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DC transport in two-dimensional electron systems under strong microwave illumination

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FLORIDA STATE UNIVERSITY
COLLEGE OF ART AND SCIENCES
DC TRANSPORT IN TWO-DIMENSIONAL ELECTRON SYSTEMS UNDER STRONG
MICROWAVE ILLUMINATION
By
SHANTANU CHAKRABORTY
A Dissertation submitted to the
Department of Physics
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Fall Semester, 2014
c 2014 Shantanu Chakraborty. All Rights Reserved.
Copyright UMI Number: 3681702
All rights reserved
INFORMATION TO ALL USERS
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and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 3681702
Published by ProQuest LLC (2015). Copyright in the Dissertation held by the Author.
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P.O. Box 1346
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Shantanu Chakraborty defended this dissertation on November 13, 2014.
The members of the supervisory committee were:
Lloyd Engel
Professor Co-Directing Dissertation
Irinel Chiorescu
Professor Co-Directing Dissertation
Naresh Dalal
University Representative
Jianming Cao
Committee Member
Nicholas Bonesteel
Committee Member
Alexander Volya
Committee Member
The Graduate School has verified and approved the above-named committee members, and certifies
that the dissertation has been approved in accordance with university requirements.
ii
ACKNOWLEDGMENTS
Over the last five years I have come across a lot people who have kindly helped to move forward
through the graduate school. First I would like to thank my research advisor Dr. Lloyd Engel for
his persistent efforts to guide me through this journey. Without his valuable insight and guidance
this work could not be accomplished so early. I would also like to thank Dr. Irinel Chiorescu and
Dr. Simon Capstick for their encouragement and valuable advice provided during the toughest
time of my graduate school life. I also want to acknowledge the excellent GaAs/AlGaAs samples
provided by Michael J. Manfra and Loren Pfieffer. I would like to specially thank Anthony Hatke
for introducing me to the area of research work presented in this thesis and his valuable advice
which was crucial to complete this work so promptly. I have learned a lot about these experimental
techniques employed in this thesis from him. I am grateful to Byoung Hee Moon for teaching me
how to operate the dilution refrigerator. I also want to acknowledge the support of my wife Pampa,
without her consistent support I would not be able to complete my research work this year. Finally,
I express my deepest gratitude to my parents for their emotional support and encouragement.
iii
TABLE OF CONTENTS
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1 Introduction
1.1 Drude Model . . . . . . . . . . . . . . . . .
1.2 Landau Levels and Energy Spectrum . . .
1.3 Shubnikov−de Haas Oscillation . . . . . . .
1.4 Microwave Induced Resistance Oscillations .
1.5 Hall-field Induced Resistance Oscillations .
1.6 Combining DC currents with Microwaves .
1.7 Thesis overview . . . . . . . . . . . . . . . .
1
2
3
5
7
9
11
13
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2 Origin of MIROs and Combined DC Current and Microwave Induced
tions
2.1 Theoretical framework of MIROs . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Displacement Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Inelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Properties of MIROs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Temperature dependence of MIROs . . . . . . . . . . . . . . . . . .
2.2.2 Power dependence of MIROs . . . . . . . . . . . . . . . . . . . . . .
2.3 Physical origin of combined DC current and Microwave Induced Oscillations
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Oscilla.
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15
15
16
18
19
19
21
23
3 Nonlinear Effects on Fractional MIROs
28
3.1 Combining DC currents with High Microwave Power . . . . . . . . . . . . . . . . . . 29
3.2 Combining DC currents with Microwaves below Cyclotron Resonance . . . . . . . . 42
4 Temperature and Power Dependence of Fractional MIROs
63
4.1 Temperature dependence of Fractional MIROs . . . . . . . . . . . . . . . . . . . . . 64
4.2 Power dependence of Fractional MIROs . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Conclusion and Outline for Future Research
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Outline for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The study of εac = 2 feature with Bichromatic Microwave Source . . . . . .
5.2.2 Non-linear DC transport study to search for the presence of Pondermotive
Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Experimental study of PIRO in a strong Microwave field . . . . . . . . . . .
iv
82
. 82
. 84
. 85
. 85
. 88
Appendix
A Experimental Details
89
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
v
LIST OF FIGURES
1.1
Landau level spectrum in B =0 and B 6=0 field for a 2DES. . . . . . . . . . . . . . . .
4
1.2
Magnetoresistance vs B in our GaAs/AlGaAs Hall bar with carrier density, n = 3.4
x1011 cm−2 and mobility µ = 6.2 x 106 cm2 /V s. . . . . . . . . . . . . . . . . . . . . .
5
Magnetoresistance vs εac , with microwave radiation of f = 31.5 GHz taken at T =
1.4 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
1.4
Typical Landau level transitions for resistivity minimum εdc = 2.5 (a) and resistivity
maximum εdc = 2(b) of HIRO between Hall-field tilted Landau levels. . . . . . . . . . 10
1.5
Typical HIRO trace obtained for I =14µA at T = 1.4 K on a 20 µm Hall bar.
1.6
Typical representitive trace of HIROs obtained at T= 1.4 K. . . . . . . . . . . . . . . 12
2.1
Simple picture of radiation-induced disorder-assisted current in displacement model,
adopted from Ref.[1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2
Oscillatory nature of DOS ν(ε) for overlapping Landau levels and the microwave
induced oscillations in the electron distribution function, f (). . . . . . . . . . . . . . 19
2.3
Landau level(tilted lines) diagrams for the combined microwave and dc excitation. . . 25
3.1
Differential resistivity r vs magnetic field B at different currents I = 0-24 µA in step
of 4 µA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2
Differential resistivity r vs εac around εac = 1 at I = 21µA. . . . . . . . . . . . . . . . 33
3.3
Overlap parameter aπ
3.4
Differential resistivity r vs εac around εac =
3.5
Differential resistivity r vs εac around εac = 1 at a dc bias of I = 21µA . . . . . . . . . 38
3.6
Differential resistivity r vs εac around εac = 1 at I = 21µA. . . . . . . . . . . . . . . . 39
3.7
Differential resistivity r vs εac around εac =
3.8
Differential resistivity r vs εdc at a fixed-B corresponding to εac =
3.9
1
Landau level diagrams at εac = for combined microwave and dc transitions involving
2
1-photon processes at +3dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
(N )
. . . . 11
vs N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vi
1
at I = 21µA. . . . . . . . . . . . . . . . 36
2
1
at I = 21µA. . . . . . . . . . . . . . . . 40
2
1
for f = 31 GHz. . 43
2
3.10
1
Additional Landau level diagrams at εac = for combined microwave and dc transi2
tions involving 1-photon processes at +3dB . . . . . . . . . . . . . . . . . . . . . . . . 45
3.11
1
Differential resistivity r vs εdc at a fixed-B corresponding to εac = for f = 31 GHz
2
for +6dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.12
1
Landau level diagrams at εac = for combined microwave and dc transitions involving
2
2-photon processes at +6dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.13
1
Additional Landau level diagrams at εac = for combined microwave and dc transi2
tions involving 2-photon processes at +6dB . . . . . . . . . . . . . . . . . . . . . . . . 49
3.14
1
Differential resistivity r vs εdc at a fixed-B corresponding to εac = for f = 31 GHz
2
for +10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.15
1
Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to εac =
3
for +3dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.16
1
Landau level diagrams at εac = for combined microwave and dc transitions involving
3
1-photon processes at + 3dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.17
1
Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to εac =
3
for +6dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.18
1
Landau level diagrams at εac = for combined microwave and dc transitions involving
3
1-photon and 2-photon processes at 6dB. . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.19
1
Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to εac =
3
for +10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.20
Landau level diagrams at (εac ,εdc )= (1/3,5/3) for combined microwave and dc transitions involving 1-photon and 2-photon processes. . . . . . . . . . . . . . . . . . . . . 57
3.21
1
Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to εac =
4
for +13dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.22
Relevant Landau Level diagrams for εac =
4.1
Resistivity ρxx vs magnetic field B at different temperature for EAC = 36.35 V/m . . . 66
4.2
Resistivity ρxx vs magnetic field B at different temperature for EAC = 119.6 V/m . . . 68
4.3
Resistivity ρxx vs magnetic field B at different temperature for EAC = 146 V/m . . . . 69
1
at +13dB . . . . . . . . . . . . . . . . . . 60
4
vii
4.4
Natural log of the normalized fractional MIRO amplitude, ln(
εac =
4.5
4.6
1
maxima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2
Natural log of the normalized fractional MIRO amplitude, ln(
εac =
δρ
) vs T 2 evaluated at
εac
1
maxima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3
Natural log of the normalized fractional MIRO amplitude, ln(
εac =
δρ
) vs T 2 evaluated at
εac
δρ
) vs T 2 evaluated at
εac
1
maxima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4
4.7
Extracted exponent α vs εac plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.8
Resistivity ρxx vs magnetic field B at different microwave electric field EAC . . . . . . 75
4.9
Overlap parameter aπ
4.10
Extraction of the MIRO amplitude from ρxx vs B trace, shown for εac = 1 and
4.11
1 1
1
MIRO amplitude (A) measured at εac = 1 , , and versus the microwave electric
2 3
4
field EAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
(N )
vs Pω for differnt number of participating photons N . . . . . 77
1
. . . 79
2
5.1
A representative spatial profile of the derivative of squared microwave field, dEx2 /dx
in the plane of the 2DES, in absence of 2DES. . . . . . . . . . . . . . . . . . . . . . . 87
A.1
A schematic diagram of the microwave circuit and the Hall bar used in the experiment. 90
A.2
A schematic diagram of the crosssection views of the CPW and the eletric field lines
inside the slot of the CPW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
viii
ABSTRACT
At low temperature (T ) and weak magnetic field (B), two dimensional electron systems (2DES) can
exhibit strong 1/B-periodic resistance oscillations on application of sufficiently strong microwave
radiation. These oscillations are known as microwave induced resistance oscillations (MIROs),
MIROs appearing near cyclotron resonance (CR) and its harmonics involve single photon processes
and are called integer MIROs while the oscillations near CR subharmonics require multiphoton
processes and are called fractional MIROs. Similar strong 1/B periodic resistance oscillations
can occur due to strong dc current, and are known as Hall-field resistance oscillations (HIROs).
Oscillations also occur for a combination of microwave radiation and strong dc current. In one
prominent theory of MIROs, known as the displacement model , electrons make impurity-assisted
transitions into higher or lower Landau levels by absorbing or emitting one or more (N ) photons.
In the presence of combined strong dc current and microwave radiation, electrons make transitions
between Landau levels by absorbing or emitting photons followed by a space transition along the
applied dc bias. The object of the dissertation is to explore how the different resistance oscillations
are affected by strong microwave radiation when multiphoton processes are relevant. We used a
coplanar waveguide (CPW) structure deposited on the sample, as opposed to simply placing the
sample near the termination of a waveguide as is more the usual practice in this field. The CPW
allows us to estimate the AC electric field (EAC ) at the sample. In much of the work presented in
this thesis we find that higher N processes supersede the competing lower N processes as microwave
power is increased. We show this in the presence and in the absence of a strong dc electric field.
ix
Finally, we look at the temperature evolution of fractional MIROs to compare the origin of the
fractional MIROs with that of integer MIROs.
x
CHAPTER 1
INTRODUCTION
The two-dimensional electron system (2DES) has been an important system in condensed matter
physics for the last three decades. Studies of 2DES at low temperature and in the presence of
magnetic field has led to the experimental discovery of remarkable effects such as integer quantum Hall effect(IQHE) [2] and fractional quantum Hall effect(FQHE) [3] both in high magnetic
field. The quantum Hall effect (QHE) is characterized by quantization of the Hall resistance, Rxy ,
accompanied by vanishing longitudinal resistance Rxx in certain ranges of magnetic fields.
In the past, a new type of oscillations were observed when a high mobility 2DES is exposed to
sufficiently strong microwave radiation in the presence of a weak perpendicular magnetic field, B,
termed microwave induced resistance oscillations (MIROs)[4] [5] [6]. In ultra-high mobility samples
(µ > 107 cm2 /Vs) under sufficiently strong microwave radiation the minima of these MIROs develop into zero-resistance states (ZRS) [6] [7]. Experimental discovery of MIRO and ZRS attracted
a great deal of attention to the field of nonequilibrium physics in high Landau levels (LL). Following
the discovery of MIRO, other closely related phenomena were also observed including Hall-field induced resistance oscillations (HIROs) [8][9], phonon induced resistance oscillations (PIROs) [10][11]
and also the oscillations in differential resistivity of a 2DES subjected to simultaneous microwave
radiation and strong dc electric field [12][13][14]. The nonlinear response of magnetoresistance
oscillations under strong microwave radiation has not been carefully explored. In what follows
1
we present some basic models of dc transport, then review the phenomena of the nonequilibrium
resistance oscillations. Finally, we proceed with an overview of the work of this thesis.
1.1
Drude Model
In this section, we briefly discuss the semiclassical theories of transport and develop the resistivity and conductivity tensor. The Drude model provides the simplest theory to explain the
transport properties of electrons in a material. According to this semiclassical approach the motion
of the electron are assumed to be classical and the electrons get scattered off the impenetrable,
immobile ion cores during their motion. The treatment here follows ref. [15]. When a static electric
~ = (Ex , Ey) and a magnetic field B
~ = (Bz ) is applied, the equation of motion of the electron
field E
can be expressed as
m∗~v˙ +
m∗~v
~ + ~v × B)
~
= −e(E
τ
(1.1)
where ~v = (vx , vy ) is the velocity of electron, m∗ is the effective mass of an electron in the presence
of magnetic field and τ is the momentum-relaxation time also known as transport lifetime. In a
steady state, h~v˙ i = 0, since the electrons emerge after a collision in random directions. Using the
~ and ~j = −ne~v we can express the conductivity tensor, σ for a steady state as
relations ~j = σ E



σo
σxx σxy 
 1
σ=
=

2
1 + (ωc τ )
ωc τ
σyx σyy
where σo =

−ωc τ 

1
(1.2)
ne2 τ
eB
is the zero field conductivity and ωc = ∗ is the cyclotron resonance (CR)
∗
m
m
frequency. The resistivity tensor can be obtained by a simple inversion of the conductivity tensor
2
and the tensor components ρxx and ρxy can be represented by
ρxx =
σxx
1
=
2
+ σxy
σo
(1.3)
σxy
B
=
2 + σ2
σxx
ne
xy
(1.4)
2
σxx
and
ρxy =
However in the high field limit σxy σxx , so σxx ∼
1.2
ρxx
∝ ρxx .
ρ2xy
Landau Levels and Energy Spectrum
In this section we follow ref. [15]. In the simplest case of perpendicularly applied B to 2DES,
the electrons experience Lorentz force and are forced into circular motion confined to the x-y plane.
~ = B k̂ using Landau gauge, the vector potential becomes A
~ = −Bxĵ and
For a perpendicular B
the Hamiltonian can be written as
1
2
~
(~
p − eA) + V (z) Ψ(x, y) = εΨ(x, y)
2m∗
(1.5)
We can safely neglect the z-direction, since the electrons are confined in the potential of the GaAs
quantum well V (z) and only occupy the lowest energy level (subband). Assuming the wavefunction
Ψn (x, y) = Φ(x)eiky y as a solution to the above Hamiltonian, these energy levels (εn ) form a discrete
series εn = ~ωc n +
1
2
, where n= 0,1,2.. separated by the cyclotron energy gap, ~ωc =
known as Landau Levels. These energy levels are highly degenerate and nφ =
~eB
also
m∗
2eB
is the number
h
of states per unit area for each Landau level. The number of occupied Landau levels below the
3
Figure 1.1: Landau level spectrum in B =0 and B 6=0 field for a 2DES.
Fermi energy is referred to as the Landau level filling factor (ν), defined as
ν=
ne
ne h
=
nφ
2eB
(1.6)
where ne is the density of carriers. The energy spectrum of the 2DES in a perpendicular magnetic
field B extended by the Zeeman term and the subband (z-direction) energy is
1
εi,n,s = εi + ~ωc n +
+ sg ∗ µB B;
2
where εi is the subband energy, µB =
1
s = ± , n = 0, 1, 2..
2
(1.7)
e~
is the Bohr magneton and g ∗ is the effective g-factor(
2me
≈ -0.44 for bulk GaAs). In GaAs the Zeeman term is much less than the ~ωc so we neglect it in
the rest of this thesis. Likewise we will consider 2DES occupying only the lowest subband so we
can neglect εi as well.
4
Figure 1.2: Magnetoresistance vs B in our GaAs/AlGaAs Hall bar with carrier density,
n = 3.4 x1011 cm−2 and mobility µ = 6.2 x 106 cm2 /V s.
In a disorder-free zero-temperature situation, the Landau levels are delta functions with zero
width,Γ=0, however in real samples they are broadened due to scattering or disorder effects and
temperature. Inclusion of any kind of scattering (electron-electron, electron-phonon, electronimpurity etc) will broaden the Landau levels and the density of states (DOS), D(ε), will broaden
from δ-functions to peaks with finite width as shown in Fig. 1.1. Using the energy-time uncertainty
relation this broadening of Landau levels can be expressed in terms of a finite quantum scattering
lifetime, τq , as Γ = ~/τq .
1.3
Shubnikov−de Haas Oscillation
In the absence of the magnetic field, B, DOS is constant: D(ε)=
m∗
, (Fig. 1.1). In the
2π~2
presence of a small perpendicular B the DOS is no longer constant and overlapping Landau levels
5
start to form. The treatment in this section follows Ref [16]. The density of states in this region
has the form, D(ε) = 1 − 2λ cos(
2πε
), where λ = exp(− ωcπτq ) is the Dingle factor. As B increases,
~ωc
the separation between the Landau levels increases and also the Landau levels move relative to the
Fermi energy (Fig. 1.1). The conductivity depends on the carrier concentration and the scattering
probability, or the number of available states nearby in energy where the carriers can scatter into.
A minimum in the longitudinal resistivity is expected at B where the Fermi energy lies between two
Landau levels. As the magnetic field is increased further, each time a Landau level passes through
the Fermi energy of the system, the longitudinal resistivity peaks. This oscillation of longitudinal
resistivity with magnetic field is known as Shubnikov-de Haas oscillation (SdHO), and is periodic
in inverse magnetic field, 1/B. The conditions to see SdH oscillations are
~ωc > kB T
and
ωc τq > 1,
(1.8)
where τq is the single particle quantum scattering time which is related to the Landau level width.
Theoretically, the amplitude of these oscillations can be expressed in the following form [17]
δρ = 4ρ0 DT λ cos(2πν), DT =
XT
,
sinh(XT )
(1.9)
where XT = 2π 2 kB T /~ωc and T is the temperature.
Fig. 1.2 shows a typical magnetoresistance trace taken at 1.4 K. Here we observe the onset of
SdHO’s for |B| > 0.3T. We can extract the density of carriers (n) in the 2DES from the period of
the SdH oscillations by
n=
∆νe
1
/(∆ ),
h
B
6
(1.10)
where ∆ν=2 for each successive peak/trough due to spin degeneracy. We can also calculate the
mobility, µ, of the 2DES using the zero field resistivity, ρ0 , as
µ=
1.4
1
.
neρ0
(1.11)
Microwave Induced Resistance Oscillations
When a high mobility (µ > 106 cm2 /Vs ) 2DES is exposed to sufficiently strong microwave
radiation (usually in the 10-100 GHz range) in the presence of a weak perpendicular magnetic field
it exhibits photoresistance oscillations termed microwave induced resistance oscillations (MIROs)[4]
[5] [6]. These oscillations in longitudinal resistance are periodic in the inverse magnetic field,
1
,
B
and occur at filling factor ν > 40. These extrema appear at particular values of the ratio of microwave radiation frequency to the cyclotron frequency:
εac =
ω
,
ωc
(1.12)
where ω = 2πf and f is the applied microwave frequency. These extrema are found to be symmetrically offset from the harmonics of cyclotron resonance by
ε±
ac = k ∓ φac ,
(1.13)
−
where, k = 1,2,3,... , ε+
ac denotes a MIRO maximum, εac denotes a MIRO minimum and φac is
called the MIRO phase, which tends to 1/4 for εac > 2 oscillations (εac ) and has been found to be
power dependent [18] [19] ,[20], [21].
7
Figure 1.3: Magnetoresistance vs εac , with microwave radiation of f = 31.5 GHz taken
at T = 1.4 K.
Such resistance oscillations are also found around fractional values of εac [18] [22] [23]. These
fractional MIROs are explained by multiphoton processes in which a single electron absorbs m
multiple photons and jumps across k Landau level spacings giving rise to MIRO at values of
εac =
k
,
m
(1.14)
where m and k are integers. These fractional MIROs occur at increasing microwave power and the
most easily observed are the εac =
1
series [18] [22] [23][24][25]. Fig. 1.3 shows the magnetoresism
tance in presence of microwave radiation of f = 30.5 GHz at T = 1.4 K as a function of εac . Here
−
ε+
ac and εac appear symmetrically about εac = 1 and εac = 2. Likewise a MIRO maximum and
1
1
minimum appear symmetrically about εac = , and εac = .
2
3
8
1.5
Hall-field Induced Resistance Oscillations
When a 2DES with sufficiently high mobility is subjected to a sufficiently strong fixed dc bias
current, Idc , and varying magnetic field, in the absence of microwave radiation, the differential
resistivity, r =
dVdc
dIdc ,
displays oscillations periodic in
1
, known as Hall-field induced resistance
B
oscillations (HIROs)[8][9]. These oscillations stem from impurity-mediated transitions between
Landau levels tilted by the Hall field and are controlled by the oscillation parameter
εdc =
Here 2Rc is the cyclotron diameter and ∆Y =
2Rc
.
∆Y
(1.15)
~ωc
is the spatial separation of the Hall-field tilted
eE
Landau levels where E is the Hall electric field [26]. Fig 1.4 shows this process with energy is on
the left axis and real space on the bottom axis. The thick tilted lines represent the Landau levels
tilted due to the Hall field. For the HIRO resistivity minimum the electron transition from the
filled Landau level just below the Fermi energy terminates in the gap between Landau levels, while
the electron transition for a HIRO resistivity maximum terminates at the center of a Landau level.
HIROs occur by virtue of a selection rule which favors electron backscattering, equivalent to the
transition of the electron guiding center by a maximum possible distance. In momentum space such
transition is associated to the largest momentum transfer ∆k ≈2kF , where 2kF is the Fermi wave
vector of the 2DES at zero magnetic field. This selection rule in momentum space can be translated
to real space as, ∆Y = 2Rc . Experimentally[8] it was observed that the HIRO resistivity maximum
occurs whenever the following condition is satisfied: l∆Y = γRc , where l is an integer and γ '2 is
9
Figure 1.4: Typical Landau level transitions for resistivity minimum εdc = 2.5 (a) and
resistivity maximum εdc = 2(b) of HIRO between Hall-field tilted Landau levels.
a experimental fitting parameter which compensates for non-uniform electric field across the Hall
bar. We can rewrite Eq 1.15 in terms of B and Idc ,
r
εdc = γ
2π Idc m∗
,
ne e2 wB
where we can immediately observe εdc depends on the ratio
(1.16)
I
.
B
Empirically, HIROs can be viewed as similar to MIROs with the microwave frequency is sub−
stituted by dc bias. Here the maxima and minima are found at εdc =ε+
dc and εdc = εdc respectively,
where
ε+
dc = m
and
1
ε−
dc = m + 2 .
(1.17)
and m is an integer. Landau level diagrams for a HIRO minimum, εdc = 2.5 and maximum εdc = 2
are shown in Fig. 1.4(a) and (b), respectively. Figure 1.5 shows the same HIRO trace for fixed I=
14 µA vs B, where ε+
dc =1,2,3 and 4 maxima are marked with their respective peaks. Figure 1.6
10
Figure 1.5: Typical HIRO trace obtained for I =14µA at T = 1.4 K on a 20 µm Hall bar.
shows a typical HIRO trace obtained by sweeping B for fixed I= 14 µA vs εdc from our w = 20 µm
Hall bar. As in Eq. 1.17 HIRO maxima and minima occur at εdc equals integers and half-integers,
respectively. Figure 1.5 and 1.6 also shows the rapid decrease of the amplitude of the differential
resistivity oscillation with decreasing B. In order to resolve higher order HIROs it is convenient to
perform I-sweep at a fixed B instead of performing B-sweep at a fixed I.
1.6
Combining DC currents with Microwaves
Experimental studies of MIRO [4] [5] [6] and HIRO [8][9] discussed in the earlier sections show
that microwave and dc induced effects exhibit several similar features. MIRO and HIRO both are
periodic in 1/B and both require a high mobility 2DES. In the separated Landau level regime(ωc τq >
11
Figure 1.6: Typical representitive trace of HIROs obtained at T= 1.4 K.
π/2), both MIRO and HIRO produce strong suppression in the resistance for small εac [22] [23][27]
and εdc [28] respectively.
Later experimental studies [12][13][14] of 2DES under simultaneous microwave radiation and dc
current produce regular oscillations in differential resistivity vs B or Idc . In the low microwave power
limit, when a 2DES is subjected to both microwave radiation and strong dc field its differential
resistivity can be represented by a simple combination of εac ± εdc parameters. In that limit, the
microwave photoresistivity is assumed to be limited to single photon processes. Maxima of these
oscillations occur when
εac + εdc ' n
(1.18)
1
εac − εdc ' m − ,
2
(1.19)
12
where n and m are both positive integers.
1.7
Thesis overview
The organization of this thesis is as follows.
In Chapter 2, we review the theoretical models which describe different nonequilibrium resistance oscillations. In addition, we discuss our initial motivation for studying the response of
2DES under strong microwave radiation to investigate the role of multiphoton processes both in
the presence and in the absence of dc electric field.
Chapter 3, presents data on the differential resistivity of 2DES with combined microwave and dc
bias excitation. The data were obtained by irradiating the 2DES with strong microwave radiation
which is neccesary to observe the effects of multiphoton processes. Our study shows additional
1
which have not been
2
1
observed before. The main topic of the chapter is the effect of microwave radiation at εac = ,
2
1
1
and
on differential resistivity vs Idc . We found that under simultaneous microwave and dc
3
4
small, high εdc -frequency oscillations in differential resistivity near the εac =
excitation the number of multiphoton processes participating in combined transitions at fractional
MIROs changes clearly with increasing microwave power.
Chapter 4 describes power dependence for Idc ∼ 0 on the integer and fractional MIROs for
a wide range of microwave power. We compare our observations with the models, based on the
displacement and inelastic mechanisms, which will be described in Chapter 2. Chapter 4 also
describes temperature dependence of fractional MIROs for a wide range of temperature at different
microwave powers.
13
The concluding Chapter 5 discusses the significance of the work and also discusses several
promising directions for future research where the work presented in this dissertation can be extended.
14
CHAPTER 2
ORIGIN OF MIROS AND COMBINED DC
CURRENT AND MICROWAVE INDUCED
OSCILLATIONS
In this chapter we will first introduce the theories which describe MIROs. We will also review
several key features of MIROs, which will help us to distinguish between the competing theories.
Later, we will introduce the theoretical framework developed to explain the differential resistivity
oscialltions in presence of microwave radiation and dc current bias.
2.1
Theoretical framework of MIROs
Theoretically, there are several approaches [1] [29] [30] [31] [32][33] [34] to explain MIROs. In
the following sections we are going to briefly discuss the two more widely accepted theories, the
displacement model [1][34] and the inelastic model [31] [32][33]. Several other less popular models
also exist which can explain the MIROs, includes the radiation driven electron orbit model [35][36]
and the non-parabolicity model [37]. However, we will introduce only the two more popularly
discussed models in the literature in the following sections.
15
2.1.1
Displacement Model
One of the more heavily studied theories to explain MIROs is based on radiation-induced
impurity-assisted scattering and is commonly known as the displacement model. The main idea
behind this model is that the dc electric field will spatially tilt the Landau levels in the direction
of the current, as shown in Fig. 2.1 in an energy-space diagram. Energy is on the left axis, real
space is on the bottom axis, V is the applied dc voltage and the thick tilted lines represents the
tilted Landau levels due to applied dc electric field in Fig 2.1(a) and 2.1(b). Similar resistance
oscillatory behavior was also predicted [29] [30] before for the separated Landau levels in presence
of strong dc current.
According to this model, an electron from the filled Landau level immediately below the Fermi
level absorbs a photon of an energy ~ω. This excited electron transitions vertically and gets
scattered horizontally off of disorder. Disorder provides the momentum needed by the electron to
make a horizontal displacement. The electron may scatter either way but prefers one direction
over the other due to available density of states. When a photoexcited electron makes a vertical
transition just below an empty Landau level (Fig. 2.1(a)), it will move to the left where the density
of states is higher. This transition is opposite to the direction of the electric field and increases the
longitudinal photoresistivity. When a photoexcited electron makes the vertical transition to just
above an empty Landau level (Fig. 2.1(b)), it scatters to the right where the density of states is
higher. Such final transition is along the electric field. Therefore the longitudinal resistivity will
decrease resulting in a MIRO minimum. As the magnetic field is swept the energy of the absorbed
photon crosses one or more Landau levels giving rise to the MIROs. In this theory, the MIROs are
16
Figure 2.1: Simple picture of radiation-induced disorder-assisted current in displacement
model, adopted from Ref.[1]. The electron transition due to the absorption of a microwave
photon, represented by the vertical arrow across the Landau levels. The electron then
scatters off an impurity, represented by a horizontal arrow. The electron transition to the
left (right) results in MIRO maximum (minimum).
explained considering only the disorder assisted scattering in the presence of MW radiation.
A systematic theoretical derivation [33] based on the displacement model in the overlapping
Landau level regime(ωc τq < π/2) assuming D(ε) ∝ 1 − 2λ cos(2πε/~ωc ), have found this oscillatory
correction to resistivity as
δρdis = −4πρ0 Pω0
3τqim
εac λ2 sin(2πεac ),
τtr
(2.1)
where τqim is the long-range impurity contribution to single particle lifetime τq , τtr is the transport
scattering time, ρ0 is the drude resistivity, λ = exp(− ωcπτq ) is the Dingle factor. and Pω0 is a
dimensionless parameter proportional to the microwave power, which can be defined as [38]
Pω0 =
(eEAC vF τem )2
,
2ef f ~2 ω 2
17
(2.2)
√
where EAC is the microwave electric field and τem = 20 ef f m∗ c/ne e2 is related to the coupling
of the microwave to the 2DES and ef f is the appropriate relative dielectric constant for widely
separated charges in the 2DES.
2.1.2
Inelastic Model
Another approach to describe the behavior of MIROs is based on electron scattering dominated
by remote impurities and is known as the inelastic model [31][33]. According to this model, MIROs
are governed by a modification in the electron distribution function in the presence of microwave
radiation. The electron distribution acquires an oscillatory part due to microwave radiation. Following Ref. [33], the modification to the Fermi distribution can be expressed as
f () = f0 () + fosc ()
ωc ∂fT
fosc () = λ
sin
2π ∂
2π
ωc
2πPω0 εac sin(2πεac )
,
1 + Pω0 sin2 (πεac )
(2.3)
(2.4)
where fT () is the smooth part of the Fermi distribution f0 () at bath temperature T . These
equations are derived for overlapping Landau level regime(ωc τq < π/2). For εac & 1 the correction
to the resistivity due to inelastic mechanism can be written as
δρin = −4πρ0 Pω0
τin
εac λ2 sin(2πεac ),
τtr
(2.5)
where τin ' EF T −2 , is the inelastic scattering time and EF is the Fermi energy. The dominant
contribution to MIRO in this model comes from the inelastic electron-electron scattering which is
strongly temperature dependent.
18
Figure 2.2: Oscillatory nature of DOS ν(ε) for overlapping Landau levels and the microwave induced oscillations in the electron distribution function, f (). Figure adopted
from Ref. [33].
2.2
2.2.1
Properties of MIROs
Temperature dependence of MIROs
In this section we will discuss the temperature dependence in the displacement and inelastic
models. While comparing their temperature dependence, we will also refer to the results of previous
experimental studies which motivate us to carry out the experimental studies presented later in
this thesis. One of the key differences between displacement model and inelastic model arises from
the temperature dependence of two different scattering times, τqim and τin respectively. The MIRO
photoresistivity due to these competing models from Eqn. 2.1 and 2.5 can be expressed in general
for overlapping Landau levels (ωc τq < π/2) as [39]:
δρi = −4πρ0 Pω0
τi
εac λ2 sin(2πεac ),
τtr
19
(2.6)
where τi = 3τqim for displacement model and τi = τin for inelastic model. Its important to recall
λ = exp(− ωcπτq ) is the Dingle factor and temperature dependence of MIRO photoresistivity can
arise from τq . Theoretical studies [31] [33] [40] suggest at low temperature (∼ 1 K) the contribution
from the inelastic mechanism should dominate, since τin τqim and τin ' EF T −2 . The inelastic
mechanism predicts a T −2 temperature dependence of MIRO photoresistivity while the displacement mechanism prediction is mostly temperature independent. The theoretical studies [31] [33]
[40] also predict that in the high microwave power limit, or in the presence of sufficiently strong dc
field, the displacement contribution can dominate even at low T .
Several experimental studies [6] [7] [41] also favor the inelastic mechanism over the displacement
mechanism to explain the observed decay of MIRO at higher temperature. On the contrary, one
experimental study [39] of temperature evolution of integer MIRO in the overlapping Landau level
regime (ωc τq < π/2 or λ 1 ) found the temperature dependence of MIRO photoresistivity to
be exponential quadratic-in-T . The study [39] suggested the different temperature dependence
might be related to the presence of sufficient short range disorder. This paper also shows that the
displacement contribution can be relevant and comparable to inelastic contribution at a much lower
temperature (∼1K) than expected from earlier studies [6] [7] [41].
The temperature dependence of integer MIRO photoresistivity is attributed to the electronelectron interaction correction of the quantum scattering time τq . τq can be written as
1
1
1
= im + ee ,
τq
τq
τq
(2.7)
where τqim and τqee are the long-range impurity and electron-electron contributions, respectively.
20
The temperature dependence of τq is introduced by the well known formula of the electron-electron
scattering rate [42] as
1
T2
,
'
κ
τqee
εF
(2.8)
where εF is the fermi energy and κ is a constant on the order of unity. Such T −2 temperature
dependence of τq was observed in earlier experiments involving tunneling spectroscopy of double
quantumwells [43] [44] and as well as in the studies of magneto-intersubband oscillations in quantum
wells [45].
Exponential quadratic-in-T dependence of integer MIRO photoresistivity orginates from τqee in
the Dingle factor. Ref [39] reported this T -dependence for a short range of temperature, T ∼ (1-4)
K and the study was also limited to the overlapping Landau level regime.
2.2.2
Power dependence of MIROs
For weak microwave intensities, the photoresistivity response for both displacement and inelastic
mechanism can be described by a simple expression [46] derivable from Eq. 2.6,
δρin
' −4πηPω0 εac λ2 sin(2πεac )
ρD
(2.9)
η is the mechanism and temperature dependent scattering parameter (displacement or inelastic)
and Pω0 is the dimensionless microwave power parameter. Eq. 2.9 predicts a linear power dependence
of photoresistivity and experimental studies [22] [19] [47] found that such linear power dependence
of photoresitivity response holds true for weak microwave intensity. The experimental studies also
reported a sublinear power dependence at large microwave power (Pω0 ∼ 1). A recent experimental
study [46] of the power dependence of integer MIRO photoresistivity at εac = 2 over a broad range
21
of microwave intensities clearly showed the existence of these two distinct power regimes. At the
lowest power(Pω0 1), the linear increase of the photoresitivity response with increasing microwave
power is attributed to single photon processes, while sublinear power dependence of MIRO at larger
Pω0 is attributed to multiphoton processes. For low microwave intensity the MIRO phase is found
to be independent of microwave power [46].
Theoretically [25] [48] [49], higher order fractional MIROs, at εac =
1
and m ≥ 2, are explained
m
in terms of multiphoton absorption processes. It has been proposed that the multiphoton process
can proceed in two ways, first the simultaneous absorption of multiple photons (via intermediate
states) [48] [49], and second, the stepwise absorption of single photons [25]. Theoretical studies
[25] [48] [49] also predicted the power dependence of fractional MIROs to be different than that of
integer MIROS.
In the simultaneous absorption model, an electron absorbs multiple photons in making a transition to a higher Landau level. In this model, for the separated Landau level regime (ωc τq > π/2)
simultaneous absorption of multiple photons require higher microwave power for fractional MIROs
with denominator m ≥ 2 as the photoresistivity response is predicted to decrease as P±m [48], where
P± is the dimensionaless power parameter related to microwave power (for circularly polarized
microwave radiation in this case) and is defined [48] by
τq
eEAC vF 2
P± =
τtr ~ω(ωc ± ω)
(2.10)
where EAC is the microwave electric field , vF is the Fermi velocity and τq /τtr 1 for typical high
mobility 2DES. Under typical experimental conditions, P± . 1, therefore higher order fractional
22
MIROs develop at higher microwave power than the integer MIROs. Previous experimental studies
[18] [22] [23] observed and reported fractional MIROs at εac = 3/2, 1/2 which can be explained in
terms of two-photon processes, and also fractional MIROs at εac = 1/3,2/3,1/4,2/5 which require
processes involving three or more photons.
On the other hand, the theory of stepwise absorption of single photons [25] predicts that in the
overlapping regime the fractional MIROs develop only for higher microwave power Pω0 > 1. The
theory suggests the photoresistivity response of fractional MIRO with denominator m scales as the
m-th power of the squared Dingle factor (λ),
δρin
∝ λ2m
ρD
(2.11)
Although the model does not predict whether higher order fractional MIROs would require progressively higher microwave power for larger m, or not. However, the model suggests fractional MIRO
amplitudes will decay faster than integer MIRO ampitudes with increasing temperature, owing to
the 2m exponent.
In Chapter 4 we will utilize these theories to explain our temperature and power dependence
studies.
2.3
Physical origin of combined DC current and Microwave
Induced Oscillations
In Chapter 1 we have already discussed the oscillatory behavior of the differential resistivity of
2DES under simultaneous microwave radiation and dc current. In the low microwave power limit
23
(Pω0 1), the microwave photoresistivity is assumed to be due to only single photon processes.
Maxima of these oscillations occur when
1
εac + εdc ' n; εac − εdc ' m − ,
2
(2.12)
where n and m are both positive integers.
A theoretical study [38] considered both the displacement and inelastic mechanism, for differential resistivity in the combined (dc + microwave) case. The study [38] showed for εdc &1 the
contribution of inelastic mechanism becomes small and the contribution of displacement mechanism dominates. In the low power limit, the oscillatory correction to the differential resistivity for
overlapping Landau Level regime(ωc τq < π/2) can be expressed as [38]
(4λ)2 τtr
εac
δr
=
sin(2πεdc ) sin(2πεac ) ],
[(1 − 2Pω ) cos(2πεdc ) + 2Pω cos(2πεdc ) cos(2πεac ) −
rD
πτπ
εdc
(2.13)
where Pω is the dimensionless power of unpolarized microwaves, which is defined as [38]
Pω = Pω+ + Pω− ,
Pω± =
Pω0
,
2 +1
(ω ± ωc )2 τem
Pω0 =
(eEac vF τem )2
,
2ef f ~2 ω 2
(2.14)
Eq. 2.13 is successful in predicting the observations of the previous experimental studies [12][13]
carried out with combined excitations for εac >1 and at sufficiently large εdc .
The first experimental study [12] of combined microwave and dc excitation verified the first
24
Figure 2.3: Landau level(tilted lines) diagrams for the combined microwave and dc excitation. Fig. (a) and (d) represent the conditions for maxima and Fig. (b) and (c) represent
the conditions for minima, adopted from Ref.[13]
25
condition for differential resistivity maxima in Eq. 2.12. The physical interpretation of Eq. 2.12
can be understood in terms of combined inter-Landau-level transition which is a combination of an
energy jump of an electron due to microwave absorption followed by a space jump in the direction
of applied dc bias due to scattering off of short range scatters. The result of the study [12] indicated
that maximum in differential resistivity occurs when the combined transition rate parallel to the
dc field is maximized. In that case of the combined transitions terminate at the center of a Landau
level where the number of available states which electrons can scatter into is maximum.
Further experimental study [13] of 2DES under combined microwave and dc excitation revealed
the second condition for differential resistivity maxima in Eq. 2.12. The study [13] showed that
while the space jump along the direction of applied dc bias has to be maximized to observe maxima
in differential resistivity, the space jump in the opposite direction of applied dc bias has be to
minimized as well.
To illustrate the physical interpretation of combined transitions in Figure 2.3 we present the
Landau level diagram for combined transitions at (εac , εdc ) = (3 ∓ 1/4, 1±1/4). Figure 2.3 shows
the combined transition processes with energy on the left axis and real space on the bottom axis.
Figure 2.3 (a) and (d) refer to the combined transitions producing differential resistivity maxima.
These combined transitions along the direction of applied dc bias terminate at the center of a
Landau level. On the other hand Figure 2.3 (b) and (c) refer to the combined transitions involving
differential resistivity minima where the combined transitions opposite to the direction of applied
dc bias terminate at the center of a Landau level. Both the studies [12][13] were done in the regime
above the cyclotron resonance i.e. εac >1.
26
In the next chapter, we will study the effects of combined (dc+ microwave) excitations. We
will point out that multiphoton effects become important in the presence of simultaneous strong
microwave radiation and dc electric field.
27
CHAPTER 3
NONLINEAR EFFECTS ON FRACTIONAL MIROS
In this chapter we present an experimental study on a 2DES under simultaneous microwave and dc
excitation. Previous experimental studies of this type [12][14] found that the differential resistivity
oscillations depend on the simple combinations of the ac and dc oscillation parameters, i.e., εac ±
εdc . More specifically, the two conditions of Eq. 1.18 and 1.19 suggest that for the maximum
in differential resistivity the transition or scattering parallel to the dc field has to be maximized
while the scattering antiparallel to the dc field has to be minimized as well. Another experimental
study [13] showed deviations from these conditions when εac and εdc were not integer pairs. The
observations of these studies [12][13][14] were explained in terms of single-photon combined (ac +
dc) transitions within the theoretical framework of the displacement model in overlapping Landau
level. As discussed earlier in chapter [2] the displacement model dominates the inelastic model
for strong dc bias i.e., 2πεdc 1. Therefore the results of these experimental studies [12][13][14]
performed under low to modest power microwave and strong dc excitation the resistivity oscillations
were explained in terms of single-photon combined transitions within the of displacement model.
Later an experimental study [20] showed that when a 2DES is subjected to high microwave power
and strong dc bias the nonlinear response of resistivity is no longer limited to processes involving a
single-photon combined transition. The study in Ref [20] found multiple extra oscillations around
cyclotron resonance i.e. εac = 1. These were interpreted in terms of combined transitions involving
28
multiphoton processes. Originally, processes involving multiple photons in absence of dc bias were
suggested [18] [22] [23] to give rise to the fractional MIROs. More recent studies[38][20][50][51][52]
suggested multiphoton processes can also become relevant near the cyclotron resonance in the
presence of intense microwave power and strong dc bias. In particular several studies[38][52][51]
predicted that intense microwave radiation in the presence of a strong dc field can give rise to
additional magnetoresistance oscillations near cyclotron resonance due to multiphoton processes.
The role of multiphoton processes near subharmonics of cyclotron resonance in the presence of
strong dc field has not been investigated prior to this work.
3.1
Combining DC currents with High Microwave Power
The experiment was performed on a 20 µm wide Hall bar etched from a 30 nm wide symmetrically doped GaAs/AlGaAs quantum well wafer. In Fig. 3.1 we present the differential resistivity
r measured using a low frequency (13Hz) lockin technique under microwave radiation of f = 30.5
GHz, at a temperature of 1.4K. Fig. 3.1 the differential resistivity plotted vs magnetic field for
different fixed dc bias currents, I= 0-24 µA in steps of 4µA plotted as a function of magnetic field.
The zero dc bias (I = 0 µA) trace in Fig. 3.1 shows MIRO with a single maximum and minimum,
appearing on either side of εac = 1 and εac = 2 and also on either side of the subharmonics of
cyclotron resonance, εac =
to εac = 2, 1,
1
1
and εac = , which are marked with vertical dotted lines. Relative
2
3
1
1
and MIRO maxima occur at ε+
ac at smaller εac (larger B) while MIRO minima
2
3
ε−
ac occur at larger εac (smaller B). Here, the presence of high microwave power can be confirmed
by the decrease in phase of the second harmonic [46]. The observed amplitude of the fractional
features at εac =
1
1
and which exceeds that of the εac = 2 oscillation at zero dc bias (I = 0 µA).
2
3
29
For non zero dc bias, at I = 4 µA we clearly observe the maximum and minimum appearing on
either side of both εac =
1
(εdc ≈ 0.25) and εac = 1 (εdc ≈ 0.5) have flipped, respectively changing
2
into a minimum and maximum, as expected according to Eqn. 1.18 and Eqn. 1.19. At higher dc
current, I = 8 µA, the maximum and the minimum on either side of εac =
1
(εdc ≈ 0.5) and εac =1
2
(εdc ≈ 1) interchange again, in agreement with Eqn. 1.18 and Eqn. 1.19. In fact the I = 8 µA trace
resembles that obtained at I = 0 µA; albeit with a reduced overall amplitude. The trace taken
at even higher dc bias of I = 16 µA displays multiple peaks and dips appearing near both εac =1
1
and εac = . We mark the position of these multiple oscillations with ↓ and ↑ around εac = 1 and
2
1
εac = on the I = 16 µA trace in Fig. 3.1. We will discuss these multiple oscillations in detail,
2
1
since multiple oscillations around εac = have not been reported before. We will refer to these
2
extra oscillations as high power high current oscillations (HPHCO). In contrast the response of the
resistance oscillation around εac =
1
for non-zero bias looks completely different and does not have
3
any extra oscillation at largest applied bias.
A previous experimental study[38] revealed the presence of such HPHCOs only around εac =1
and εac =2. Theory of the HPHCOs has been limited to εac ≥ 1 was formulated in the overlapping
Landau level regime and did predict features near fractional εac . In order to further explore the
effect of high dc current at high microwave field, we refer to the theoretical model proposed by
M. Khodas et al. in Ref. [20] [38] for arbitrary microwave power in the overlapping Landau level
regime (ωc τq < 1). The proposed model uses multiphoton processes near the cyclotron resonance
and its harmonics, and predicts that intense microwave radiation in the presence of a strong dc
field will give rise to magnetoresistance oscillations induced by multiphoton processes. The model
considered the displacement mechanism in the overlapping Landau level regime and is based on
30
Figure 3.1: Differential resistivity r vs magnetic field B at different currents I = 0-24 µA
1
in step of 4 µA. Multiple oscillations near εac =1 and εac = are marked with ↓ and ↑.
2
All traces are vertically offset for clarity.
31
quantum kinetics in the presence of arbitrary microwave power and DC fields. The result of the
calculation[38] in presence of strong dc bias and arbitrary microwave power is:
p
δr
(4λ)2 τtr
=
[cos(2πεdc )J0 (4 Pω sin(πεac ))
rD
πτπ
p
p
2εac
Pω sin(2πεdc ) cos(πεac )J1 (4 Pω sin(πεac ))],
−
εdc
(3.1)
where Jn (x) is the Bessel function of n-th order, λ= exp(−π/ωc τq ) is the Dingle factor, rD is
the Drude resistivity, τπ and τtr are the electron backscattering and transport time and Pω is the
dimensionless microwave power defined as,
Pω = Pω+ + Pω− ,
Pω± =
Pω0
,
2 +1
(ω ± ωc )2 τem
Pω0 =
(eEAC vF τem )2
,
2ef f ~2 ω 2
(3.2)
√
where EAC is the microwave electric field and τem = 20 ef f m∗ c/ne e2 is related to the radiative
decay of the microwave in presence of 2DES and ef f is the appropriate relative dielectric constant
for widely separated charges in the 2DES.
The magnetoresistance in Eq. 3.1 is non-linear in εdc and Pω . Owing to the oscillatory behavior
of the Bessel function J0,1 , Eq. 3.1 predicts these additional high power high current oscillations
HPHCOs around the vicinity of the cyclotron resonance and its harmonics. The model also predicts
a nonlinear power dependence of these resistance oscillations under intense microwave radiation.
These predictions were confirmed experimentally in Ref. [20], where these multiphoton induced
magnetoresistance oscillations were observed near the cyclotron resonance.
32
Figure 3.2: Differential resistivity r vs εac around εac = 1 at I = 21µA. Figure shows both
the theoretical curve and experimental data. Up (down) arrows indicate the MIRO maximum (minimum) and triangles indicate the additional HPHCO resistance osciallations.
In Fig. 3.2 we compare our experimental result to the calculations of the theoretical model in
Eq. 3.1. Here we plot the experimental (left axis) differential resistivity measured at I = 21 µA from
a supplementary experiment under the same conditions and the theoretical (right axis) differential
resistivity at I = 21 µA in the vicinity of εac = 1. The theoretical curve in Fig. 3.2 is generated
using the following parameters: τq ' 9ps, τtr /τπ ≈ 0.34 and Pω0 ' 0.15 in Eq. 3.1. There is good
agreement in period, number and amplitude between the theoretical and experimental curves for
Pω0 ' 0.15 as calculated from our microwave field inside the CPW slot, EAC = 47 V/m(peak). The
oscillations around εac = 1 of the theoretical curve and those of our experimental data have the
same number of resistance oscillations. The extrema of the theoretical and experimental curves
occur nearly at the same εac .
33
The work [20][38] by Khodas and co workers suggests that multiphoton processes play a crucial
role in inducing these HPHCOs near cyclotron resonance. Strong microwave radiation can cause
a time-periodic phase change in the electron wave function and the wave function will acquire
components which have energies E, E±~ω, E±2~ω, ..etc, where E is the electron energy in absence
of microwave radiation. These subbands separated by ±~ω are known as Floquet bands [53].
Electrons can be scattered between Floquet bands of levels accompanied by absorption/emission
of multiple photons. The overlap between initial and final Floquet bands for an inelastic scattering
with an angle θ is expressed by the following parameter:
(N )
aθ
p
2
= JN
[ 2Pω (1 − cos θ)]
(3.3)
The scattering amplitude of electrons accompanied by absorption/emission of microwave photons
(N )
is proportional to the overlap parameter aθ . In the presence of strong dc bias, we consider the
main contribution to the scattering amplitude comes from electrons being backscattered (θ = π).
(N )
For the case of backscattering, the overlap parameter in Eq. 3.3 reduces to aθ
(N )
Fig. 3.3 shows the calculated aπ
√
2 (2 P ).
= JN
ω
vs N for our estimated value of dimensionless microwave power
parameter, Pω0 ' 0.15. For Pω0 ' 0.15 which corresponds to Pω ' 0.96 in this experiment, the
(N )
overlap parameter aθ
peaks around |N |=1 and rapidly decays for |N | >2 in Fig. 3.3. As JN (x)
quickly decays for |N | >x, theory [38] predicts that the number of absorbed or emitted photon by
an electron is limited by:
| N |≤ Nm ≈ 2
p
Pω
(3.4)
where Nm is the maximum number of photons which can be absorbed or emitted by an electron for
34
(N )
Figure 3.3: Overlap parameter aπ vs N. The figure shows overlap between the initial
and final Floquet bands for Pω ' 0.96 (i.e. Pω0 ' 0.15).
a given power Pω . For Pω ' 0.96 in presence of strong dc bias, we estimate the maximum number
of participating photons (absorbed or emitted).
Now we compare our measured differential resistivity centered around εac =
1
to the theoretical
2
model in Eq. 3.1. In Fig. 3.4 we plot the experimental (left axis) and the theoretical (right axis)
1
differential resistivity at I = 21 µA in the vicinity of εac = . The theoretical curve in Fig. 3.4 was
2
calculated using Eq. 3.1 for the following parameters: τq ' 9 ps, τtr /τπ ≈ 0.34 and Pω0 ' 0.15. The
experimental curve in Fig. 3.4 shows additional high power high current oscillations (HPHCOs) near
1
1
. These multiple oscillations near εac = imply that the multiphoton processes are also
2
2
1
relevant near εac = . However, the theoretical model in Ref. [38] fails to generate a trace similar
2
εac =
to our observed HPHCOs, shown in Fig. 3.2. We stress that the theoretical model in Ref. [38] was
developed for arbitrary microwave power in the overlapping Landau level regime where ωc τq <
35
π
.
2
1
at I = 21µA. Figure shows
2
both the theoretical curve and experimental data. Up (down) arrows indicate the MIRO
maximum (minimum) and triangles indicate the additional resistance osciallations.
Figure 3.4: Differential resistivity r vs εac around εac =
Moreover the model [38] predicted the observed HPHCOs in magnetoresistance only near cyclotron
resonance and its harmonics i.e. εac ≥1. The model correctly predicts HPHCOs around εac = 1
where ωc τq ≈ 1.6 in our experiments. However, as we move further into the separated Landau level
1
regime, to εac = , where ωc τq ≈ 3.2 in our experiments, the model appears to need modification.
2
To investigate these additional oscillations in magnetoresistance further we now refer to the
theoretical studies carried out by X.L.Lei in Ref. [50][51][52] for arbitrary microwave electric field
considering both the separated and overlapping Landau level regime. The theory used an electron
drift model with force balance-equation approach, and predicted the observed additional oscillations in magnetoresistance near cyclotron resonance in the presence of strong dc bias and high
36
microwave electric field. Lei et al.[52][51] suggested that in the presence of microwave radiation
the electrons absorbing microwave energy dissipate the energy to the host lattice not only via
electron-phonon scattering but also through inter and intra-Landau level transitions. In presence
of strong enough microwave radiation the inelastic scattering time will become much smaller than
the transport scattering time due to rapid thermalization of electrons in high-mobility 2DES. Under
these circumstances, the inelastic mechanism is again dominated by the displacement mechanism
contribution in the overlapping Landau level regime. In the case of strong dc bias (2πεdc 1),
in the overlapping Landau level regime, the theoretical differential resistivity calculated by Lei et
al.[51][52] is,
1
δr
π
√
2
= 8λ2 εdc
[ cos(2πεdc + )J0 [2 eω η sin(πεac )]
rD
4
π εac
√
√
− sin(2πεdc − )
cos(πεac )( eω η)J1 [2 eω η sin(πεac )] ],
4 εdc
(3.5)
where the effective radiation power parameter, eω is defined as
eω =
e2 kF2 Eω2
m2 ω 4
(3.6)
and η is a polarization related dimensionless coefficient for linearly x-polarized radiation defined
as,
η=
3 3a2 + 3c2 + b2 + d2
.
4
(a2 + b2 )2
(3.7)
Here, a = 1 − ( ε1ac )2 + γω2 , b = 2γω ( ε1ac ), c = (1 + ( ε1ac )2 + γω2 )γω and d = (1 − ( ε1ac )2 − γω2 )γω . The
radiation damping factor γω and effective incident radiation Eω for a 2DES located under a semiinfinite semiconductor with a refractive index ns are defined as γω =
37
ne e 2
and Eω =
[1 + ns ]m∗ 0 c ω
Figure 3.5: Differential resistivity r vs εac around εac = 1 at a dc bias of I = 21µA.
Figure shows both the theoretical curve generated by Eq. 3.5 for Eiω = 95V /cm and the
experimental curve. Up (down) arrows indicate the MIRO maximum (minimum) and
triangles indicate the additional resistance osciallations.
2Eiω
respectively[51][52], where Eiω is the incident microwave radiation field. The differential
(1 + ns )
resistivity in Eq. 3.5 is non-linear εdc and Eiω . The model also predicts these additional high
power high current oscillations HPHCOs around the vicinity of the cyclotron resonance and its
harmonics. The model in Ref.[52] also showed a nonlinear power dependence of these multiphotoninduced resistance oscillations under intense microwave radiation.
In Fig. 3.5 we compare our experimental result to the calculations of Lei et al. given here in
Eq. 3.5. In Fig. 3.5 we plot the experimental (left axis) and the theoretical (right axis) differential
resistivity at I = 21 µA in the vicinity of εac = 1. Here, the incident radiation field parameter, Eiω
in Eq. 3.5 is adjusted to predict our observed additional HPHCOs near εac = 1. The theoretical
curve in Fig. 3.4 is calculated using the following parameters: τq ' 9ps,Eiω = 95 V /cm(peak)
38
Figure 3.6: Differential resistivity r vs εac around εac = 1 at I = 21µA. Figure shows
both the theoretical curve for Eiω = 0.47 V /cm and experimental data. Up (down) arrows
indicate the MIRO maximum (minimum) and triangles indicate the additional resistance
osciallations.
and ns =
√
12.8 ' 3.6 is the refractive index of GaAs. We observe that the calculated and the
experimental curves in Fig. 3.4 closely resemble each other near εac = 1 in terms of the period,
number and amplitude of the oscillations. The oscillations around εac = 1 of the calculated curve
and those of our experimental data have the same number of resistance oscillations. The extrema
of the theoretical and experimental curves also occur nearly at the same εac in Fig. 3.5.
We found the theoretical model in Eq. 3.5 predicts these observed HPHCOs in magnetoresistance
around εac = 1 for incident microwave electric field (Eiω = 95 V /cm). However, the highly intense
incident microwave electric field Eiω = 95 V /cm required by this model to generate our observed
HPHCOs is several orders higher in magnitude than our estimated microwave electric field EAC =
0.47 V /cm (peak) inside the CPW slot.
39
1
at I = 21µA. Figure shows both
2
0 = 0.47 V /cm. Up (down)
the experimental data the theoretical curve calculated for Eiω
arrows indicate the MIRO maximum (minimum) and triangles indicate the additional
resistance osciallations.
Figure 3.7: Differential resistivity r vs εac around εac =
In Fig. 3.6 we depict the experimental (left axis) and the theoretical (right axis) differential
resistivity for Eiω = 0.47 V /cm at I = 21 µA in the vicinity of εac = 1. The theoretical curve
shown in Fig. 3.6 is generated using our measured microwave electric field inside the CPW slot
Eiω = 0.47 V /cm instead of Eiω = 95 V /cm. As shown in Fig. 3.6, the model fails to generate the
observed HPHCOs in magnetoresistance around εac = 1 for Eiω = 0.47 V /cm.
Lastly, in Fig. 3.7 we compare the experimental (left axis) and the theoretical (right axis)
1
1
. The multiple oscillations near εac = in the
2
2
1
experimental curve in Fig. 3.4 imply that the multiphoton processes are also relevant near εac = .
2
1
Eq. 3.5 however fails to generate our observed HPHCOs in magnetoresistance around εac = even
2
differential resistivity at I = 21 µA around εac =
for a highly intense incident microwave electric field of Eiω = 95V /cm.
In this section we presented our experimentally observed multiphoton induced HPHCOs near
40
εac = 1 and εac =
1
. In addition we compared our experimental results with the existing theo2
retical models[38][52][51] where observation of multiphoton induced additional oscillations in magnetoresistance near εac = 1 and its harmonics has been predicted. Theoretical models developed
by M.Khodas et al. in Ref. [38] and by X.L. Lei et al. in Ref. [52][51] both suggested that in
the overlapping Landau level regime, in the presence of strong dc bias, the inelastic mechanism
contribution is negligible compared to the displacement mechanism contribution. Both of these
theories predicted multiphoton-induced additional oscillations in magnetoresistance near εac = 1
and its harmonics only in the overlapping Landau level regime. We found the model proposed by
M.Khodas et al. in Ref. [38] is able to reproduce our observed multiphoton induced additional
oscillations in magnetoresistance near εac = 1 for an estimated microwave field from our measure(N )
ment. The overlap parameter aθ
in this theoretical model[38] also predicts multiple participating
photons for our experimental results. On the contrary, the theoretical model developed by X.L. Lei
et al. in Ref. [52][51] is also able to reproduce the multiphoton-induced additional oscillations in
magnetoresistance near εac = 1 but requires intense microwave electric field which is much larger
than our estimated microwave electric field of EAC = 0.47 V /cm. However, as we move further
into the separated Landau level regime, to εac =
1
, we found both the theories were inadequate
2
and fail to capture the multiphoton induced additional oscillations in magnetoresistance observed
1
in our experiments near εac = .
2
41
3.2
Combining DC currents with Microwaves below Cyclotron
Resonance
In Fig. 3.1 we have observed that in presence of high microwave power the differential resistivity
extrema on either side of εac = 1 and εac =
1
kept changing from maxima to minima and vice
2
versa as dc bias was increased. In a previous study [13] the effects of dc bias on the resistivity
1
and εac = 1 were compared. The resistivity oscillation extrema on
2
1
either side of εac = 1 and εac =
kept changing from maxima to minima and viceversa as the
2
1
applied DC current was increased. The differential resistivity oscillation around εac =
closely
2
1
replicated the behavior of the oscillation around εac = 1, though the magnetic field at εac =
2
oscillations around εac =
is twice that of the εac = 1. The currents were the same that flip maxima into minima for εac
1
. Such a result was unexpected because one would assume that the current
2
1
needed to flip the oscillation obtained at εac =
would be twice that of the εac = 1 oscillation
2
1
because εdc ∝ BI . This response of differential resistivity oscillation around εac = with respect to
2
1
increasing dc current is referred as ’frequency doubling’ around εac = . Such observation of HIRO
2
1
εdc frequency doubling for the nonlinear response of εac = has not yet been fully understood. In
2
1
1
contrast, the nonlinear response of εac = was found to evolve differently from those at εac =
3
2
1
and εac = 1. The effect of dc bias on the εac = oscillation has not yet been investigated prior to
3
just above 1 and
this work. In this section we present the results of our experiments to study the microwave power
dependence of fractional MIROs and to investigate the possible role of multiple photon processes
near subharmonics of cyclotron resonance.
In Fig. 3.8 we present differential resistivity r vs εdc , for microwave radiation of f = 31 GHz
42
1
Figure 3.8: Differential resistivity r vs εdc at a fixed-B corresponding to εac = for f = 31
2
GHz. Dotted and solid trace represent the microwave power 0dB and +3dB respectively.
at two different power. The measurement was performed at a fixed-B corresponding to εac =
1
2
while sweeping applied dc bias at a constant bath temperature of T = 1.5K. The dotted line is
for the lower power (0 dB) differential resistivity r vs εdc trace while the solid line is for higher
power +3dB. The 0 dB trace shows well defined HIROs upto εdc = 3 with maxima and minima in
r appearing at integer and half-integer values of εdc respectively. In contrast to the 0 dB trace, the
trace taken at +3dB shows maxima in r near fractional εdc = 1/2, 3/2 and 5/2. In addition, maxima
in the +3dB trace also appear near integer εdc = 1,2 and 3, albeit with a reduced amplitude in
comparison to 0dB trace.
We notice that the observed maxima in +3dB trace at εac =
1
and integer εdc can no longer be
2
explained in terms of single photon combined transitions. As discussed earlier in Chapter 2 the set
43
1
Figure 3.9: Landau level diagrams at εac = for combined microwave and dc transitions
2
involving 1-photon processes at +3dB. Vertical arrows represent the electron transition
due to absorption(up arrows) or emission(down arrows) of a photon, the horizontal arrows represent the backscattering transition of 2Rc and the inclined arrows represent the
combined transitions.
44
1
Figure 3.10: Additional Landau level diagrams at εac = for combined microwave and dc
2
transitions involving 1-photon processes at +3dB
of conditions described by Eq. 1.18 and 1.19 were developed [12][13][14] for single photon combined
transitions fails to predict the observed maxima at εac =
1
and integer εdc . We assume that only
2
0-photon and 1-photon processes are relevant at lower power (0dB) and are competing with each
other. We also suggest that the observed maxima at integer εdc in the +0dB trace appear as a
result of combined transitions involving 0-photon processes while the slight maximum at εdc = 1/2
appear due to transitions involving 1-photon processes. In addition, the absence of maxima at εdc
= 3/2 and 5/2 indicates that at higher oscillation order (εdc ) 0-photon processes win over 1-photons
processes. Now we attempt to extend this argument of competing transitions involving different
number of photons to higher power. At +3dB we assume 1-photon processes will become more
relevant and dominate over the 0-photon processes. The dominant role of 1-photon processes even
at higher oscillation order can explain the presence of maxima εdc = 3/2 and 5/2. On the other
45
hand at +3dB the reduced amplitude of maxima at integer εdc in comparison to 0dB indicates that
0-photon processes cannot be fully disregarded.
In Fig. 3.9 and 3.10 we show the relevant energy-space Landau level (LL) diagrams for combined
transitions involving 1- photon processes at +3dB. The Hall-field tilted LLs are represented by tilted
thick lines, while vertical arrows and horizontal arrows represent the microwave and backscattering
transitions respectively. The inclined arrows depict the combined transitions, with solid and dotted
line representing a maximum or minimum respectively. At weak power (0dB), for 0-photon processes
the backscattering transitions due to dc field terminate at the center of a LL and thus result in a
maxima at εac =
1
and integer εdc , as shown in Fig. 3.9(b),(d) and Fig. 3.9(a). However, at higher
2
power (+3dB), 1-photon processes weaken the εdc = n maxima and also strengthen the εdc = n- 1/2
maxima. In Fig. 3.9(a),(c) and Fig. 3.10(a) we observe that the combined transitions involving 1photon processes parallel to the dc field terminate at the center of a LL and thus result in maxima.
In Fig. 3.9(b),(d) and Fig. 3.10(b) the combined transitions involving 1-photon processes parallel
to the dc field end in the gap between two LLs and thus result in minima.
In Fig. 3.11 we plot r vs εdc trace taken at increased microwave power +6dB and the low power
0dB trace as a baseline. The solid line represents the higher power +6dB r vs εdc trace while the
dotted line represents the 0dB r vs εdc trace. The +6dB r vs εdc trace shows the maximum in
r at εdc =1/2 has a little variation in strength compared to +3dB trace. The maximum seen for
+3dB at εdc =3/2 is greatly reduced in amplitude and can be observed as shoulder. In contrast to
the +3dB trace the maximum at εdc =5/2 completely changes and turns into a minimum. We also
notice that at +6dB the amplitudes of maxima near integer εdc become stronger in comparison
46
1
Figure 3.11: Differential resistivity r vs εdc at a fixed-B corresponding to εac = for f =
2
31 GHz for +6dB. Dotted trace represents the data take at baseline weak power 0dB and
solid trace represents the data taken at the microwave power of +6dB.
to the +3dB trace. The entire +6dB trace more closely resembles the 0dB trace rather than the
+3dB trace.
In Fig. 3.12 we show the energy-space Landau Level(LL) diagrams for combined transitions
involving 1-photon and 2-photon processes at +6dB. At higher power (+6dB) where 2-photon
processes rather than 1-photon processes become relevant,the 2-photon processes will weaken the
εdc = n-1/2 maxima and strengthen the εdc = n maxima. In Fig. 3.9(b),(d) and 3.10 (b) we observe
that the combined transitions involving 2-photon processes parallel to the dc field terminate at the
center of a LL and therefore maximize r. On the other hand, in Fig. 3.9(a),(c) and 3.10(a) we
observe that the combined transitions involving 2-photon processes parallel to the dc field end in
the gap between two LLs and thus result in minima of r.
We suggest that with increasing power 2-photon processes become more important and dominate
47
1
Figure 3.12: Landau level diagrams at εac = for combined microwave and dc transitions
2
involving 2-photon processes at +6dB. Vertical arrows represent the electron transition
due to absorption(up arrows) or emission(down arrows) of a photon, the horizontal arrows represent the backscattering transition of 2Rc and the inclined arrows represent the
combined transitions.
48
1
Figure 3.13: Additional Landau level diagrams at εac = for combined microwave and dc
2
transitions involving 2-photon processes at +6dB.
over 1-photon processes at +6dB. For εac =1/2, 2-photon processes strengthen the maxima at εdc =
n and weaken the maxima at εac = n- 1/2. This is confirmed by the transformation of the maximum
at εdc =5/2 into a minimum and also by the loss of the maximum at εdc =3/2. In addition, the
amplitudes of the maxima at integer εdc also increased at +6dB relative to their +3dB values due
to 2-photon processes. However the presence of the maximum at εdc =1/2 suggests that 1-photon
processes are still present at this power. The overall evolution of the r vs εdc trace at +6dB clearly
shows that 2-photon processes remain dominant not only at εdc = n but also at εdc = 5/2. On the
other hand, at εdc =3/2 we see a competition between 1-photon and 2-photon processes which can be
observed as shoulder. At (εac ,εdc )= (1/2,3/2) the combined transition involving 1-photon processes
result in a maximum while the combined transition for 2-photon processes predict a minimum as
shown in Fig. 3.9(c). Similarly the strong maximum at εdc =1/2 suggests the 1-photon processes
49
1
Figure 3.14: Differential resistivity r vs εdc at a fixed-B corresponding to εac = for f =
2
31 GHz for +10dB. Dotted and solid trace represent the data take at baseline weak power
0dB and at the highest microwave power +10dB respectively.
are still relevant at lower oscillation order as shown in Fig. 3.9(a).
In Fig. 3.14 we plot r vs εdc trace measured at the highest microwave power +10dB, and also
at the low power 0dB as a baseline. A local minimum appears at εdc = 3/2 in the +10dB trace.
However the maximum at εdc = 1/2 continues to persist even at the highest power. The +10dB
trace looks more like the 0dB trace than the +3dB and +6dB traces.
For the highest power +10dB, the observed behavior of differential resistivity r can be viewed
as a result of mostly 2-photon processes. At +10 dB the dominating role of 2-photon processes
over 1 photon processes makes the maxima at εdc more stronger. The amplitude of the maxima
at integer εdc is observed to increase at the +10dB trace in comparison to the +6dB one. We
interpret the observed minimum at εdc = 3/2 as a result of the dominant contribution of 2-photon
processes. Nevertheless the continued observation of the maximum at εdc =1/2 even at the highest
50
power indicates that 1 photon processes are still present but only at lower oscillation order. Apart
from the maximum at εdc =1/2 we can clearly see the role of 1 photon processes at +10dB less
important than it is at +6dB. The increased amplitude of the maxima at εdc = n in the +10dB
trace clearly demonstrates the significance of 2-photon processes at the highest power.
The observed microwave power dependence of the fractional HIROs at εac = 1/2 can be explained
in terms of the combined transitions involving multiphoton processes. With increasing power we
see the role of multiphoton processes become more important and find that it predicts the behavior
of fractional HIROs. We also observed possible HIRO εdc -frequency doubling at εac = 1/2 for a
microwave power of +3dB.
Now we turn our attention to the nonlinear response of εac = 1/3 and investigate the effect
of dc bias at εac = 1/3. In Fig. 3.15 we present differential resistivity r measured at a fixed-B
corresponding to εac =
1
for f = 31 GHz. Again the low power 0dB trace is dotted and the +3dB
3
traces is solid. The 0dB trace shows only a single HIRO maximum at εdc = 1 accompanied by a
strong zero bias peak(ZBP). The higher power +3dB trace clearly shows a reduced amplitude of
the maximum at εdc = 1. In addition, the +3dB trace shows maxima appearing at εdc = 2/3 and
4/3. At +3dB presence of maxima in r at εdc = 1,εdc = 2/3 and εdc = 4/3 would indicate a possible
εdc frequency tripling. However the absence of maxima at εdc =1/3 and 5/3 fails to confirm such
observation at this power.
In Fig. 3.15, at 0dB the single HIRO maximum appearing at εdc = 1 can be viewed as a consequence of 0-photon processes. The absence of any observed maximum expect at εdc = 1 demonstrates
that only 0-photon processes dominate at this power (0dB). With increasing power at +3dB emergence of maxima at εdc = 2/3 and 4/3 clearly indicates the presence of 1-photon processes. The
51
Figure 3.15: Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding
1
to εac = for +3dB. Dotted and solid trace depict the microwave power 0dB and +3dB
3
respectively.
effect of 1-photon processes at +3dB can be also be confirmed by the reduced amplitude of the
maximum at εdc = 1. The presence of the maximum at εdc = 1 with reduced amplitude at +3dB
also suggests a possible competition between 0-photon and 1-photon processes at this power.
In Fig. 3.15, we present the relevant energy-space Landau level(LL) diagrams for combined
transitions involving 1-photon processes at +3dB. In Fig. 3.15 (b) (d)the combined transitions can
result in a maximum for 1-photon processes. At (εac ,εdc )= (1/3,2/3) and (εac ,εdc )= (1/3,4/3) the
combined transitions parallel to the dc field terminate at the center of a LL and give rise to a
maximum. On the other hand, in Fig. 3.15 (b)(d) the combined transitions antiparallel to the
dc field end in the gap between two LLs and are minimized. Fig. 3.15 (a)(c) represents 1-photon
processes where the combined transitions end in the gap between the two LLs and result in a
52
1
Figure 3.16: Landau level diagrams at εac = for combined microwave and dc transitions
3
involving 1-photon processes at + 3dB. The inclined arrows represent the combined transitions involving 1-photon processes, solid and dotted line represent differential resistivity
maximum and minimum respectively.
53
Figure 3.17: Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to
1
for +6dB. Dotted and solid trace depict the weak baseline power 0dB and the
εac =
3
higher microwave power +6dB respectively.
minimum.
In Fig. 3.17, we present r vs εdc trace for increased microwave power +6dB and the low power
0dB trace as a baseline. The solid line represents the higher power +6dB trace while the dotted
line represents the 0 dB trace. The higher power +6dB trace shows maxima appearing at εdc =
1/3, 2/3 and 4/3. At +6dB the amplitude of the maximum appearing at εdc = 1 is greatly and
can be observed as a local maximum. The maxima at εdc = 1/3 and 4/3 are considerably stronger
compared to the maxima observed at εdc = 2/3 and 1. In addition, the presence of maxima in r
at εdc = 1/3,2/3,1 and εdc = 4/3 indicates possible εdc -frequency tripling at +6dB. We suggest a
combination of 1-photon and 2-photon processes can result in such observed εdc tripling. Both
1-photon and 2-photon processes are relevant at +6dB. While the maximum appearing at εdc = 1/3
can be interpreted by 2-photon processes, the maximum appearing at εdc = 2/3 indicates 1-photon
54
1
Figure 3.18: Landau level diagrams at εac = for combined microwave and dc transitions
3
involving 1-photon and 2-photon processes at 6dB. Inclined arrows show the combined
transitions involving 1-photon and 2-photon processes. Differential resistivity maximum
and minimum correspond to solid and dotted inclined arrows respectively.
55
processes are still present. At εdc = 4/3 the amplitude of the maximum is increased where both
1-photon and 2-photon processes are likely to contribute.
Fig. 3.18 illustrates the typical combined transitions for increased microwave power (+6dB) at
εac = 1/3. Fig. 3.18 (a)(d), displays the LL diagrams of combined transitions involving 2-photon
processes which result in maxima. At (εac ,εdc )= (1/3,1/3) and (εac ,εdc )= (1/3,4/3) the combined
transitions parallel to the dc field end at the center of a LL and give rise to the observed maximum
at +6dB. For (εac ,εdc )= (1/3,4/3) the combined transition involving 1-photon processes also result
in a maximum if a photon is emitted instead of being absorbed as shown in Fig. 3.18(d). We
suggest that at (εac ,εdc )= (1/3,4/3) the increased amplitude of the maximum for 6dB trace occurs
due to the contribution of 1 and 2-photon processes. Both 1 and 2-photon processes can result in a
minimum at (εac ,εdc )= (1/3,1) as shown in Fig. 3.18(c). This is confirmed by the reduced amplitude
of the maximum at (εac ,εdc )= (1/3,1) at +6dB relative to its +3dB value. In Fig. 3.18 (b) where the
LL diagram at (εac ,εdc )= (1/3,2/3) shows that the combined transitions involving only 1-photon
processes parallel to the dc field terminate at the center of a LL and results in maximum. At the
same time combined transitions involving 2-photon processes at (εac ,εdc )= (1/3,2/3) parallel to the
dc field end at the gap between two LLs and are minimized. This continued presence of maximum at
+6dB albeit with a reduced amplitude suggests that 1-photon processes are still present at +6dB.
In the entire +6dB trace we observe a competition between 1-photon and 2-photon processes while
2-photon processes mostly dominate over 1-photon processes everywhere expect at εdc = 2/3.
For the highest microwave power +10dB, in Fig. 3.19 we plot the r vs εdc trace with the
0dB trace as a baseline. At the highest power, we observe maxima appearing at εdc = 1/3,1 and
4/3. The maxima at εdc = 2/3 in both the +3dB and +6dB traces transform into a minima at
56
Figure 3.19: Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to
1
εac =
for +10dB. The dotted trace depict the weak baseline power 0dB and the solid
3
trace depict the highest microwave power +10dB.
Figure 3.20: Landau level diagrams at (εac ,εdc )= (1/3,5/3) for combined microwave and
dc transitions involving 1-photon and 2-photon processes. Fig. (a) shows maximum in
differential resistivity due to 1-photon absoprtion proces and Fig. (b) shows maximum in
differential resistivity due to 2-photon absoprtion process at the highest power +10dB.
57
the highest power(+10dB). At the highest power the 2-photon processes are likely to dominate
1-photon processes.
We again refer to the Hall-field tilted LL diagrams shown in Fig. 3.18 to explain the observed
maxima in the r vs εdc trace at +10dB. At (εac ,εdc )= (1/3,1/3) and (εac ,εdc )= (1/3,4/3) combined transitions related to 2-photon processes parallel to the dc field are maximized as shown in
Fig. 3.18 (a)(d). The observed minimum at (εac ,εdc )= (1/3,2/3) also confirms the importance of
2-photon processes at increased microwave power. The combined transition in the Hall-field tilted
LL digram at (εac ,εdc )= (1/3,2/3) is shown in Fig. 3.18 (b). The minimum at (εac ,εdc )= (1/3,2/3)
occurs when the 2-photon processes become dominant over 1-photon processes and the combined
transition parallel to dc field ends at the gap between two LLs. Finally we focus at the feature
appearing (εac ,εdc )= (1/3,5/3) at +10dB. The LL diagrams in Fig. 3.20 (a)(b) suggest a maximum
at (εac ,εdc )= (1/3,5/3) for both 1-photon and 2 photon processes. Fig. 3.20 (a) shows the combined
transition for 1-photon absorption and scattering ends at the center of a LL and thus maximized. In
Fig. 3.20(b) the combined transition related to 2-photon emission also results in a maximum. The
observed local maximum at (εac ,εdc )= (1/3,5/3) can be viewed as a result of combined contribution
of both 1-photon and 2-photon processes. Thus the observed features in the +10dB trace clearly
shows the importance and the dominant role of 2-photon related combined transitions at highest
microwave power.
Lastly we attempt to explore the nonlinear response of εac =
1
to see the effect of dc bias at
4
εac = 1/4 in presence of intense microwave radiation. In Fig. 3.21 we plot differential resistivity r
vs εdc for f = 31 GHz obtained at a fixed B corresponding to εac =
1
. Again the low power (0dB)
4
trace and the high power (+13dB) trace are shown by the dotted and the solid line respectively.
58
Figure 3.21: Differential resistivity r vs εdc for f = 31 GHz at a fixed-B corresponding to
1
εac =
for +13dB. The dotted trace depict the weak baseline power 0dB and the solid
4
trace depict the highest microwave power +13dB.
In addition to a strong zero bias peak (ZBP), the 0dB trace also shows a single HIRO maximum in
r near εdc = 1. In contrast to the 0dB trace, the +13dB trace clearly shows maxima appearing at
εdc =
1 3
, and 1. At 0dB, the single HIRO maximum which appears near εdc = 1 can be explained
4 4
in terms of combined transitions involving 0-photon processes. On the other hand, emergence of
maxima at εdc =
1
3
and clearly suggest the presence of 3-photon processes and presence of possible
4
4
4-photon process near εdc = 1.
Fig. 3.22 shows the relevant energy-space Landau level (LL) diagrams at εac = 1/4 for combined
transitions involving 3 and 4-photon processes at +13dB. At (εac ,εdc )= (1/4,1/4) the combined
transitions parallel to the dc field end at the center of a LL only for 3-photon processes and result
in a maximum, shown in Fig. 3.22(a). Fig. 3.22(b) shows that at (εac ,εdc )= (1/4,1/2) combined
transitions involving 3-photon processes terminate in the gap between two LLs and result in a
59
Figure 3.22: Relevant Landau level diagrams for εac =
60
1
at +13dB.
4
minimum. On the other hand, at (εac ,εdc )= (1/4,1/2) combined transitions involving 2-photon
processes should result in maximum. The observed minimum at (εac ,εdc )= (1/4,1/2) suggests
at +13dB, 3-photon processes dominate over 2-photon processes. Fig. 3.22(c) shows (εac ,εdc )=
(1/4,3/4) the combined transitions involving a 2-photon absorption accompanied by a 1-photon
emission and scattering ends at the center of a LL and result in a maximum. The observed
maximum at (εac ,εdc )= (1/4,3/4) further confirms the significance of 3-photon processes at +13dB.
Finally, at (εac ,εdc )= (1/4,1) combined transitions parallel to the dc field terminate at a LL for 4photon processes and thus maximized. However, combined transitions involving 3-photon processes
result into minimum at (εac ,εdc )= (1/4,1). The observed maximum in r appearing at (εac ,εdc )=
(1/4,1) indicates that 4-photon processes can also become relevant at +13dB.
In this section we have presented our experimental results the nonlinear response of 2DES at
εac = 1/2, 1/3 and 1/4 at high microwave power and strong dc bias. We investigated 1) the effect
of dc bias on the subharmonics of the cyclotron resonance and 2) the possible role of multiphoton
processes at subharmonics of the cyclotron resonance. We performed fixed-B measurement while
sweeping the dc current at different microwave powers to investigate the previously observed frequency doubling at εac = 1/2 [?]. Our experimental data not only demonstrates the εdc -frequency
doubling at εac = 1/2 but also shows possible εdc -frequency tripling at εac = 1/3. We suggest that
the observed εdc -frequency doubling at εac = 1/2 and εdc -frequency tripling at εac = 1/3 can be
explained in terms of combined transitions involving multiphoton processes. In a previous theoretical study in Ref. [50] the observed εdc -frequency doubling at εac = 1/2 of Ref [13] was numerically
reproduced in terms of multiphoton processes in the seperated Landau level regime. That study
also predicted possible εdc -frequency tripling at εac = 1/3 due to the multiphoton processes in the
61
seperated Landau level regime. However, the study did not provide any analytical result in terms
of microwave power or the degree of overlap of LLs to compare with our experimental data. Finally,
we have shown that combined transition mechanism involving multiphoton processes also holds at
εac = 1/4 and reasonably explains our observation in the presence of strong microwave power.
Our observed nonlinear response of 2DES at εac = 1/2, 1/3 and 1/4 can be interpreted in
terms of combined transitions involving scattering processes accompanied by different number of
photon absorption and emission within the scope of displacement mechanism. To summarize our
experimental results we suggest that the maxima in r due to such combined transition can be
described by
εdc + N εac = m
(3.8)
where m is a non-negative integer, N = 0, ±1, ±2,.... where positive and negative sign represents
photon absorption and emission processes respectively. We also note that the Eq. 3.8 was originally
developed by Ref. [50] to explain the observed εdc -frequency doubling in Ref. [13].
62
CHAPTER 4
TEMPERATURE AND POWER DEPENDENCE OF
FRACTIONAL MIROS
In this chapter we address the temperature and power dependence of fractional MIROs. As discussed earlier in Chap 2 the temperature dependence studies of fractional MIRO will allow us to
compare the displacement and inelastic mechanisms and allow us to figure out which mechanism is
dominant at low temperature. While the MIRO amplitude for the inelastic model is proportional to
the inverse square of the temperature, the displacement model is mostly temperature independent.
A previous experimental study[54] of the fractional MIROs reported that the inelastic mechanism
is dominant only at low temperature T < 3K. Fractional MIROs at εac =
1
with m ≥2 require
m
multiphoton absorption process and appear at the strong microwave power. Two competing models
exist for these multiphoton processes which lead to fractional MIROs. The first is a simultaneous
absorption of multiple photons (via virtual intermediate states)[24] and the second is a stepwise
absorption of single photons[25]. The simultaneous absorption of multiple photons requires high
microwave power for higher order fractional MIROs. The stepwise absorption of single photons predicts that the photoresistivity response of a fractional MIRO with denominator m is proportional
to λ2m while the integer MIROs are proportional to λ2 , where λ is the Dingle factor. Therefore
the fractional MIROs should decay exponentially faster than the integer MIROs with increasing
temperature [25]. In addition it suggest the fractional MIRO amplitude will be exponentially sup-
63
pressed as compared to that of integer MIRO. A temperature dependence study of fractional MIROs
over a wide range would provide some insight about their underlying mechanism. In addition, we
also perform a power dependence of integer and fractional MIROs to investigate the importance
of multiphoton processes at high microwave power. We suggest that power dependence study of
the fractional MIROs will allow us to distinguish between the multiphoton model and the stepwise
single photon model.
4.1
Temperature dependence of Fractional MIROs
We performed measurements in the absence of dc fields for a fixed microwave radiation frequency
of f = 30.5 GHz at different temperatures, on the same sample used for the work in Chap 3, with a
20 µm wide Hall bar etched from a 30 nm wide symmetrically doped GaAs/AlGaAs quantum well
wafer. We also used different microwave powers. In this section we present the results obtained at
three different microwave powers similar results were obtained at other power levels. In Fig. 4.1
we plot resistivity ρxx as function of B obtained at different temperatures staring from 2 K upto
17 K for a fixed microwave power which corresponds to the microwave electic field inside the CPW
slot, EAC = 36.4 V/m. The lowest temperature (T = 2 K) trace in Fig. 4.1 shows fractional
MIROs with a single maximum and minimum appearing on either side of the subharmonics of
cyclotron resonance, εac =
1
1
1
, εac = and εac = , marked with vertical dotted lines. In addition
2
3
4
to the fractional MIROs, we also observe strong integer MIROs appearing around εac = 1 and
εac = 2 at the lowest temperature (T = 2 K). There is monotonic decrease of the fractional MIRO
amplitudes with increasing temperature until completely disappearing at highest temperature (T =
17 K) trace. Moreover, the integer MIROs also weaken with increasing temperature and eventually
64
disappear at higher temperature (T > 10 K). Although MIROs are more immune [7] to the thermal
smearing of the Fermi surface than SdH oscillations, both integer and fractional MIROs weaken
with increasing temperature in Fig. 4.1. Previous experimental studies [6] [7] [41] [39] also reported
strong suppression of MIRO amplitudes with increasing temperature and complete disappearance
of MIROs at T ∼ 4 − 7K. However, in our experiments we observe fractional MIROs upto 10 K. In
Fig. 4.1 the zero field resistivity also increases monotonically due to excitation of thermal acoustic
phonons with increasing temperature.
In Fig. 4.2 we present the temperature evolution of fractional MIROs measured at a higher
fixed microwave power which corresponds to microwave electic field, EAC = 119.6 V/m inside the
CPW slot. Resistivity, ρxx vs B traces plotted in Fig. 4.2 are vertically offset for clarity. The
lower temperature traces in Fig. 4.2 show integer MIROs around εac = 1 and εac = 2, that become
invisible at T ∼ 5K. In addition to integer MIROs, we again observe well-developed fractional
MIROs centered at the subharmonics of cyclotron resonance, εac =
1
1
1
, εac =
and εac = .
2
3
4
The amplitude of the fractional MIROs is larger in comparison to the fractional MIROs shown in
Fig. 4.1. At EAC = 119.6 V/m, the fractional MIRO amplitudes continue to decrease with increasing
temperature, however all the fractional features are still present at highest temperature (T = 14.7K)
trace. The temperature evolution shown in Fig. 4.2 also shows the integer MIROs are damped more
heavily than the fractional MIROs with increasing temperature. Such observation is in contrast
to a previous experimental study [54] where fractional MIROs were found to be damped similarly
to the integer MIROs and both eventually disappear at the same temperature. A close inspection
of the data presented in Fig. 4.2 reveals that the fractional MIRO maxima slightly shift to the
higher magnetic field for the higher temperature traces. We suggest that such observation might
65
Figure 4.1: Resistivity ρxx vs magnetic field B at different temperatures EAC = 36.35 V/m.
Vertical dotted lines mark the cyclotron resonance, its second harmonic and subharmonics.
66
be related to the phonon contribution which is likely to be present at these higher temperature. We
also suggest that for higher microwave power, multiphoton processes become relevant and dominate
over 1-photon processes.
In Fig. 4.3 we plot the ρxx vs B traces measured at the fixed microwave power which corresponds
to a microwave electric field, EAC = 146 V/m at different temperatures. All traces in Fig. 4.3 are
vertically offset for clarity. In Fig. 4.3 we observe well-developed fractional MIROs appearing at
1
1
1
εac = , εac = and εac = . At EAC = 146 V/m the fractional MIROs have increased amplitude in
2
3
4
comparison to that for EAC = 119.6 V/m. In Fig. 4.3 only the lowest temperature (T = 2.27K) trace
shows integer MIRO around εac = 1. The amplitude is reduced for EAC = 146 V/m in comparison
to that for EAC = 119.6 V/m. We suggest that with increasing power multiphoton processes become
more important and dominate over 1-photon processes. Nevertheless the presence of integer MIRO
around εac = 1 for lowest temperature even at highest power indicates that 1-photon processes are
still present. With increasing temperature the amplitude of the fractional MIROs start to decrease
for highest power as shown in Fig. 4.3. However all three fractional MIROs still continue to persist
even at the highest temperature (T = 14K) trace which indicates that the microwave induced
multiphoton processes are still present and overcome thermally excited phonon contirbtion at these
higher temperature. At the highest microwave power, all three fractional MIRO maxima shift also
slightly to higher magnetic field.
Now we analyze the temperature evolution of fractional MIROs measured at three different
microwave powers shown in Fig. 4.1, 4.2 and 4.3. As mentioned earlier in Chapter 2 the correction
to the resistivity due to displacement model and inelastic model differ in their predicted temperature
dependence. Using Eqn. 2.1 and 2.5 we can express the MIRO photoresistivity due to these
67
Figure 4.2: Resistivity ρxx vs magnetic field B at different temperatures EAC = 119.6 V/m.
Vertical dotted lines mark the cyclotron resonance, its second harmonic and subharmonics.
All traces are vertically offset for clarity.
68
Figure 4.3: Resistivity ρxx vs magnetic field B at different temperatures for EAC = 146
V/m. Vertical dotted lines mark the cyclotron resonance, its second harmonic and subharmonics. All traces are vertically offset for clarity.
69
competing models in general as:
δρi = −4πρ0 Pω0
τi
εac λ2 sin(2πεac ),
τtr
(4.1)
im for displacement model and
where λ2 = exp(−2 ωπεc ac
τq ) is the Dingle factor squared, τi = 3τq
τi = τin for inelastic model. While the τqim is mostly temperature independent on the other hand
τin ' EF T −2 is inversely proportional to temperature squared. Regardless of the model one would
expect the normalized MIRO amplitudes
δρ
to scale with λ2 = exp(−2 ωπεc ac
τq ). From Eqn. 2.8 we
εac
also recall that the quantum scattering time τq contains a temperature dependent electron-electron
1
1
1
1
1
T2
and
can
be
written
as
=
+
[43]
[44][45],
with
'
κ
[42].
τqee
τq
τqim
τqee
τqee
εF
δρ
Therefore we expect the normalized MIRO amplitudes
to be expressed as function of T 2 by a
εac
interaction term
simple expression
2πεac
δρ
∝ A exp(−
) ' A exp(−αT 2 ),
εac
ωc τq
where α =
2πκ
ωεF
εac and A = −4πρ0 Pω0
(4.2)
τi
sin(2πεac ). For displacement mechanism, the prefτtr
actor A is mostly temperature independent, i.e. A = const(T ). On the other hand, for inelastic
mechanism, the prefactor A contains the inelastic scattering time, τi = τin which has a inverse
squared temperature dependence.
To compare our experimental results presented in Fig. 4.1, 4.2 and 4.3 with Eqn. 4.2, we
begin by extracting the normalized fractional MIRO amplitudes evaluated at the MIRO maxima
1
1
1
, εac = and εac = for different temperature. In Fig. 4.4 we plot natural Log of the
2
3
4
δρ
1
normalized fractional MIRO amplitude, ln ( ) evaluated at εac = maxima as a function of T 2
εac
2
for εac =
obtained at three different microwave powers. For all three microwave powers in Fig. 4.4 clearly
70
Figure 4.4: Natural log of the normalized fractional MIRO amplitude, ln (
δρ
) vs T 2
εac
1
evaluated at εac = maxima. Solid lines denote the fits to exp(-αT −2 ) lines for different
2
microwave powers.
Figure 4.5: Natural log of the normalized fractional MIRO amplitude, ln (
δρ
) vs T 2
εac
1
evaluated at εac = maxima. Solid lines denote the fits to exp(-αT −2 ) lines for different
3
microwave powers.
71
Figure 4.6: Natural log of the normalized fractional MIRO amplitude, ln (
δρ
) vs T 2
εac
1
maxima. Solid lines denote the fits to exp(-αT −2 ) lines for different
4
microwave powers.)
evaluated at εac =
1
monotonically decreasing as an exponential
2
δρ
function of T 2 . To confirm our observation we fit the all three ln ( ) vs T 2 data sets to a linear
εac
1
1
fitting assuming a constant exponent α for εac = . At εac = we find that normalized fractional
2
2
shows the normalized MIRO amplitudes at εac =
MIRO amplitudes for all three microwave powers clearly suggest that the temperature dependence
is exponential and is in good agreement with the Eqn. 4.2. Fig. 4.4 also indicates the prefactor
A in Eqn. 4.2 is mostly temperature independent and depends on microwave power in agreement
with displacement model.
Now we focus on the temperature dependence of the fractional MIROs at εac =
1
and compare
3
our experimental results to Eqn. 4.2. We plot natural log of the extracted normalized fractional
MIRO amplitude, ln (
δρ
1
) vs T 2 evaluated at εac =
maxima for all three different microwave
εac
3
72
Figure 4.7: Extracted exponent α vs εac plot. Dotted line presents the linear fit to α vs
εac line for an estimated κ ' 4.1 obtained for integer MIROs in Ref [39].
δρ
) vs T 2 data sets measured at three different microwave
εac
δρ
1
powers. Remarkably the ln ( ) vs T 2 data sets at εac = for all three microwave powers can
εac
3
1
also be reasonably described by Eqn. 4.2. The temperature evolution at εac = also appears to
3
powers in Fig. 4.5. We linearly fit the ln (
be in contrast to the prediction of inelastic model.
Lastly we turn our attention to the temperature evolution of fractional MIROs appearing at
1
δρ
1
. In Fig. 4.6 we present the extracted ln ( ) vs T 2 evaluated at εac =
maxima for
4
εac
4
1
all three different microwave powers. The normalized fractional MIRO amplitude at εac = also
4
δρ
roughly scales with exp(-αT 2 ). To confirm our observation we fit all three ln ( ) vs T 2 data sets
εac
δρ
to a linear function of T 2 for a constant exponent α= 0.0053. Fig. 4.6 shows that ln ( ) vs T 2
εac
1
evaluated at εac = maxima can also be described by Eqn. 4.2.
4
εac =
In Fig. 4.7 we plot the extracted exponent α vs εac to estimate the constant κ for fractional
73
MIROs from our experimental results. A previous experimental study [39] on the temperature
dependence of integer MIROs reported the linear dependence of α on εac with the constant κ
2πκ
2
εac for κ ' 4.1
estimated to be ' 4.1 Watts/K . In Fig. 4.7 the dotted line is α =
ωεF
Watts/K2 observed for integer MIROs [39]. Our temperature study on fractional MIROs doesnt
have enough points to provide an accurate estimation of κ.
To summarize our experimental results we suggest that the temperature dependence of fractional
1
1
1
MIROs at εac = , εac = and εac = for three microwave powers can be well described by
2
3
4
2πεac
δρ
' A exp(−
),
εac
ωc τq
where the quantum scattering time τq is
dependence via
1
τqee
1
τq
=
1
τqim
(4.3)
+ τ1ee , and contains an exponential quadratic-in-T
q
2
' κ TεF . Our experimental observation is in contrast to the inelastic model for
MIROs which predicts a δρ ∝ T −2 dependence. We have performed temperature dependence study
on fractional MIROs for three different powers for a wide range of temperature. The temperature
1 1
1
evolution at εac = , and shows a similar exponential quadratic-in-T dependence for all three
2 3
4
different microwave powers. However we noticed that with increasing microwave power the fractional MIROs appeared with an increased amplitude. In addition we observed that the amplitudes
of integer MIROs are also strongly suppressed with increasing microwave power.
4.2
Power dependence of Fractional MIROs
We performed our microwave power dependence experiment on a sample from a different wafer
with higher mobility compared to the sample used for earlier section in this chapter. We made a
74
Figure 4.8: Resistivity ρxx vs magnetic field B at different microwave electric field EAC .
Vertical dotted lines mark the cyclotron resonance and its subharmonics. All traces are
shifted vertically for clarity.
75
50µm wide Hall bar etched from a 29 nm wide GaAs/AlGaAs quantum well wafer with density
ne = 2.8 × 1011 cm−2 and mobility µ = 1.6 × 107 cm2 /Vs. For this study sample was irradiated
with f = 22 GHz for a wide range of microwave power at constant sample-holder temperature T '
1.5K, without dc field. Microwave power was varied using different 50 Ω impedance precise fixed
attenuators at the He-3 cryostat top. The resistivity was recorded using a standard low frequency(13
Hz) lockin technique. In Fig. 4.8 we present the magnetoresistivity ρxx vs B measured at various
microwave electric field (EAC ) inside the CPW slot. All the traces in Fig. 4.8 are labeled with
the corresponding microwave electric field and are vertically offset for clarity. The zero-EAC trace
(Fig. 4.8) shows only SdH oscillations appearing at B∼ 0.25 T. While power dependence at other
frequencies show similar result we choose to present the power dependence for f = 22GHz since
the fractional MIROs upto εac =
1
appear at a magnetic field below the onset of SdH oscillations.
4
The lowest microwave power, trace which corresponds to microwave electric field, EAC = 57.1
V/m inside the CPW slot, clearly shows strong integer MIROs appearing around εac = 1 and 2.
We also notice a weakly developed fractional MIRO appearing around εac =
1
. Upto about 167
2
V/m the integer MIRO amplitude monotonically increases with increasing EAC . In addition, the
maximum and minimum appearing on either side of εac = 1 become more prominent and sharper
as they move closer to εac = 1 with increasing microwave power showing the known tendency for
the MIRO phase to be reduced with increasing power. The integer MIRO amplitude at εac = 1
saturates for microwave electric field EAC = 167 V/m and eventually starts to decrease at higher
microwave intensities, possibly due to microwave heating. We also observe the zero field resistivity
increases with higher microwave power which confirms the increase of electron temperature due
to strong microwave radiation. Although the fractional MIRO feature near εac =
76
1
shows power
2
(N )
Figure 4.9: Overlap parameter aπ
vs Pω for differnt number of participating photons N .
dependence similar to that of εac = 1, the MIRO amplitude of εac =
EAC = 248.8 V/m. The MIRO amplitude of εac =
power. The εac =
1
saturates at a higher at
2
1
also starts to decrease at higher microwave
2
1
fractional feature starts to develop at EAC = 124.6 V/m and continues to
3
increase with increasing microwave power until it saturates for EAC = 278.5 V/m. Lastly, the
MIRO amplitude at εac =
1
starts to appear at a higher microwave power in comparison to other
4
fractional MIROs and shows a similar evolution with increasing power eventually disappearing at
the highest microwave power. Fig. 4.8 clearly suggests that the MIRO amplitudes appearing at
1 1
1
, and start to saturate at different microwave power and particularly that fractional
2 3
4
1
MIROs at εac =
require more power to saturate with increasing m.
m
εac = 1 ,
Previous studies [19] [22] [47] also reported that the photoresistivity response for integer MIROs
deviates from linear dependence with increasing microwave power and starts to saturate even under
77
modest microwave intensities. A more recent power dependence study [46] of integer MIRO at
εac = 2 over a broad range of microwave intensities also suggested the existence of two distinct
power regimes. The study found the photoresitivity response at low microwave intensities grow
linearly with increasing microwave intensity, but that above a certain intensity the photoresistivity
response becomes saturated and starts to decrease with increasing microwave intensity.
Theoretical studies [38] [20][46] developed for arbitrary microwave power in the overlapping
Landau level regime (ωc τq < π/2) predicted the importance of multiphoton processes at high
microwave intensities. In this study we compare our data with this model even though the experiment were done in the well separated Landau level regime (ωc τq > π/2). The model [38] [20][46]
is based on quantum kinetics in the presence of arbitrary microwave power and considered both
inelastic and displacement contributions for a mixed disorder system. The study [46] showed in
the presence of high microwave power the inelastic contribution is dominated by displacement contribution. It also suggested the observed deviations from linear microwave power dependence of
the MIRO amplitude originated from the multiphoton processes which require strong microwave
radiation. Theoretically, the electrons can be scattered between Floquet bands with energies E,
E ± ~ω,E ± 2~ω,..etc, which appear in the presence of high microwave radiation. As discussed
earlier in Chap. 3 (Eqn. 3.3), electrons can scatter between the Floquet bands by either absorbing
or emitting N -photons and the rate of such an N -photon scattering process is proportional the
(N )
overlap parameter aθ
p
2 [ 2P (1 − cos θ)]. Since, the Bessel function J 2 becomes negligibly
= JN
ω
N
small when its argument exceeds N , the number of photons in the dominant scattering process
√
at a certain microwave power can be expressed as N ≈ 2 Pω where the dimensionless microwave
78
Figure 4.10: Extraction of the MIRO amplitude from ρxx vs B trace, shown for εac = 1 and
1
.
2
power, Pω is defined as
Pω = Pω+ + Pω− ,
Pω± =
Pω0
,
2 +1
(ω ± ωc )2 τem
Pω0 =
(eEAC vF τem )2
.
2ef f ~2 ω 2
(N )
Fig. 4.9 shows the overlap parameter, aπ
(4.4)
plotted against arbitrary dimensionless microwave
power, Pω evaluated for different numbers of participating photons, N . Fig. 4.9 clearly shows that
(N )
the 1-photon aπ
vs Pω trace peaks at the low microwave power (Pω ), and the ranges of dominance
of higher N occur at higher Pω . To compare our experimental observation with previous studies
[19] [20] [22] [46] [47][38] we extract and analyze the power dependence of the MIRO amplitudes
79
1 1
1
Figure 4.11: MIRO amplitude (A) measured at εac = 1 , , and versus the microwave
2 3
4
electric field EAC . The solid lines represent the fits to a Gaussian distribution.
appearing at εac = 1 ,
1 1
1
,
and
(Fig. 4.8). In order to compute the MIRO amplitude more
2 3
4
accurately and also to takecare of the B-dependent background we extracted the peak-to-peak
amplitude at both the MIRO maximum (2C) and the MIRO minimum (2B) shown in Fig. 4.10.
Using the data shown in Fig. 4.8 we extract the average MIRO amplitude (A) for εac = 1 ,
and
1 1
,
2 3
1
, where A = (B + C)/2.
4
In Fig. 4.11 we present the average MIRO amplitude (A) vs EAC for εac = 1 ,
1 1
1
,
and .
2 3
4
This plot also shows peak in A occur at larger EAC for larger fractional denominator m. A vs EAC
can be fitted to Gaussians for each m although the error for m = 3 or 4 is considerable. Fig. 4.11
shows that for εac = 1 the MIRO amplitude peaks around EAC ' 200 V/m. The fractional MIRO
amplitude at εac =
1
1
1
peaks near EAC ' 240 V/m and the amplitudes for εac = and exhibit
2
3
4
peak near EAC ' 280 V/m and 320V/m respectively.
80
A previous study [46] observed a crossover from linear to sublinear microwave power dependence
of the MIRO amplitude only at εac = 2. This was explained as a transition from single photon
processes to multiphoton processes at higher microwave power. In our experimental observation
such a crossover for εac = 1 ,
1 1
1
, and would occur for low EAC . 100 V/m.
2 3
4
While the integer MIRO appearing at εac = 1 is a result of single photon process, the fractional
MIROs (m ≥2) are explained in terms of multiphoton absorption processes [25] [48] [49]. A fractional MIRO feature with denominator m is produced by m-photon processes. Fig. 4.11 shows at a
particular EAC processes involving certain number of photons will dominate the other competing
photon processes and that larger photon number succeed smaller as EAC is increased.
To summarize the experimental results of the power dependence of MIRO, the MIRO amplitude
1
1 1
of εac = 1 , , and shows similar power dependence behavior. With increasing microwave power
2 3
4
all the MIRO amplitudes deviate from linear power dependence. The MIRO amplitudes with higher
denominator m peaks at higher EAC which favors the simultaneous multiphoton absorption model
[48] [49]. We do not observe the exponential suppression fractional MIRO amplitudes predicted
by the stepwise absorption model [25]. The microwave power dependent behavior of the MIRO
amplitudes clearly suggest the number of photons in dominant processes increases with increasing
microwave power. In the chapter 3, we showed that in presence of dc electric field the nonlinear
response differential resistivity varies with microwave power due to multiphoton processes. The
study presented in this chapter further demonstrates the effect of multiphoton processes due to
high microwave power in the absence of dc electric field.
81
CHAPTER 5
CONCLUSION AND OUTLINE FOR FUTURE
RESEARCH
5.1
Conclusion
In this thesis, we have presented magnetotransport studies of 2DES under strong microwave
radiation. Our studies mainly focused on multiphoton processes to which the fractional MIROs are
ascribed.
One aspect of multiphoton process is the high power high current oscillations (HPHCO) phenomenon. We have found under strong microwave field and strong dc bias the differential resistivity
shows HPHCO around εac = 1 and
1
which appears as a result of multiphoton processes. While
2
these HPHCOs were predicted[38][52][51] and observed [20] for εac = 1 and 2, previous studies
did not predict the presence of HPHCOs around the subharmonics of cyclotron resonance. These
HPHCOs appearing at εac =
1
in our investigation under strong combined (ac + dc) excitation
2
suggest multiphoton processes are also important at subharmonics of cyclotron resonance. We
have compared our experimentally observed HPHCOs to the theoretical models [38][52][51] which
predicts such multiphoton induced addition oscillations near cyclotron resonance and its harmonics
in the regime of overlapping Landau levels. Both the theories put forward by M. Khodas et al. [38]
and by X. L. Lei et al. [52][51]. While the theory of M. Khodas can recreate our observed HPHCOs
near εac = 1 for an estimated microwave field (EAC ) inside the CPW slot, the theory of X.L. Lei et
82
al. requires a much larger EAC than our estimated EAC = 47 V/m. Around εac =
1
both theories
2
fail to generate our observed HPHCOs in magnetoresistance. We stress that both the models were
developed for the overlapping Landau level regime. At εac =
into the separated Landau level regime (ωc τq >
1
where ωc τq ≈ 3.2, we are already
2
π
).
2
The significance of multiphoton processes in 2DES is clarified by iur study of the εdc -frequency
with varying microwave power. We found that for fractional MIROs appearing at εac =
1 1
, and
2 3
1
in the presence of dc bias the number of participating photons in combined (ac + dc) transitions
4
increases consistently with increasing microwave power. Our study also suggests that the observed
change in the εdc -frequency at fractional MIROs can be interpreted in terms of the combined (ac
+ dc) transitions within the framework of the displacement mechanism. The observed maxima
in differential resistivity for combined (ac + dc) transitions can be predicted by the condition,
εdc + N εac = m, where m is a non-negative integer and N is the number of participating photons.
1
It also explains the previously observed [13] εdc -frequency at εac = . The results of our combined
2
(ac + dc) excitation studies verifies the significance of multiphoton processes at subharmonics of
cyclotron resonance and constrains any future theoretical model for high microwave power in the
separated Landau level regime.
In addition to the combined (ac + dc) excitation studies, we have also investigated the temperature and power dependence of fractional MIROs in the absence of dc bias. Our initial motivation
for these studies was to compare the displacement and inelastic mechanism, to find out which one
1
1 1
is dominant at low temperature. We found that the fractional MIROs at εac = , and exhibit
2 3
4
an exponential quadratic-in-T amplitude dependence for all three different microwave powers. The
83
T -dependence of the fractional MIROs originates from the temperature dependent electron-electron
interaction term (
1
) of the quantum scattering time. The observed exponential quadratic-in-T
τqee
dependence of MIRO amplitude is in contrast to the T −2 law predicted by the inelastic model and
also in agreement with a previous temperature study [39] on integer MIROs.
Finally, we explored the power dependence of fractional MIROs to distinguish between the
simultaneous multiphoton absorption and the stepwise absorption of single photon mechanism.
Our experimental results are in favor of simultaneous multiphoton absorption mechanism, which
predicts higher order fractional MIROs will require higher microwave power. We found that MIRO
amplitudes at εac =
1
1 1
,
and
peak at a higher EAC for higher order fractional MIROs. The
2 3
4
integer MIRO at εac = 1, which is related to 1-photon processes peaks, at a lower EAC than that
1
of εac = .
2
5.2
Outline for Future Research
The work carried out in this dissertation opens up promising directions for future research. The
CPW technique employed in this thesis to deliver microwave to the sample allows comparatively
large and accurately estimable local microwave electric field EAC at the sample. The evidence
of multiphoton processes at large microwave electric field EAC can be confirmed from the work
presented in earlier chapters. We suggest that this CPW technique can also be used to investigate
of several unresolved observations in the 2DES photoresponse to microwaves.
84
5.2.1
The study of εac = 2 feature with Bichromatic Microwave Source
Recent experimental studies [55][21] on high mobility 2DES revealed a giant microwave photoresistivity spike appearing near εac = 2. This peak was found to be an order larger in magnitude
compared to other MIRO maxima. Previous studies indicate this peak at εac = 2 is of completely
different origin than other MIROs and appears only at high microwave power. However, the reason
behind the appearance of the giant peak at εac = 2 is not been well understood. It was suggested
[55] that near ω ' 2ωc in presence of strong microwave field both dipole and quadrupole transitions
can occur and that the constructive quantum interference between the two types of transitions
might have a crucial role in the formation of the giant peak. Using etched Hallbar inside the CPW
slot we can try to generate strong enough EAC to see the giant peak at εac = 2. This will require
much higher frequency (f ∼ 40 GHz) to observe the εac = 2 peak at higher B. Frequency dependence study of the ω ' 2ωc feature with bichromatic microwave source with higher frequency range
(f ∼ 40−120 GHz), will be of interest. According to Ref [56], this may produce electromagnetically
induced transparency of the 2DES due to constructive quantum interference of the different transition processes. In addition, power dependence of the giant peak near εac = 2 will help determine
whether the feature arises due to possible multiphoton processes at strong microwave power, like
those at εac = 1 and
5.2.2
1
in our work.
2
Non-linear DC transport study to search for the presence of
Pondermotive Potential
Theoretical studies [57][58][59] by Mikhailov et al. were developed to explain microwave induced
zero-resistance states (ZRS). The model suggests the metallic contacts and any metal surface near
85
the 2DES significantly produce large gradients of microwave electric field on the 2DES. Near a thin
metal surface or contacts the local in-plane electric field can in principle be highly inhomogeneous
on the scale of cyclotron radius (Rc ). Therefore the electrons near the metal surface or contact
region experience a non-linear, time-independent ponderomotive force, Fpm (r) which is expressed
as
Fpm (r) = − 5 Upm (r),
(5.1)
where the ponderomotive potential Upm is proportional to the squared local in-plane electric field
E(r). For a metallic CPW, the electric field E(r) is nearly linearly polarized, perpendicular to
the propagation direction and can be expressed as, E(r) ≈ Ex (x). Near the metallic edge the
√
inhomogeneous electric field Ex (x) diverges as, 1/ x. In Fig. 5.1 we present the spatial profile of
the derivative of squared microwave field, dEx2 /dx, in the plane of 2DES near the edge of center line
and the ground plane of the CPW calculated with the 2DES conductivity and the host dielectric
loss neglected. Fig. 5.1 shows the large possible dEx2 /dx near the metallic edge which makes the
pondermotive force likely to be present in these 2DES.
According to the Ref [57] the maximum of the ponderomotive potential experienced by the
2DES near a metallic edge is,
max
Upm
'
2
eµEmax
32πf
(5.2)
where Emax is the amplified inhomogeneous electric field near the metallic edge which is at least
22 times larger than the incident electric field, EAC . For a strong microwave electric field of
EAC ' 500V/m with a mobility of µ ∼ 16 x 106 cm2 /Vs and radiation frequency, f = 22 GHz,
86
Figure 5.1: A representative spatial profile of the derivative of squared microwave field,
dEx2 /dx in the plane of the 2DES, in absence of 2DES. Figure on the left and right
shows the abrupt change in dEx2 /dx near the edge of the center line and the ground plane
respectively. The spatial profiles are calculated for a 150µm wide straight CPW with
100µm separation from the ground plane and 2DES depth of 190nm in presence of thin
metal.
using Eq. 5.2 the ponderomotive potential is
max
Upm
' 224 meV
(5.3)
which is at least an order larger than the Fermi energy, EF ≈ 11.4 meV for an electron density of
ns ≈ 2.85 x 1011 cm−2 . The ponderomotive potential should be able to nearly deplete the electrons
near CPW and any metal surface close to the 2DES and virtually cut off these contacts from the
Hall bar resulting in a ZRS. We would like to examine the current-voltage(I-V) characteristics of the
contacts and the Hall bar at different microwave powers to see whether the predicted ponderomotive
force induced ZRS can be seen in our high mobility 2DES.
In addition, the theory [57] suggested that the inhomogeneous nature of the in-plane electric
field can lead to generation of the subharmonic radiation(f /2) for the original frequency(f ). We
87
would like to investigate the possible subharmonic generation by 2DES and its dependence on the
depth of 2DES from the sample surface with CPW technique.
5.2.3
Experimental study of PIRO in a strong Microwave field
We would like to couple acoustic phonons with the microwave induced photons and explore
the possible effect of the combined (acoustic phonon + ac) transitions in the 2DES. Previous
experimental studies [10][11][60] [61][62][63] showed that similar to MIRO and HIRO, electrons
in 2DES can make indirect transitions between Landau levels either by absorbing or emitting
acoustic phonons, in a phenomenon known as phonon-induced resistance oscillation (PIRO). The
characteristic parameter which controls PIRO is defined as
εph =
ωH
2kF s
=
=m
ωc
ωc
(5.4)
where, 2kF is the momentum carried by the acoustic phonons, s is the sound velocity and a
resistivity maximum in PIRO occurs for an integer value of εph . Studies [10][11][60][62] revealed
that the PIRO require higher temperature, in contrast to MIRO and HIRO since the phonons are
thermal. Since the phonons are thermal, PIROs are typically observed at T ∼ 5-20 K and are
strongly suppressed for both low and high T [62]. Our temperature dependence study shows that
using the CPW technique the fractional MIROs can be observed at a much higher temperature,
T ∼15 K than before . We would like to probe the possible mixing of MIROs and PIROs. In
particular we propose checking theoretical studies [64][65] which predicted a possible new class of
resistance oscillations in 2DES controlled by εph ± εac parameter.
88
APPENDIX A
EXPERIMENTAL DETAILS
All measurement of 2DES were performed on Hall bar structures like the one shown in Fig. A.1.
Hall bar mesa structures require several steps in terms of fabrication but provide a well defined
path for current flow. In magnetotransport measurements the voltages VXX shown in Fig. A.1
is measured using a low frequency lock-in technique. The oscillator output of the lock-in is used
as a constant current (iac ) source by placing a large resistor in series. Usually a current of 1µA
or lower is used for low temperature measurements. Typically a Hall bar of width w = 20 µm or
50µm is lithographically defined and etched from a GaAs/AlGaAs quantum well. Ohmic contacts
are created from GeAu/Ni/Au alloy deposited on the contact pads and annealed at 4400 C for 10
minutes. A coplanar waveguide (CPW)[66] is then defined on the top of the mesa again by means
of optical lithography, followed by deposition of Cr/Au in a thermal evaporator. The CPW consists
of a center conductor separated from a pair of groundplanes by a distance called the slot width, s.
The width of the central conductor is 150 µm and s is 100 µm. These parameters were chosen such
that the characteristic impedance of the CPW is Z0 = 50 Ω in the absence of the 2DES. In Fig. A.1
the blue region represents the metal deposited on the sample surface in order to make the CPW.
The metal is deposited so that the Hall bar (red region) is positioned at the center of the slot.
A vector network analyzer (HP-8722D) is used as a generator for the microwave signal which
produces a nearly linearly polarized E field as shown in Fig. A.2. The contacts to the 2DES for
89
Figure A.1: A schematic diagram of the microwave circuit and the Hall bar used in the
experiment. Blue color indicates the metal film of the CPW and red color shows the Hall
bar inside the slot.
90
magnetotransport measurements are placed far away (∼ 6s) from the slots in order to isolate them
from the influence of the AC electric field. Since the AC electric field is more concentrated in
the slots of the CPW, we are able to avoid any unwanted modification in the density of 2DES
around the contact region in the presence of intense microwave radiation, which has been suggested
in a recently proposed theory in Ref.[57]. This CPW technique enables us to probe the 2DES
magnetotransport properties over a frequency range up to several tens of GHz and achieves high
EAC at the Hall bar with minimal total power applied to the sample. The voltages between the
contacts are measured using a low frequency lock-in technique. The lock-in frequency is kept low
(a few Hz) to avoid capacitive coupling between the leads. The oscillator output of the lock-in is
used as a constant current source by placing a large adjustable resistor in series. Usually a current
of 1µA or lower is used for low temperature measurements.
Since the coaxial cable lengths from both the transmitter and the receiver to the sample are
identical with this experimental setup, we are able to calculate the AC electric field in the slot
from the microwave power applied. We measure the power both at the sending (Pin ) and receiving
ports (Pout ) at the 3 He cryostat top, which enables us to calculate the roundtrip power loss due to
the coaxial cable inside the cryostat. A calibrated single channel microwave powermeter (AgilentE4418B) is used to measure the power of the microwave signal generated by the vector network
analyzer (HP-8722D). We can calculate the power incident at the sample P2DES as,
P2DES = Pin +
91
(Pout − Pin )
,
2
(A.1)
Figure A.2: A schematic diagram of the crosssection views of the CPW and the eletric
field lines inside the slot of the CPW.
from which we estimate the microwave electric field (EAC ) inside the slot as
EAC
V
=
=
s
√
where s is the slotwidth of the CPW.
92
P2DES Z0
,
s
(A.2)
BIBLIOGRAPHY
[1] A. C. Durst, S. Sachdev, N. Read, and S. M. Girvin, “Radiation-induced magnetoresistance
oscillations in a 2d electron gas,” Phys. Rev. Lett., vol. 91, p. 086803, Aug 2003.
[2] K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination
of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett., vol. 45,
pp. 494–497, Aug 1980.
[3] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the
extreme quantum limit,” Phys. Rev. Lett., vol. 48, pp. 1559–1562, May 1982.
[4] P. D. Ye, L. W. Engel, D. C. Tsui, J. A. Simmons, J. R. Wendt, G. A. Vawter, and J. L. Reno,
“Giant microwave photoresistance of two-dimensional electron gas,” Applied Physics Letters,
vol. 79, no. 14, pp. 2193–2195, 2001.
[5] M. A. Zudov, R. R. Du, J. A. Simmons, and J. L. Reno, “Shubnikov˘de haas-like oscillations
in millimeterwave photoconductivity in a high-mobility two-dimensional electron gas,” Phys.
Rev. B, vol. 64, p. 201311, Oct 2001.
[6] R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky,
“Zero-resistance states induced by electromagnetic-wave excitation in gaas/algaas heterostructures,” Nature, vol. 420, pp. 646–650, 2002.
[7] M. A. Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, “Evidence for a new dissipationless
effect in 2d electronic transport,” Phys. Rev. Lett., vol. 90, p. 046807, Jan 2003.
[8] C. L. Yang, J. Zhang, R. R. Du, J. A. Simmons, and J. L. Reno, “Zener tunneling between landau orbits in a high-mobility two-dimensional electron gas,” Phys. Rev. Lett., vol. 89, p. 076801,
Jul 2002.
[9] W. Zhang, H.-S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Magnetotransport in
a two-dimensional electron system in dc electric fields,” Phys. Rev. B, vol. 75, p. 041304, Jan
2007.
[10] M. A. Zudov, I. V. Ponomarev, A. L. Efros, R. R. Du, J. A. Simmons, and J. L. Reno, “New
class of magnetoresistance oscillations: Interaction of a two-dimensional electron gas with leaky
interface phonons,” Phys. Rev. Lett., vol. 86, pp. 3614–3617, Apr 2001.
[11] C. Yang, M. Zudov, J. Zhang, R. Du, J. Simmons, and J. Reno, “Magnetophonon resonance of
two-dimensional electrons by leaky interface-acoustic phonons,” Physica E: Low-dimensional
Systems and Nanostructures, vol. 12, no. 14, pp. 443 – 445, 2002. Proceedings of the Fourteenth
International Conference on the Ele ctronic Properties of Two-Dimensional Systems.
93
[12] W. Zhang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Resistance oscillations in twodimensional electron systems induced by both ac and dc fields,” Phys. Rev. Lett., vol. 98,
p. 106804, Mar 2007.
[13] A. T. Hatke, H.-S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Nonlinear magnetotransport in microwave-illuminated two-dimensional electron systems,” Phys. Rev. B, vol. 77,
p. 201304, May 2008.
[14] A. T. Hatke, H.-S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Microwave photoresistance in dc-driven 2d systems at cyclotron resonance subharmonics,” Phys. Rev. Lett.,
vol. 101, p. 246811, Dec 2008.
[15] T. Ihn, Semiconductor Nanostructures: Quantum states and electronic transport. Semiconductor Nanostructures: Quantum States and Electronic Transport, OUP Oxford, 2009.
[16] W. De Lange, F. Blom, and J. Wolter, “Effective electron mass in gaas/alxga1-xas heterostructures under hydrostatic pressure,” Semiconductor science and technology, vol. 8, no. 3, p. 341,
1993.
[17] L. Shubnikov and W. de Haas Leiden Communication, vol. 207a, no. 3, 1930.
[18] M. A. Zudov, “Period and phase of microwave-induced resistance oscillations and zeroresistance states in two-dimensional electron systems,” Phys. Rev. B, vol. 69, p. 041304, Jan
2004.
[19] R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky,
“Demonstration of a 1/4-cycle phase shift in the radiation-induced oscillatory magnetoresistance in gaas/algaas devices,” Phys. Rev. Lett., vol. 92, p. 146801, Apr 2004.
[20] M. Khodas, H. S. Chiang, A. T. Hatke, M. A. Zudov, M. G. Vavilov, L. N. Pfeiffer, and
K. W. West, “Nonlinear magnetoresistance oscillations in intensely irradiated two-dimensional
electron systems induced by multiphoton processes,” Phys. Rev. Lett., vol. 104, p. 206801,
May 2010.
[21] A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Microwave photoresistance in a
two-dimensional electron gas with separated landau levels,” Phys. Rev. B, vol. 84, p. 241304,
Dec 2011.
[22] R. L. Willett, L. N. Pfeiffer, and K. W. West, “Evidence for current-flow anomalies in the
irradiated 2d electron system at small magnetic fields,” Phys. Rev. Lett., vol. 93, p. 026804,
Jul 2004.
[23] S. I. Dorozhkin, J. H. Smet, K. von Klitzing, L. N. Pfeiffer, and K. W. West, “Microwave
induced magnetoresistance oscillations at the subharmonics of the cyclotron resonance,” Soviet
Journal of Experimental and Theoretical Physics Letters, vol. 86, pp. 543–547, Dec. 2007.
94
[24] I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, “Theory of fractional microwave-induced
resistance oscillations,” Phys. Rev. Lett., vol. 99, p. 206805, Nov 2007.
[25] I. Pechenezhski, S. Dorozhkin, and I. Dmitriev, “Fractional features in radiation-induced oscillations of the magnetoresistance of two-dimensional electron systems,” JETP Letters, vol. 85,
no. 1, pp. 86–91, 2007.
[26] M. G. Vavilov, I. L. Aleiner, and L. I. Glazman, “Nonlinear resistivity of a two-dimensional
electron gas in a magnetic field,” Phys. Rev. B, vol. 76, p. 115331, Sep 2007.
[27] M. A. Zudov, R. R. Du, L. N. Pfeiffer, and K. W. West, “Multiphoton processes in microwave
photoresistance of two-dimensional electron systems,” Phys. Rev. B, vol. 73, p. 041303, Jan
2006.
[28] W. Zhang, H.-S. Chiang, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Magnetotransport in
a two-dimensional electron system in dc electric fields,” Phys. Rev. B, vol. 75, p. 041304, Jan
2007.
[29] V. Ryzhii, “Photoconductivity characteristics in thin films subjected to crossed electric and
magnetic fields,” Soviet Physics Solid State, vol. 11, p. 2078, 1970.
[30] V. Ryzhii, R. A. Suris, and B. S. Shchamkhalova, “Photoconductivity characteristics in thin
films subjected to crossed electric and magnetic fields,” Soviet Physics Solid State, vol. 20,
p. 1299, 1986.
[31] I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, “Cyclotron-resonance harmonics in the ac
response of a 2d electron gas with smooth disorder,” Phys. Rev. Lett., vol. 91, p. 226802, Nov
2003.
[32] M. G. Vavilov and I. L. Aleiner, “Magnetotransport in a two-dimensional electron gas at large
filling factors,” Phys. Rev. B, vol. 69, p. 035303, Jan 2004.
[33] I. A. Dmitriev, M. G. Vavilov, I. L. Aleiner, A. D. Mirlin, and D. G. Polyakov, “Theory of
microwave-induced oscillations in the magnetoconductivity of a two-dimensional electron gas,”
Phys. Rev. B, vol. 71, p. 115316, Mar 2005.
[34] X. L. Lei and S. Y. Liu, “Radiation-induced magnetoresistance oscillation in a two-dimensional
electron gas in faraday geometry,” Phys. Rev. Lett., vol. 91, p. 226805, Nov 2003.
[35] J. Iñarrea and G. Platero, “Temperature effects on microwave-induced resistivity oscillations and zero-resistance states in two-dimensional electron systems,” Phys. Rev. B, vol. 72,
p. 193414, Nov 2005.
[36] J. Iñarrea and G. Platero, “Theoretical approach to microwave-radiation-induced zeroresistance states in 2d electron systems,” Phys. Rev. Lett., vol. 94, p. 016806, Jan 2005.
95
[37] A. A. Koulakov and M. E. Raikh, “Classical model for the negative dc conductivity of ac-driven
two-dimensional electrons near the cyclotron resonance,” Phys. Rev. B, vol. 68, p. 115324, Sep
2003.
[38] M. Khodas and M. G. Vavilov, “Effect of microwave radiation on the nonlinear resistivity of
a two-dimensional electron gas at large filling factors,” Phys. Rev. B, vol. 78, p. 245319, Dec
2008.
[39] A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Temperature dependence of
microwave photoresistance in 2d electron systems,” Phys. Rev. Lett., vol. 102, p. 066804, Feb
2009.
[40] I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, “Microwave photoconductivity of a twodimensional electron gas: Mechanisms and their interplay at high radiation power,” Phys. Rev.
B, vol. 75, p. 245320, Jun 2007.
[41] S. A. Studenikin, A. S. Sachrajda, J. A. Gupta, Z. R. Wasilewski, O. M. Fedorych,
M. Byszewski, D. K. Maude, M. Potemski, M. Hilke, K. W. West, and L. N. Pfeiffer,
“Frequency quenching of microwave-induced resistance oscillations in a high-mobility twodimensional electron gas,” Phys. Rev. B, vol. 76, p. 165321, Oct 2007.
[42] G. F. Giuliani and J. J. Quinn, “Lifetime of a quasiparticle in a two-dimensional electron gas,”
Phys. Rev. B, vol. 26, pp. 4421–4428, Oct 1982.
[43] S. Q. Murphy, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Lifetime of two-dimensional
electrons measured by tunneling spectroscopy,” Phys. Rev. B, vol. 52, pp. 14825–14828, Nov
1995.
[44] J. Eisenstein, D. Syphers, L. Pfeiffer, and K. West, “Quantum lifetime of two-dimensional
holes,” Solid State Communications, vol. 143, no. 89, pp. 365 – 368, 2007.
[45] N. C. Mamani, G. M. Gusev, T. E. Lamas, A. K. Bakarov, and O. E. Raichev, “Resonance
oscillations of magnetoresistance in double quantum wells,” Phys. Rev. B, vol. 77, p. 205327,
May 2008.
[46] A. T. Hatke, M. Khodas, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Multiphoton
microwave photoresistance in a high-mobility two-dimensional electron gas,” Phys. Rev. B,
vol. 84, p. 241302, Dec 2011.
[47] S. Studenikin, M. Potemski, P. Coleridge, A. Sachrajda, and Z. Wasilewski, “Microwave radiation induced magneto-oscillations in the longitudinal and transverse resistance of a twodimensional electron gas,” Solid State Communications, vol. 129, no. 5, pp. 341 – 345, 2004.
[48] I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, “Theory of fractional microwave-induced
resistance oscillations,” Phys. Rev. Lett., vol. 99, p. 206805, Nov 2007.
96
[49] X. L. Lei and S. Y. Liu, “Multiple and virtual photon processes in radiation-induced magnetoresistance oscillations in two-dimensional electron systems,” Applied Physics Letters, vol. 88,
no. 21, pp. –, 2006.
[50] X. L. Lei, “Nonlinear theory of fractional microwave-induced magnetoresistance oscillations in
a dc-driven two-dimensional electron system,” Phys. Rev. B, vol. 79, p. 115308, Mar 2009.
[51] X. Lei, “Magnetoresistance oscillations in high-mobility two-dimensional semiconductors: A
unified description with balance-equation model,” Materials Science and Engineering: R: Reports, vol. 70, no. 36, pp. 126 – 150, 2010.
[52] X. L. Lei and S. Y. Liu, “Multiphoton-induced nonlinear magnetoresistance oscillations in a
dc-driven two-dimensional electron system irradiated by intense microwaves,” Phys. Rev. B,
vol. 84, p. 035321, Jul 2011.
[53] P. K. Tien and J. P. Gordon, “Multiphoton process observed in the interaction of microwave
fields with the tunneling between superconductor films,” Phys. Rev., vol. 129, pp. 647–651,
Jan 1963.
[54] S. Wiedmann, G. M. Gusev, O. E. Raichev, A. K. Bakarov, and J. C. Portal, “High-order
fractional microwave-induced resistance oscillations in two-dimensional systems,” Phys. Rev.
B, vol. 80, p. 035317, Jul 2009.
[55] Y. Dai, R. R. Du, L. N. Pfeiffer, and K. W. West, “Observation of a cyclotron harmonic spike in
microwave-induced resistances in ultraclean GaAs/AlGaAs quantum wells,” Phys. Rev. Lett.,
vol. 105, p. 246802, Dec 2010.
[56] D.-H. Lee and J. M. Leinaas, “Role of interference in millimeter-wave-driven dc transport in
a two-dimensional electron gas,” Phys. Rev. B, vol. 69, p. 115336, Mar 2004.
[57] S. A. Mikhailov, “Microwave-induced zero-resistance states and second-harmonic generation
in an ultraclean two-dimensional electron gas,” Phys. Rev. B, vol. 89, p. 045410, Jan 2014.
[58] S. A. Mikhailov, “Theory of microwave-induced zero-resistance states in two-dimensional electron systems,” Phys. Rev. B, vol. 83, p. 155303, Apr 2011.
[59] S. A. Mikhailov and N. A. Savostianova, “Influence of contacts on the microwave response of
a two-dimensional electron stripe,” Phys. Rev. B, vol. 74, p. 045325, Jul 2006.
[60] A. A. Bykov, A. V. Goran, and S. A. Vitkalov, “Interference of magneto-intersubband and
phonon-induced resistance oscillations in single gaas quantum wells with two populated subbands,” Phys. Rev. B, vol. 81, p. 155322, Apr 2010.
[61] W. Zhang, M. Zudov, L. N. Pfeiffer, and K. W. West, “Resonant phonon scattering in quantum
hall systems driven by dc electric fields,” Phys. Rev. Lett., vol. 100, p. 036805, Jan 2008.
97
[62] A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Phonon-induced resistance
oscillations in 2d systems with a very high electron mobility,” Phys. Rev. Lett., vol. 102,
p. 086808, Feb 2009.
[63] A. T. Hatke, M. A. Zudov, L. N. Pfeiffer, and K. W. West, “Phase of phonon-induced resistance
oscillations in a high-mobility two-dimensional electron gas,” Phys. Rev. B, vol. 84, p. 121301,
Sep 2011.
[64] O. E. Raichev, “Microwave-induced magnetoresistance of two-dimensional electrons interacting
with acoustic phonons,” Phys. Rev. B, vol. 81, p. 165319, Apr 2010.
[65] O. E. Raichev, “Magnetoresistance oscillations in two-subband electron systems: Influence of
electron-phonon interaction,” Phys. Rev. B, vol. 81, p. 195301, May 2010.
[66] C. Wen, “Coplanar waveguide, a surface strip transmission line suitable for nonreciprocal
gyromagnetic device applications,” pp. 110–115, May 1969.
98
BIOGRAPHICAL SKETCH
Shantanu Chakraborty was born in Dhaka, Bangladesh. He completed his Bachelor of Science
degree in Physics and Master of Science degree in Theoretical Physics from University of Dhaka.
Currently he is enrolled into the Physics doctoral studies program at Florida state university. He is
working under the supervision of Dr Lloyd Engel at the NHMFL, Tallahassee, Florida. His area of
research interest includes the microwave spectroscopy of two dimensional electron systems at low
temperature.
99
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