. Weiss oscillations of the photocurrent
2
关e Dyy␯ 兴B2共⌬␻/2VN兲. 共70兲
2
*
The expression 共70兲 is given in terms of the function
B2共⌬␻/2VN兲 = −
再
Asymptotically, this function behaves as
VN
−VN+兩⌬␻兩
d⑀
1
˜␯*共⑀ − 兩⌬␻兩兲.
关˜␯*共⑀兲兴2
共71兲
This integral is elementary and we obtain 共see Fig. 3兲
The Weiss oscillations arising from the oscillatory behavior of the Vn as function of LL index n or magnetic field have
two effects on the photocurrent. First, they lead to a modulation of the amplitude of the photocurrent. This amplitude
modulation is similar to that of the dark conductivity as the
photoconductivity is proportional to the dark conductivity. A
difference may arise from the additional prefactor ␶in / ␶tr* entering the photocurrents, see Eqs. 共66兲 and 共70兲. Specifically,
if the ineleastic relaxation rate ␶in depends in a different way
on the LL DOS ␯* compared to the transport time ␶tr, there
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MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
may be a distinct difference between the Weiss oscillations in
the dark and the photoconductivity. Depending on whether
the Weiss oscillations in the photoconductivity are more or
less pronounced than those in the dark conductivity, this may
help or impede reaching negative conductivities and hence
observing the zero-resistance state.
A second effect of the Weiss oscillations is associated
with the modulation of the LL width Vn. The expressions for
the photocurrent 共66兲 and 共70兲 involve a factor which depends on the ratio of the detuning ⌬␻ and the LL width VN at
the Fermi energy. This implies that in the limit of wellseparated LLs considered here, the range in the detuning
over which there is a significant photocondictivity also oscillates with magnetic field or Fermi energy.
In the photoconductivity, the 1 / B-periodic Weiss oscillations are superimposed on the microwave-induced oscillations which are also periodic in 1 / B. The periods in 1 / B of
these oscillations are ea / mvF and e / m␻, respectively, which
can be of comparable magnitude.
VI. POLARIZATION DEPENDENCE OF THE
PHOTOCONDUCTIVITY
In this section, we discuss the dependence of the photoconductivity on the polarization of the microwave field E.
We begin by calculating the transition matrix elements for a
microwave field polarized in the y direction. In this case
␾=−
eE i␻t −i␻t
y共e + e 兲 = ␾+e−i␻t + ␾−ei␻t .
2
共74兲
The matrix elements of the operator ␾± are given by
具n ± mk⬘兩␾±兩nk典 = −
冉
1 ⳵
␦共k⬘ − k兲
i ⳵k
⫻e−共k − k⬘兲
⫿ ␦m,±1
2l2 /4
B
再
冊
␦m,0L0n
共k⬘ − k兲lB
冑2n
L1n
冉
冉
共k⬘ − k兲2lB2
2
共k⬘ − k兲2lB2
2
冊
冊冎
共75兲
for m 艌 0. Using
and
we obtain for the
transition matrix element T± the large-n result
L1n共0兲 = 1
具n ± 1k⬘兩T±兩nk典 = −
L1n共0兲 = n + 1,
冉 冊
1 eERc
关具n ± 1k⬘兩U兩n ± 1k典
i 4⌬␻
− 具nk⬘兩U兩nk典兴
共76兲
valid for ⌬␻ Ⰶ ␻c. This matrix element differs from the corresponding matrix element for polarization in the x direction
by a multiplicative prefactor of unit modulus. As the photoconductivity depends only on the modulus of the transition
matrix elements, we find that the photoconductivity is the
same for microwave fields linearly polarized in the x and y
directions. We also find that more generally, the photoconductivity remains unchanged for any linear polarization of
the microwaves.
We now turn to irradiation by circularly polarized microwave fields which are described by
␾ ␴± = −
eE
2 冑2
关共x ± iy兲e−i␻t + 共x ⫿ iy兲ei␻t兴.
共77兲
Combining the transition matrix elements for microwaves
linearly polarized in the x and y directions, one finds zero
photoconductivity for ␾␴+. In this case, the E vector rotates
opposite to the circular cyclotron motion of the electrons in
the magnetic field. For ␾␴−, both E and the cyclotron motion
rotate in the same direction and the photoconductivity is
double that for linearly polarized microwave fields. We note
that this dependence on the type of circularly polarized light
is specific to the cyclotron resonance ␻ ⬃ ␻c. For higher harmonics ␻ ⬃ n␻c 共n ⬎ 1兲 of the cyclotron resonance, no such
dependence exists.23
VII. CONCLUSIONS
We have computed the microwave-induced photocurrent
in the regime of high Landau level filling factors, in a model
in which the Landau levels are broadened into a band due to
a static periodic modulation. We assume that the static modulation is small compared to the spacing between LLs, but
large compared to the Landau-level broadening due to residual disorder. In this case, the eigenstates are still given to
leading order by the Landau level oscillator states in the
Landau gauge. The localization properties of these states allow us to compute the dark conductivity as well as the
microwave-induced photoconductivity using Fermi’s golden
rule. The Fermi’s golden rule expression for the current directly suggests that there are two distinct mechanisms contributing to the photocurrent, analogous to previous
results11,14 for disorder-broadenend Landau levels. 共i兲 The
displacement mechanism relies on the spatial displacements
associated with disorder-assisted microwave absorption and
emission. This contribution can be associated with an additional, microwave-induced contribution to the transition matrix element in the Fermi’s golden rule expression. 共ii兲 The
distribution-function mechanism by contrast, relies on the
microwave-induced change in the electronic distribution
function, again due to disorder-assisted microwave absorption and emission.
For the photocurrent parallel to the modulation direction,
we find that the distribution-function mechanism 关see Eq.
共66兲兴 dominates by a large factor ␶in / ␶s* over the displacement contribution 关see Eq. 共51兲兴, in agreement with earlier
results for disorder broadened Landau levels.11,14 The sign of
the photocurrent changes with the sign of the detuning
⌬␻ = ␻c − ␻ of the microwaves. For the dominant distribution
function mechanism, there are additional sign changes associated with the divergence of the density of states at the edge
of the Landau level. Remarkably, the situation is rather different for the transverse photocurrent perpendicular to the
modulation direction. In this case, we find that the displacement mechanism is in some sense singular with the result
that both contributions to the photocurrent 关see Eqs. 共57兲 and
共70兲兴 are of the same order. We find that our results remain
unchanged for any linear microwave polarization. For circular polarization, we find a nonzero photoconductivity only
when the microwave electric field rotates in the same direc-
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PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
tion as the cyclotron rotation of the electrons in the magnetic
field.
ACKNOWLEDGMENTS
We thank M.E. Raikh for illuminating discussions. L.I.G.
thanks the Free University of Berlin and FvO the Weizmann
Institut for hospitality 共supported by LSF and the Einstein
Center兲 while part of this work has been performed. This
work has been supported by NSF Grants Nos. DMR0237296 and EIA02-10736 共L.I.G.兲, the Institut Universitaire
de France 共F.W.J.H.兲, the DFG-Schwerpunkt Quanten-HallSysteme 共J.D. and F.v.O.兲, and the Junge Akademie 共F.v.O.兲.
APPENDIX A: SMOOTH DISORDER POTENTIALS
As opposed to a white-noise potential, the 共zeromagnetic-field兲 single particle time ␶s is different from the
transport time ␶tr for a smooth disorder potential. Specifically, the single-particle time can be expressed in terms of
the correlator W as
1
= 2␲
␶s
兺q
1
W̃共q兲具␦共⑀k − ⑀k+q兲典FS =
␲vF
冕
⬁
e−q
2l2 /4
B
Ln
冉 冊 冑
q2lB2
⯝
2
I0 =
1
2 ␲ 2R c
冕
⬁
0
0
dqW̃共q兲具␦共⑀nk
− ⑀nk⬘兲典k⬘ =
0
兺q 共1 − cos ␪q兲W̃共q兲具␦共⑀k − ⑀k+q兲典FS
1
␲ vF
冕
共A6兲
The same integral is involved in the computation of the dark
conductivity ␴yy.
The transport time involves the integral
I2 =
冕
冋 冉 冊册
q2lB2
d2q q2 −q2l2 /2
B
e
L
n
2
共2␲兲2 2kF2
冕
I2 =
J0 =
0
冋 冉 冊册
q2lB2
d2q −q2l2 /2
B
e
L
n
2
共2␲兲2
2
0
0
W̃共q兲␦共⑀nk
− ⑀nk+q
兲.
y
In the limit of large N, the Laguerre polynomial has oscillations on the q scale of ␭F / lB2 and falls off on the scale of
Rc / lB2 . The correlator falls off on the scale 1 / ␰ and finally, the
characteristic scale of the argument of the ␦ function is a / lB2 .
Thus, unlike for white-noise potential, it is now the correlator W̃共q兲 which cuts off the integral at large q. Under the
condition ␭F Ⰶ a Ⰶ lB2 / ␰, we can still factorize the q integration as
I0 =
冕
冋 冉 冊册
d q −q2l2 /2
e B Ln
共2␲兲2
q2lB2
2
y
共A8兲
冕
冋 冉 冊 冉 冊册
˜ 共q兲␦共⑀0 − ⑀0 兲.
⫻W
nk
nk+q
2
共A9兲
y
For smooth disorder, the difference of Laguerre polynomials
is suppressed relative to a single Laguerre polynomial. Using
⯝ qR共n兲
that qR共n+1兲
c
c + q / kF, we find that the difference effectively introduces a small factor q2 / kF2 into the integrand and
thus 共see the calculation for I2 above兲
J0 =
␯ *共 ⑀ 兲 1
.
␯ ␲␶tr
共A10兲
photoI
, we need to conFor the displacement photocurrent ␴xx
sider
J2 =
冕
冋 冉 冊
q2lB2
d2q 2 2 −q2l2 /2
B
共q
/2k
兲e
L
n+1
F
2
共2␲兲2
− Ln
2
0
W̃共q兲具␦共⑀nk
− ⑀nk⬘兲典k⬘ .
␯ *共 ⑀ 兲 1
.
␯ 2␲␶tr
q2lB2
q2lB2
d2q −q2l2 /2
B
e
L
−
L
n+1
n
2
2
共2␲兲2
共A3兲
2
0
0
W̃共q兲␦共⑀nk
− ⑀nk+q
兲,
The same integral appears in the calculation for the dark
conductivity ␴xx.
A similar integral appears in the calculation for the disI
, namely,
placement photocurrent ␴ photo
yy
共A2兲
where ␪q denotes the scattering angle. Note that
␶tr / ␶s ⬃ 共kF␰兲2.
We start by considering the elastic scattering times for
smooth disorder. For ␶s*, we need to reconsider the q integration in Eq. 共19兲,
I0 =
2
共A7兲
⬁
dq共q2/2kF2 兲W̃共q兲,
␯ *共 ⑀ 兲 1
.
␯ 2␲␶s
where we used that 1 − cos ␪q ⯝ q2 / 2kF2 for q Ⰶ kF. An analogous analysis as for I0 above gives the result
dqW̃共q兲,
0
where the average is over the Fermi surface and ⑀k denotes
the zero-field dispersion. Likewise, the transport time can be
expressed as
=
共A5兲
in the remaining integral. 共Strictly speaking, we need a
slightly more accurate approximation. However, this changes
only the argument of the cosine24 which does not affect the
results.兲 This yields
共A1兲
1
= 2␲
␶tr
2
cos共qRc − ␲/4兲
␲qRc
冉 冊册
q2lB2
2
2
0
0
W̃共q兲␦共⑀nk
− ⑀nk+q
兲.
y
共A11兲
0
共A4兲
Since the integral is now cut off at large q by W, it is sufficient to employ the semiclassical equation
Again, the difference of Laguerre polynomials introduces a
small factor q2 / kF2 into the integrand. The resulting integral
can no longer be related directly to either ␶tr or ␶s. However,
noting that every factor 共q / kF兲2 reduces the integral by a
factor of order 1 / 共kF␰兲2, we can estimate
045329-12
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MICROWAVE PHOTOCONDUCTIVITY OF TWO-…
J2 ⬃
1 ␶s*
1
␯ *共 ⑀ 兲 1
⬃
.
␯ ␲␶tr 共kF␰兲2 ␲␶tr* ␶tr*
共A12兲
for the antisymmetric part as 共⳵␣ / ⳵t兲dis = −␣nk / ␶s*共⑀nk兲. Inserting the second of these equations into the first, we obtain
The precise numerical prefactor depends on the detailed nature of the smooth potential.
APPENDIX B: DISTRIBUTION FUNCTION FOR LARGE
dc ELECTRIC FIELDS
*
For large dc electric fields, Edc Ⰷ Edc
, Joule heating effects
become important. In this appendix, we study the distribution function in this limit. We start by decomposing the distribution function f nk into symmetric and antisymmetric parts
␴nk and ␣nk under k → −k, i.e.,
f nk = ␴nk + ␣nk .
共B1兲
共Recall that the electron dispersion is symmetric under this
transformation.兲 Specifically, we can write ␴nk = 共1 / 2兲关f nk
+ f n−k兴 and ␣nk = 共1 / 2兲关f nk − f n−k兴. Inserting this decomposition into the kinetic equation in the presence of a dc electric
field in the y direction 共and without microwaves兲, we obtain
the two equations
− eEdc
⳵␣nk ␴nk −
=
⳵k
␶in
− eEdc
0
␴nk
,
⳵␴nk
␣nk
.
= *
⳵k
␶s 共⑀nk兲
共B2兲
Here, we have used the inequality ␶in Ⰷ ␶s* and the fact that
the disorder collision integral vanishes for the symmetric
part ␴nk. In addition, we have rewritten the collision integral
f nn⬘共k − k⬘兲 =
冦
冉 冊
冉 冊
2nn!
1/2
2 n⬘n ⬘!
2 n⬘n ⬘!
2nn!
共eEdc兲2␶s*共⑀nk兲
APPENDIX C: EXPLICIT CALCULATION OF
DISPLACEMENT PHOTOCURRENT jy
In this appendix, we derive the displacement photocurrent
in the y direction more formally. In order to derive the quantum version of the meandering equipotential lines, we consider the Schrödinger equation, including the dc electric field
in the y direction, in LL representation
0
␦nn⬘␦kk⬘ − eEdc共− i兲
具nk兩H0兩n⬘k⬘典 = ⑀nk
2
n−n
k⬘兲n−n⬘e−共k − k⬘兲 /4Ln⬘ ⬘
冉
冉
共k − k⬘兲2
2
共k − k⬘兲2
2
冊
冊
if n⬘ 艌 n,
if n 艌 n⬘ .
再冕
冧
0
k
␺nk = ␺0n exp i
共C3兲
with
兩nE典 =
兺k ␺nk兩nk典.
dk⬘
E − ⑀nk⬘
eEdc
冎
.
共C5兲
To count the number of such states, we assume periodic
boundary conditions in the x direction ␺nk+Lx/l2 = ␺nk, and
B
thus 共with l 苸 Z兲
El =
共C4兲
This is readily solved and gives the quasiclassical meander
states
共C2兲
.
0
⳵
␺nk = E␺nk
⳵k
⳵
␦kk f nn⬘共k − k⬘兲
⳵k ⬘
with
Neglecting LL mixing, we find the Schrödinger equation in
the quasiclassical limit
0
⑀nk
␺nk + eEdc共− i兲
冉 冊
共C1兲
2
共k −
共B3兲
Note that this equation reproduces the estimate of the char*
. This follows immediately from
acteristic electric field Edc
the fact that the characteristic scale of the k dependence is
a / lB2 .
0
*
, we therefore find ␴nk ⯝ ␴nk
. This is the
For Edc Ⰶ Edc
starting point of the calculation leading to the expression
共37兲 for the dark conductivity ␴yy. In the opposite limit
*
, we write ␴nk = ¯␴n + ␦␴nk, where ¯␴n is the average of
Edc Ⰷ Edc
␴nk over k. Note that ␣nk which determines the current and
hence the conductivity is directly related to ␦␴nk. Then, Eq.
0
*
/ Edc兲2␴nk
. As a re共B3兲 shows that in magnitude ␦␴nk ⬃ 共Edc
sult, we expect that heating reduces the dark conductivity
according to Eq. 共40兲 in Sec. III.
共k − k⬘兲n⬘−ne−共k − k⬘兲 /4Lnn⬘−n
1/2
0
⳵2␴nk ␴nk − ␴nk
=
.
⳵k2
␶in
2␲leEdclB2
,
Lx
共C6兲
where El is measured relative to the LL energy. As the energies El fall into the range eEdcLy, the total number of states is
LxLy / 2␲lB2 , in agreement with the LL degeneracy. Requiring
045329-13
PHYSICAL REVIEW B 71, 045329 共2005兲
DIETEL et al.
1 = 具nEl 兩 nEl典, we find the normalized meander states
冑
␺nk = 具nk兩nEl典 =
再冕
2␲lB2
exp i
L xL y
0
k
dk⬘
E − ⑀nk⬘
0
eEdc
冎
jphotoI
=
y
. 共C7兲
⫻
To verify that these states are indeed the meander states, we
evaluate the expectation value of y = i⳵ / ⳵k, and find
具nEl兩y兩nEl典 =
2␲lB2
L xL y
兺k
冉
−
0
Enl − ⑀nk
eEdc
冊
=−
e
LxLy nn
El
eEdc
⬘
l l⬘
␻
兩␥nE →n⬘E 兩2 = 2␲兩具n⬘El⬘兩T␴兩nEl典兩2 .
共C9兲
e ⌬␻
兩␥␻
兩2␦共El − El⬘ + ⌬␻兲.
LxLy eEdc El,El NEl→N+1El⬘
⬘
共C11兲
The relevant transition matrix element is
l⬘
l
冉 冊
eERc
4⌬␻
2
兩具NEl兩U兩NEl⬘典
− 具N + 1El兩U兩N + 1El⬘典兩2 .
冏
共E 兲* 共E ⬘兲 iq 共k+q /2兲
␺k+q
␺k e
兺
兺
E ,E
k
l l⬘
=
兩具NEl兩U兩NEl⬘典 − 具N + 1El兩U兩N + 1El⬘典兩2
2
冕
d 2q
共2␲兲2
冏兺
k
冏␦
2
共El − El⬘ + ⌬␻兲.
x
y
y
Ly
eEdc
冏冕
冏␦
2
共El − El⬘ + ⌬␻兲
冏
dk i共V/eE 兲兰k+qydk˜ cos共Qk˜兲+i⌬␻k/eE +iq 共k+q /2兲 2
dc k
dc
x
y
e
.
2␲
The k sum can be turned into an integral which in the limit
Edc → 0 can be evaluated in the stationary-phase approximation. This gives the result
冏
共E 兲* 共E ⬘兲 iq 共k+q /2兲
␺k e
兺 兺k ␺k+q
E ,E
l l⬘
=
l
l
=
jphotoI
y
L xL y
共2␲兲2VlB2
冏
Inserting this into the expression for the current, we obtain
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y
冋
冏␦
2
共El − El⬘ + ⌬␻兲
1
Qq
⌬␻
y
−
sin2
2
2V
冉 冊册
冉 冊
eERc
⌬␻
2
2␲VlBEdc 4⌬␻
− LN共q2/2兲兴2
共C13兲
x
y
2 1/2 .
共C16兲
Note that Edc drops out of this expression. This can be interpreted as follows. The length over which the electron can
jump between meander lines is proportional to 1 / Edc. On the
other hand, the electron density along the meander line is
proportional to Edc. Thus, the overall probability to jump is
independent of the dc electric field. In this way, we finally
arrive at
共El兲* 共El⬘兲 iqx共k+qy/2兲 2
␺k+q
␺ e
y k
⫻e−q /2关LN+1共q2/2兲 − LN共q2/2兲兴2 .
1 R.
l
l
共C12兲
After disorder averaging, the matrix element becomes
1
=
2␲␯␶
y
Performing the energy sums yields
By carrying out the summation over n, n⬘ for ␻c Ⰷ T Ⰷ V we
get
␻
2
兩␥NE
→N+1E 兩 = 2␲
x
y
共C14兲
共C10兲
l⬘
兺
l
l
共C15兲
where the transition matrix element is given by
=
jphotoI
y
冏
共E 兲* 共E ⬘兲 iq 共k+q /2兲
␺k e
兺 兺k ␺k+q
E ,E
⫻
l
⫻关f共Enl兲 − f共En⬘l⬘兲兴␦共Enl − En⬘l⬘ − ␻兲,
l
1
2␲␯␶
共C8兲
兺 E兺,E 兩␥nE␻ →n⬘E ⬘兩2关ȳ共El⬘兲 − ȳ共El兲兴
l
冕
2
d2q −q2/2
e
关LN+1共q2/2兲 − LN共q2/2兲兴2
共2␲兲2
l l⬘
with Enl = ␻c共n + 1 / 2兲 + El, in agreement with the classical expectation.
Following the same arguments as for the displacement
photocurrent in the x direction, the current in the y direction
can now be expressed as
=
jphotoI
y
冉 冊
e 2␲⌬␻ eERc
LxLy eEdc 4⌬␻
冋
2
1
2␲␯␶
1
冕
d2q −q2/2
e
关LN+1共q2/2兲
共2␲兲2
冉 冊册
Qqy
⌬␻
−
sin
2
2V
2
2 1/2
共C17兲
for the displacement photocurrent in the y direction. Up to
the sums over Landau levels, this is just the expression for
the displacement photocurrent in the y direction in Eq. 共55兲.
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045329-15