experimental studies of electron transport and …we have investigated the electron transport...

EXPERIMENTAL STUDIES OF ELECTRON TRANSPORT AND

THERMOPOWER IN STRONGLY CORRELATED

TWO-DIMENSIONAL ELECTRON SYSTEMS

A dissertation presented

by

Anish Mokashi

to

The Department of Physics

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Physics

Northeastern University

Boston, Massachusetts

December, 2011

1

EXPERIMENTAL STUDIES OF ELECTRON TRANSPORT AND

THERMOPOWER IN STRONGLY CORRELATED

TWO-DIMENSIONAL ELECTRON SYSTEMS

by

Anish Mokashi

ABSTRACT OF DISSERTATION

Submitted in partial fulfillment of the requirement

for the degree of Doctor of Philosophy in Physics

in the Graduate School of Northeastern University

December, 2011

2

Abstract

The discovery of the Metal-Insulator Transition at B = 0 in strongly correlated two-

dimensional electron systems with low disorder has lead to investigations into related phe-

nomena in this regime of significant electron-electron interactions. Diverging spin suscep-

tibility has been observed in parallel magnetic fields which was traced to an enhancement

of the effective mass (albeit to values only up to 4 times band mass of electrons in Si.) at

low electron densities corresponding to strong electron-electron interactions. Below a certain

density nχ close to the critical density of the MIT, the electron spins are spontaneously

polarized which is interpreted as a phase transition to either a Wigner crystal or a ferromag-

netic electron liquid.

We have performed experiments to measure the diffusion thermopower in low disordered

Si-MOSFETs with high electron mobilities. The measured values of thermopower are ob-

served to diverge at a particular disorder-independent electron density, nt. The thermopower

is linear with temperature, consistent with the Mott formula for diffusion thermopower. The

effective mass values are seen to be enhanced up to 25 times the band mass as the density

nt is approached.

The two-parameter (disorder and interactions) scaling theory by Punnoose and Finkel’stein

accurately describes the metallic behavior near the MIT without any fitting parameters. We

have extended the earlier results to even lower temperatures and we observe that once the

effects of changes in the valley degeneracy due to splitting and intervalley scattering are

taken into account, the two-parameter theory still provides accurate predictions.

We have investigated the electron transport properties of strongly correlated 2D systems at

temperatures of the order of the Fermi temperature and we have found qualitative agreement

with the analogy with hydrodynamics of liquid He relating the viscosity to the resistivity.

3

Acknowledgements

I thank my advisor Prof. Sergey Kravchenko. It was absolutely wonderful and inspiring to do

the experiments because of his spirited ingenuity and great sense of troubleshooting complex

situations with elegant clarity and directness. I would also like to thank Prof. Alexander

Shashkin who was a visiting scientist in our lab. I thank Prof. Myriam Sarachik and my

colleagues from her group: Shiqi Li, Bo Wen and Lukas Zhao. I thank Prof. Donald Heiman

for his encouragement and his patience while we were working in his lab and significant help

from time to time. I thank Prof. Latika Menon for her cheerful attitude and for helping us

with the graphene experiments with Dr. Adam Friedman. Prof. Jeffrey Sokoloff’s introduc-

tory course on Condensed Matter Physics was very helpful and I thank him for making the

advanced course exciting and enjoyable. I would also like to thank Prof. Yogendra Srivastava,

Prof. Alain Karma, Prof. Pran Nath and Prof. Timothy Sage for their beautiful courses. I

thank Tim Hussey for his great sense of humor and his wizardry with instruments (in his

own words - ‘making the crappiest piece of equipment work’).

I thank my colleagues for their support and their amazing friendships: Tanmoy Das, Ashenafi

Dadi, Gina Escobar, Baris Altunkaynak, Susmita Basak, Siddhartha Mal, Pui Yin Pang,

Yung-Jui Wang, David Drosdoff, Svetlana Anissimova, Thayaparan Paramanathan, Fei Wang,

Eugen Panaitescu, Daniel Feldman, Evin Gultepe, Adam Friedman, Hasnain Hafiz, Arda

Halu, Pradeep Murugesan, Anup Singh and others.

I thank Prof. George Alverson for his help when he was the Graduate Advisor. I thank

Prof. Mark Williams for his course on experimental physics and for making it possible for

me to do a Teaching Assistantship in the final stages of my work. I also thank Suzanne

4

Robblee, Chantal Cardona, Alina Mak and Moki Smith for their help and assistance. I thank

Thomas Hamrick from the Introductory Physics Laboratory and Prof. Oleg Batishchev for

making the teaching duties a very enjoyable experience.

I thank my professors from IIT Bombay: Prof. Sharad Patil, Prof. P. Ramadevi, Prof. Alok

Shukla, Prof. Kushal Deb and Prof. Ranjeev Misra from IUCAA Pune.

I thank all my friends and dear ones from everywhere.

I thank Suryasarathi Dasgupta, Vinay Bhat, Tathagata Sengupta, Tejaswini Madabhushi,

Manish Verma, Preeti Rao, Aman, Dhritiman Nandan, Aditee Dalvi, Priyanka Dalvi, Prashant

Sable, Chinmay Belthangady, Tenzin Tsephel, Preetha Mahadevan, Rebecca Albrecht, Nidhi

Agarwal, Melliyal Annamalai, Umang Kumar, Kavita Sukerkar, J C Prasad, Aravind Prasad,

Somnath Mukherji, Mona Mandal, Adrita and all her friends for making Boston special.

I thank my parents - Suniti and Avinash and my grandmother, Inni.

I thank my partner Junuka for all the beautiful times gone by and for those to come.

Anish Mokashi

Dec 2011

5

Contents

Abstract 2

Acknowledgements 4

Table of Contents 6

1 Introduction and Experimental Set-up 9

1.1 Low Temperature Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Metal-Insulator Transition in Two Dimensions . . . . . . . . . . . . . . . . . 10

1.3.1 Localization in two dimensions . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Discovery of the Metal-Insulator Transition in Strongly Correlated 2DES 12

1.4 Effect of Parallel Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Diverging Spin Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Dilution Refrigerator and Superconducting Magnet . . . . . . . . . . . . . . 18

1.7 Si-MOSFET devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7.1 Split Gates in Si-MOSFETs . . . . . . . . . . . . . . . . . . . . . . . 25

1.8 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Thermopower and Divergence of Effective Mass 28

6

CONTENTS

2.1 Diverging χ and m∗ with decreasing ns in strongly correlated 2DES . . . . . 28

2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Thermopower results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Comparison with thermopower in high disorder systems . . . . . . . . . . . . 40

2.5 Enhanced effective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Effect of Parallel Magnetic Fields on Thermopower signal . . . . . . . . . . . 43

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Extension of Scaling Theory to include Valley Effects 46

3.1 Two valleys in the [100] direction in Si conduction band . . . . . . . . . . . . 47

3.2 Predictions of the Two-parameter Scaling Theory . . . . . . . . . . . . . . . 48

3.2.1 Main results from theory and experimental verification . . . . . . . . 48

3.2.2 Extention to lower temperatures . . . . . . . . . . . . . . . . . . . . . 50

3.3 Exact agreement with predicted crossover without fitting parameters . . . . 53

3.4 Comparison of theory and experiment considering the changing valley degen-

eracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Transport in Strongly Correlated Regime at intermediate temperatures 58

4.1 Broad range of temperatures unexplored . . . . . . . . . . . . . . . . . . . . 58

4.2 Accessing the regime of strong correlations . . . . . . . . . . . . . . . . . . . 60

4.3 System two-dimensional even at high temperatures . . . . . . . . . . . . . . 61

4.4 Temperature dependence of resistivity for T > TF . . . . . . . . . . . . . . . 62

4.4.1 Disorder as an effective medium for hydrodynamic flow of the electron

liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.2 Two temperature regimes for strong correlations above TF . . . . . . 64

4.4.3 Resistivity beyond Tph . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7

CONTENTS

4.5 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6.1 ρxx versus T at B = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6.2 Magnetoresistance in parallel field . . . . . . . . . . . . . . . . . . . . 82

4.6.3 Results in perpendicular field . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Bibliography 94

8

Chapter 1

Introduction and Experimental Set-up

1.1 Low Temperature Physics

Dr. D. K. C. MacDonald in his book “Near Zero: An Introduction to Low Temperature

Physics” [1] gives the following analogy as a motivation for studying low temperature phe-

nomena. He asks the reader to imagine looking at a tree from a window. If there is a strong

wind blowing, we know it is a tree all right, but we cannot tell much about the shape of the

leaves and so on, even if we try to look through a telescope. But when everything is calm

and the tree is still, we can see the leaves easily and if we used a telescope, we could even

pick out the veins and other details. Similarly by using low temperatures to increase the

order, we can examine the finer details of matter.

1.2 Quantum Phase Transitions

From our everyday life experience, we are familiar with phase transitions such as water

freezing into ice. A crude description of this would be: decreasing the temperature of water

9

1.3. METAL-INSULATOR TRANSITION IN TWO DIMENSIONS

to 0C progressively reduces the thermal fluctuations of the molecules to a point at which

they are not enough to overcome the attractive forces and the increased order manifests in a

transition to the energetically/entropically favored crystalline ice phase. Pressure, magnetic

fields are among other variables that induce phase transitions just like temperature does.

However, it has been argued that there exist phase transitions that occur at absolute zero,

i.e. in the complete absence of thermal fluctuations [2]. Heisenberg’s Uncertainty Principle

would forbid matter becoming completely ‘still’ even at absolute zero (Zero Point Energy).

Furthermore the particles interact with each other, have either Bose-Einstein or Fermi-Dirac

statistics and there also exist crucial many-body effects. Tuning these quantum fluctuations

by changing certain physical parameters (obviously other than temperature) effectively gives

rise to Quantum Phase Transitions (QPT) at 0K.

The effects of a QPT are expected to extend to experimentally accessible finite tempera-

tures approximately up to the region where the quantum fluctuations, ~ω and the thermal

fluctuations, kBT become comparable. These effects are especially important as the ordering

and properties of the ground state of the phases at absolute zero determine the nature of

the excitations/quasiparticles through which the particles interact with each other.

1.3 Metal-Insulator Transition in Two Dimensions

1.3.1 Localization in two dimensions

The presence of impurities in crystalline structures are a source of disorder. In Si-MOSFETs,

for example, there exist imperfections in the crystal structure or charge traps in the oxide

and at the Si− SiO2 interface created during the process of growing the oxide on the bulk

10


Figure 1.1: A conceptual sketch of Metals with itinerant electrons and Insulators in whichthe delocalized electrons become localized around ions. From Ref.[3].

semiconductor and while growing the semiconductor crystal itself, that give rise to random

potential fluctuations.

As the temperature is decreased, the various modes for electrons to exchange energy with

their environment such as lattice vibrations/phonon scattering cease. Coherent backscat-

tering from impurities makes the electron wavefunctions scatter back to their origin and

interfere constructively giving rise to localized states and so the electrons cannot diffuse

away. This phenomenon is called weak localization.

The scaling theory of localization [4] predicts that electron wavefunctions in 2D are al-

ways localized and that there can exist no metallic states in 2D. It was an accepted notion

for many decades that when electrons in two dimensions are cooled to absolute zero, the

resistance increases invariably and all the experimental evidence confirmed this prediction

till the discovery of the Metal-Insulator Transition in 2D.

11


Figure 1.2: Predictions of the Scaling theory of localization for non-interacting electrons;adapted from Ref.[4].

1.3.2 Discovery of the Metal-Insulator Transition in Strongly Correlated 2DES

The discovery of the Metal-Insulator Transition (MIT) (Fig 1.3) at zero magnetic fields in the

two-dimensional electron systems (2DES) of high mobility “clean” samples of Si-MOSFETs

[5][6] (later also observed in 2DES of other devices) opened up further investigations into

the physics of strongly interacting electrons in two dimensions. The discovery of a metallic

state in two dimensions was initially met with skepticism as it violated the scaling theory

of localization [4] which rules out the presence of metallic state in two dimensions due to

weak localization of electrons as a result of quantum interference. Progress in semiconductor

fabrication made available samples in which the electrons had higher mobilities, making it

possible to access the regime of low electron densities that was not accessible in high disor-

dered samples due to Anderson localization. This regime is also that of strong interactions.

Fig 1.4 illustrates the fact that because the Fermi energy is proportional to the electron

12

1.4. EFFECT OF PARALLEL MAGNETIC FIELD

density, ns and the Coulomb energy to n1/2s , decreasing ns beyond a point makes it possible

to access the strongly interacting regime. It is because of the significantly stronger electron-

electron interactions (which the scaling theory did not account for) in such a regime that a

metallic state was observed.

The strength of interactions is given by the interaction parameter (the dimensionless Wigner-

Seitz radius)

rs =ECEF

=1

(πns)1/2aB

where aB(∝ εr/mb) is the Bohr radius in a semiconductor, εr and mb being the relative per-

mittivity and the electron band mass respectively. The values for εr and mb in Si-MOSFETs

(plus the presence of two degenerate valleys) lead to a larger value of rs compared to other

structures.

There exists a separatrix (shown in red in Fig 1.3) between the metallic and insulating states

with a critical density nc, at which the resistivity (∼ 3h/e2) is independent of temperature

at low temperatures.

1.4 Effect of Parallel Magnetic Field

On applying a parallel in-plane magnetic field B‖, the MIT disappears (insulating behavior

seen for all ns) and a huge magnetoresistance is observed (Fig 1.5) that was established to

be due to coupling between the electron spins and the parallel field. The magnetoresistance

is observed to saturate at a particular parallel field Bc (or B∗) corresponding to complete

spin polarization, depending on the electron density beyond which it remains constant. The

critical field was observed to be proportional to the deviation of the electron density from a

disorder independent density value, nχ close to the critical density nc [9] as can be clearly

13

1.4. EFFECT OF PARALLEL MAGNETIC FIELD

0 2 4 6 8

T (K)

10−1

100

101

102

103

104

ρ (h/e

2)

ns=7.12x10

10 cm

−2 ....... 13.7x10

10 cm

−2

Figure 1.3: Temperature dependence of the B = 0 resistivity in a dilute low-disordered SiMOSFET for 30 different electron densities ranging from 7.12×1010 cm−2 to 13.7×1010 cm−2;adapted from Ref.[7].

14

1.5. DIVERGING SPIN SUSCEPTIBILITY

EF

EC

Electron density

EF,E

C

Figure 1.4: Strongly interacting regime EC EF at low electron densities

seen from the plot of fields corresponding to complete spin polarization extrapolating to zero

at nχ (Fig 1.6). i.e., B∗ ∝ (ns − nχ) and below nχ, the spins are spontaneously polarized.

The spontaneous spin polarization below nχ can be interpreted as either being due to a

transition to a Wigner crystal or to a strongly interacting ferromagnetic liquid (Suggested

Phase diagrams in Fig. 1.7).

1.5 Diverging Spin Susceptibility

The dashed line in Fig 1.6 shows the expected dependence for non-interacting electrons where

the critical field for complete spin polarization is proportional to the electron density.

B∗ =π~2nsµBg0mb

where g0 and mb (mb = 0.19me for the conduction band valleys of Si in the [100] direction)

are the band values of the Lande g-factor and the effective mass of the electrons and µB is

the Bohr magneton.

15


0.01

0.1

1

10

100

0 2 4 6 8 10 12

1.55

1.60

1.65

1.70

1.80

2.0

2.2

2.6

ρ (

h/e

2)

Magnetic Field (T)

B||T = 0.29 K

(a)

Figure 1.5: Resistivity versus parallel magnetic field measured at T = 0.29 K in a Si MOS-FET shown at different densities. Adapted from Ref.[8]

0

1

2

3

4

5

6

7

0 2 4 6 8 10

µBB

* (m

eV

)

ns (10

11 cm

-2)

nχ

Figure 1.6: Magnetic field corresponding to full spin polarization as a function of electrondensity. Adapted from Ref.[9]

16


Figure 1.7: Phase diagrams suggested for the Transition

17

1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET

Thus, the measured values of B∗ are smaller than the expected ones for the non-interacting

case. From the expression for B∗ we can see that the mass m and/or the g-factor g in the

denominator have to be larger than the non-interacting values. Fermi liquid theory predicts

that electron-electron interactions enhance the effective mass and g-factor to renormalized

values m∗ and g∗. Combining this with the fact that the spin susceptibility χ = d∆ns/dB∗,

we can write an expression for the renormalized spin susceptibility,

χ

χ0

=g∗m∗

g0mb

The diverging behavior of spin susceptibility indicates that the system is nearing a phase

transition (Fig. 1.8).

1.6 Dilution Refrigerator and Superconducting Magnet

Dilution Refrigerator

The dilution refrigerator works on the principle that when a correctly chosen mixture of 3He

and 4He is cooled below 0.86K, it separates into two phases. One of these phases (phase 1)

has a greater proportion of 3He than the other (phase 2). The enthalpy of 3He is different

in the two phases. Cooling can be achieved by allowing 3He from phase 1 to “evaporate”

into phase 2. (The 3He in the concentrated phase can be considered to be ‘liquid’ and those

in the dilute phase to be a ‘gas’ - since it does not interact with the 4He liquid) This cooling

occurs in the mixing chamber of the dilution refrigerator - which is in thermal contact with

the experimental space (the ‘cold finger’ of the fridge).

18


0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.50

2

4

6

8

10

µBB

c (

meV

)

Bc (

tesla

)

nχ

nc

ns (10

11 cm

-2)

1

2

3

4

5

6

7

0.5 1 1.5 2 2.5 3 3.5 4

χ/χ

0

ns (10

11 cm

-2)

nc

Figure 1.8: The Pauli spin susceptibility as a function of electron density obtained by thermo-dynamic methods: direct measurements of the spin magnetization (dashed line), dµ/dB = 0(circles), and density of states (squares). The dotted line is a guide to the eye. Also shownby a solid line is the transport data of Ref.[10]. Inset: Field for full spin polarization as afunction of the electron density determined from measurements of the magnetization (circles)and magnetocapacitance (squares). The data for Bc are consistent with a linear fit whichextrapolates to a density nχ close to the critical density nc for the B = 0 MIT. Adaptedfrom [11]

.

19


Since the proportions of 3He in the two phases have to be maintained constant in the

continuous operation of the fridge, the extra 3He in the dilute phase 2 has to be removed

and restored to phase 1. This is made possible by pumping on the liquid surface in the ‘still’

which is maintained at around 0.6 to 0.7K because at this temperature, the vapor pressure

of 3He is 1000 times that of 4He - so it is almost only 3He that is evaporating. The reduced

concentration of 3He in the still, makes the 3He from phase 2 to flow to the still - thus

balancing the extra molecules that have “evaporated” into it from phase 1.

The mixture has to be condensed before the continuous cycle can start. The ‘1K pot’

which draws 4He gas from the main bath is connected to a separate pump and it reaches

a temperature of about 1.2K. A flow impedance is used to get a high enough pressure in

the 1K pot region to allow the mixture to condense. The vapor pressure of the liquid in the

still side is then reduced by the other pump, which makes it go below 1.2K. This gradually

lowers the temperature of the rest of the system up to the phase separation point (0.86K)

beyond which the continuous cycle can begin and we can reach base temperature (in our

case, the base temperature is ∼ 30mK).

The 3He gas coming from the still is used to cool the returning gas through a series of

heat exchangers. The vacuum pump works at room temperature and the gas passes through

filters and cold traps to remove impurities and comes back to the cryostat where it gets

cooled first in the main bath and then in the 1K pot. The gas circulation system can be

controlled manually or through a LabView computer program supplied by the manufacturer.

The sample is inserted at the bottom of the cryostat, a radiation shield is carefully fit-

ted on, over which the Inner Vacuum Chamber (IVC) is fastened and vacuum-sealed using

In wire. The IVC is then evacuated and checked for leaks with a mass spectrometer leak

20


detector (that detects He) by spraying He gas near the seal and near all the connections

going into the cryostat. The cryostat is lowered into the main bath and all tubes and wires

are connected. The various lines connecting the assembly and the control module have to

be evacuated (generally by pumping overnight). The evacuated Outer Vacuum Chamber

(OVC) shields the main bath from the room temperatures.

The system is pre-cooled with liquid N2 overnight and is reaches about 125K. Before trans-

ferring the liquid N2, a small amount of He (exchange gas) has to be introduced in the IVC

so that the insert can be cooled down. The liquid N2 then has to be forced out by pressuring

the main bath with He gas. The transfer tube has to reach the funnel-like structure at the

base of system (above the superconducting magnet) to ensure that all the liquid N2 has

been removed from the system. Any liquid N2 left by accident at the bottom of the bath can

freeze when we fill liquid He and this could be dangerous for the magnet. The liquid He is

filled in the main bath using the He transfer tube (evacuated) by pressuring the dewar with

He gas. After the bath is filled, the IVC has to be thermally insulated from the liquid He in

the main bath by evacuating the exchange gas, to allow it to cool down further by operating

the dilution unit.

The temperature can be varied from the base temperature of ∼ 30mK to around 1K by

means of the temperature control panel of the control unit that can supply different heating

powers to carefully provide heat to the mixing chamber. The sample is normally cooled by

means of the 16 electrically-insulated copper wires (going to the 16 pins of the sample holder)

that have been thermally anchored to the mixing chamber. However, for the thermopower

measurements, we rewired the system and used constantan wires (that have extremely low

thermal conductivity) and even Nb-Ti superconducting wires to ensure that there are no heat

leaks from the contacts. In this case, the source and drain of the MOSFET sample were

21


separately thermally anchored to the mixing chamber as mentioned in the next chapter.

Superconducting Magnet

The magnet is made up of coaxial solenoid sections wound using multifilamentary supercon-

ducting wire. It is located at the base of the main bath and it surrounds the lower portion

of the cryostat. It is constructed to be physically and thermally stable and can withstand

the the large Lorentz force generated when it is being operated.

An advantages of the superconducting magnet is that it can be operated in the persis-

tent mode. In this mode, the superconducting circuit is closed to form a continuous loop,

and the power supply can be switched off, to leave the magnet at a particular field. This is

done by by using the superconducting switch, which is in parallel with the main windings.

When the magnet field has to be changed, the superconducting switch is warmed by a heater

to make it non-superconducting. The resistance of the switch increases to a few Ohms which

is greater than than the resistance of the main superconducting magnet windings. As a

result almost all the current flows through the magnet. Soon after the magnet reaches the

desired field the induced voltage across the switch drops to zero and all the current then

flows through the magnet. The switch is closed by turning off the heater. After a few tens of

seconds the current in the magnet leads is slowly reduced by running down the power supply.

As the current in the leads drops, the current flowing through the switch gradually rises,

until it carries the full current of the magnet. The magnet current leads are optimized to

carry the maximum operating current of the magnet and introduce as little heat as possible

to the liquid He.

22


Figure 1.9: A schematic diagram of the Oxford Kelvinox dilution refrigerator. Adapted fromRef.[13]

23

1.7. SI-MOSFET DEVICES

Figure 1.10: The structure of electron bands in the semiconductor gets modified when a gatevoltage is applied. When a strong enough voltage is applied, a 2D electron layer is formedat the oxide-semiconductor interface.

1.7 Si-MOSFET devices

Two-dimensional electron (or hole) systems are realized in metal-oxide-semiconductor field-

effect transistors (MOSFETs). The devices used in our experiments are made of bulk p-type

Si with SiO2 oxide layer between the semiconductor substrate and metallic Al gate. The

contacts are n-doped with phosphorus. When a positive voltage is applied to the metallic

gate, first a depletion layer is formed between the bulk Si and the oxide (in which only

negatively charged acceptor atoms are left since the positive holes have been pushed away

from the semiconductor-oxide interface). When the gate voltage is further increased, an

’inversion layer’ or two-dimensional sheet of electrons (that are supplied by the n-doped

contacts) is formed (Fig. 1.10) at the interface of the oxide and the semiconductor (actually

the depletion layer). The devices used in our experiments are in the form of a Hall bar with

typical dimensions of tens of microns (Fig 1.11).

24

1.8. OUTLINE OF THESIS

1.7.1 Split Gates in Si-MOSFETs

Split-gate technique has been used in the fabrication of the high mobility Si-MOSFET devices

(prepared by Prof. T. M. Klapwijk’s group [12]) used in our experiments. Submicron gaps

have been introduced in the gate metallization to separate the experimentally interesting

low density region of the 2DES from the high density region so that gate voltages can be

applied independently (electron densities can be controlled separately). This enables us to

avoid the high contact resistances typically associated with low electron densities. The split

gate technique proves crucial to access the very low electron densities corresponding to the

regime of strongly interacting electrons as it significantly reduces the contact resistances at

low temperatures.

1.8 Outline of thesis

In the next chapter, we discuss our latest experiment on thermopower measurements in the

2DES of strongly correlated electrons near the MIT in high mobility Si-MOSFETs. A brief

description of the experimental set-up is followed by the main results, viz. the divergence

of the diffusive thermopower signal itself and the unprecedentedly huge enhancement in the

effective mass of electrons observed near the transition and the implications thereof.

In chapter three, we discuss a recent paper on simultaneous measurements of resistivity and

parallel field magnetoconductance at low temperatures T < Tv ≈ 0.5K, where Tv is the

valley splitting temperature below which the two valleys in the [100] direction in conduction

band of Si are no longer degenerate. These measurements have allowed us to extend the

two-parameter scaling theory of Punnoose and Finkel’stein to lower temperatures where it

is still successful in describing the metallic state without any fitting parameters.

25


Figure 1.11: Wire-bonding layout of Hall bar Si-MOSFET devices used in our experiments.Numbers 1, 5, 8 and 12 indicate the high density gate, while number 6 is the low densitygate to tune the electron densities in the central portion of the Hall bar. All other numbersrefer to contacts - with numbers 9 and 16 used as source-drain.

26


In chapter four, we discuss results from experiments performed in Prof. Heiman’s lab to

probe a temperature regime of relatively higher temperatures (8K to 70K) that has not

been studied in detail previously in strongly correlated 2DES.

27

Chapter 2

Thermopower and Divergence of Effective Mass

This chapter is adapted from: A. Mokashi, S. Li, Bo Wen, S. V. Kravchenko, A. A. Shashkin,

V. T. Dolgopolov, and M. P. Sarachik, 2011, Divergence of the effective mass in a strongly-

interacting 2D electron system. (Manuscript in preparation)

2.1 Diverging χ and m∗ with decreasing ns in strongly correlated

2DES

The behavior of strongly-interacting electrons in two dimensions has attracted a great deal

of recent interest. The interaction strength in these two-dimensional (2D) systems is char-

acterized by the Wigner-Seitz radius, rs, the ratio of the Coulomb energy to the Fermi

energy in the case of a single valley. The Wigner-Seitz radius is proportional to n−1/2s and

increases with decreasing electron density, ns. Wigner crystallization is expected [14, 15] in

the strongly-interacting limit (rs 1), while Fermi liquid behavior with effective mass, m,

and Lande g factor renormalized by interactions has been established when rs . 1, at least

28

2.1. DIVERGING χ AND M∗ WITH DECREASING NS IN STRONGLYCORRELATED 2DES

in finite systems [16]. It was not until recently that the electron effective mass and the spin

susceptibility χ ∝ gm [17, 10, 18, 19] were found to increase dramatically with decreasing

electron density in strongly correlated 2D electron systems (rs & 10). This unexpected be-

havior was attributed to the proximity of the electron liquid to Wigner crystallization, or the

possible existence of intermediate phases [20, 21, 22, 23, 24]. A divergence of the effective

mass was found in a number of theories: using an analogy with He3 near the onset of Wigner

crystallization [21, 22]; extending the Fermi liquid concept to the strongly-interacting limit

[23]; solving an extended Hubbard model using dynamical mean-field theory [24]; renormal-

ization group analysis for multi-valley 2D systems [25]; Monte-Carlo simulations [26, 27].

Particularly strong many-body effects have been observed in silicon metal-oxide-semiconductor

field-effect transistors (MOSFETs), where the effective mass was found to increase sharply.

Determined from measurements of the temperature dependence of conductivity [28], the

damping of Shubnikov-de Haas oscillations with temperature [29, 30], magnetocapacitance

[31], and the magnetization [32], the value of the mass was found to be independent of dis-

order within the experimental uncertainties, in disagreement with theoretical expectations

[25, 26, 27]. Although the effective mass increases sharply at low electron densities, mass

enhancements were observed up to, but no more than, a factor of about four (Fig. 2.1); it

remained unclear whether the value of the mass indeed diverges in the critical region. We

report measurements of the diffusion thermopower at low temperatures in a low-disordered

strongly-interacting 2D electron system in silicon. We find that in the metallic regime, the

thermopower is proportional to temperature and increases with decreasing electron density,

tending to infinity at a finite density nt. The critical density nt is close to the density

nc = 8 × 1010 cm−2 for the metal-insulator transition in this electron system. However, in

contrast with the value of nc determined from measurements of the resistivity, the critical

density nt determined by measurement of the thermopower is independent of disorder. The

29

2.1. DIVERGING χ AND M∗ WITH DECREASING NS IN STRONGLYCORRELATED 2DES

0

1

2

3

4

5

6

7

8

0.5 1.5 2.5 3.5

g

*/2

, m

*/m

b,

an

d

χ/χ

0

ns (10

11 cm

-2)

nc

g*/2

χ/χ0

m*/mb

Figure 2.1: Diverging χ and m∗ at low electron densities. Ref. [32]. The effective g factor(circles) and the cyclotron mass (squares) as a function of the electron density. The solid andlong-dashed lines represent, respectively, the g factor and effective mass, previously obtainedfrom transport measurements, and the dotted line is the Pauli spin susceptibility obtainedby magnetization measurements in parallel magnetic fields. The critical density nc for themetal-insulator transition is indicated.

30

2.2. EXPERIMENTAL SETUP

divergence of the thermopower indicates a diverging effective mass, signaling the approach

to a phase transition.

2.2 Experimental setup

Measurements were made in a sample-in-vacuum Oxford Kelvinox dilution refrigerator with

a base temperature of ≈ 30 mK on (100)-silicon MOSFETs similar to those previously

used in Ref. [33]. As mentioned before, the advantage of these samples is a very low con-

tact resistance (in “conventional” silicon samples, high contact resistance becomes the main

experimental obstacle in the low-density low-temperature limit). To minimize contact resis-

tance, thin gaps in the gate metallization have been introduced, which allows for maintaining

high electron density near the contacts regardless of its value in the main part of the sample.

Samples were used with Hall bar geometry of width 50 µm and distance 120 µm between

the central potential probes and measurements of the thermoelectric voltage were obtained

in the main part of the sample (shaded in the inset to Fig. 2.5). A Hall contact pair between

the central probes and either source or drain (i.e. either 1-5 or 4-8) was employed as a heater:

the 2D electrons were locally heated by passing an ac current at a low frequency f through

either Hall contact pair. This was achieved by means of a decoupling circuit Fig 2.2(to cre-

ate an independent reference ground for the heating current - implemented using an ‘Ultra

Low Input Bias Current Instrumentation Amplifier’, INA116PA on a breadboard cased in

a metallic box) and solder-in ‘pi filters’ (that heavily attenuate any external unwanted high

frequency noise/pick-up).

Low ac currents ∼ 0.1 − 1nA were obtained from the lock-in voltage output using a

simple 1/100 potential divider and a 10MΩ resistor. Both source and drain contacts were

thermally anchored by connecting the corresponding pins of the chip holder with thick elec-

31


Figure 2.2: Decoupling circuit to provide independent reference ground for the heater.

trically insulated copper wires to the ‘cold finger’ which is at the temperature of the mixing

chamber of the dilution refrigerator. In such an arrangement it was possible to reverse the

direction of the temperature gradient induced in the central region of the sample. Tem-

peratures of the central probes were determined using two thermometers (resistors that are

calibrated to lowest temperatures) externally connected to corresponding pins on the chip

holder; the temperature gradients reached were 1–5 mK over the distance. The average

temperature in the central region was checked using the calibrated sample resistivity. The

sample resistivity was measured as a function of temperature by changing the temperature

of the mixing chamber very slowly. The resistivity was measured using a standard 4-probe

measurement with current going from source to drain and voltage measured between con-

tacts in the central part of the sample. Constantan and superconducting (bare Nb-Ti) wiring

was employed to minimize heat leaks from the sample. Possible RF pick-up was carefully

suppressed, and the thermoelectric voltage was measured using a low-noise low-offset LI-

32


Bare Nb-Ti wires without a Cu

matrix (replaced Constantan wires

–had a tiny heat leak)

Heat shrinking tube

Gold wire

Cryogenic-grade calibrated thermometer

(sensor chip)

A pin of the 16-pin sample holder

VGE-7031 varnish used as an electrically

insulating adhesive -excellent thermal conductor

Figure 2.3: The thermometer assembly at the contact pins.

33


SR830 Lock-in Amplifier

1/100 potential divider

10 MΩ resistor

Decoupling circuit box

Pi filters

LI-75A Pre-Amp

A - B 2f mode measurement

RF filters

Figure 2.4: A schematic diagram of thermopower measurement set-up. The lock-in amplifieris operated to measure the 2nd harmonic of the frequency of the input heating current signal.

34

2.3. THERMOPOWER RESULTS

75A preamplifier and a lock-in amplifier in the 2f mode in the frequency range 0.01–0.1 Hz.

Any non-zero first harmonic signal indicates the presence of pick-up/noise which has to

be tracked and eliminated. The sample resistance was measured by a standard 4-terminal

technique at a frequency 0.4 Hz. Excitation current was kept sufficiently small (0.1–1 nA)

to ensure that measurements were taken in the linear regime of response for each value of

temperatures used to ensure that the electrons are not overheated. This was achieved by

recording the increase in the average temperature (from sample resistance calibration) as a

function of excitation current. The power dependence of the thermopower signal itself was

also meticulously verified to be linear at the different excitations chosen for each value of

mixing chamber temperatures. Below, we show results obtained on a sample with a peak

electron mobility close to 3 m2/Vs at T = 0.1 K.

2.3 Thermopower results

The thermopower is defined as the ratio of the thermoelectric voltage and the temper-

ature difference, S = −∆V/∆T . In the low-temperature metallic regime, the diffusion

thermopower is determined by the relation

S = −α2πk2BmT

3e~2ns, (2.1)

where −e is the negative electron charge and the parameter α depends on both disorder

[34, 35, 36] and interaction strength [37]. According to Ref. [37], the dependence of α

on electron density is rather weak, and the main effect of electron-electron interactions is

the suppression of S values. At high temperatures, the phonon drag contribution to the

thermopower, which is proportional to T 6 for silicon MOSFETs [34], becomes dominant

over the diffusion contribution.

35


0

30

60

90

120

150

0.7 0.8 0.9 1 1.1 1.2

800 mK

600 mK

400 mK

300 mK

200 mK

ns (1011 cm-2)

-S (

µV/K

)

nc

(a) 1 2 3 4

5 6 7 8

S D

Figure 2.5: Change of the thermopower with electron density at different temperatures. Notall measured data points are shown to avoid overcomplicating the figure. The density nc forthe metal-insulator transition is indicated. The inset shows a schematic view of the sample.The contacts include four pairs of potential probes, source, and drain. The main part of thesample is shaded.

36


0

0.03

0.06

0.09

0.12

0.15

0.7 0.8 0.9 1 1.1 1.2

800 mK600 mK400 mK300 mK

ns (1011 cm-2)

-1/S

(K

/µV

)

(b)

nc

nt

Figure 2.6: The inverse thermopower as a function of electron density at different temper-atures. The solid lines are linear fits to the data which extrapolate to zero at a density nt.The density nc for the metal-insulator transition is also indicated.

37


Figure 2.5 shows data for the thermopower as a function of electron density at different

temperatures. The value −S increases strongly with decreasing electron density and becomes

larger as the temperature is increased. The divergent behavior of the thermopower at low

ns is more evident when plotted as −1/S versus electron density in Fig. 2.6. The inverse of

the thermopower −1/S tends to zero in a linear fashion at a density nt which is close to the

critical density nc for the metal-insulator transition in this electron system. The slope of the

fits to the data is proportional to the inverse temperature 1/T , which corresponds to S ∝ T ,

as expected for the diffusion thermopower. This confirms that the phonon drag contribution

is small in the temperature range of our experiments, and our measurements have yielded

the contribution of interest, namely, the diffusion thermopower.

The main experimental result is shown in Fig. 2.7, where −T/S is plotted as a function

of ns: the data fall on a straight line with intercept −T/S → 0 at ns = nt. According

to Eq. (2.1), the value T/S is proportional to ns/m and, therefore, the data indicate a

divergence of the mass m at the density nt: m ∝ ns/(ns − nt). It is interesting to compare

these results with data for the effective mass m∗ obtained earlier for the same samples by

combining measurements of the slope of the temperature dependence of the conductivity

and measurements of the parallel magnetic field for full spin polarization [28]; this allows a

separate determination of m∗ and the g-factor. As seen from the figure, the two data sets

show similar behavior. However, the thermopower data do not yield the absolute value of

m because of uncertainty in the coefficient α in Eq. (2.1). It is important to note that the

current experiment includes data for the thermopower for densities that are much closer to

the critical point.

38


0

0.2

0.4

0.6

0.8

0.6 0.8 1 1.2 1.4 1.6 1.8

800mK700mK600 mK400mK300mK

ns (1011 cm-2)

nc

-(k B

2 /e)

T/S

; π

h2 n s/2m

* (m

eV)

nt

-T/S:

ns/m*

Figure 2.7: The value −T/S versus electron density for different temperatures. The solidline is a linear fit which extrapolates to zero at nt. The metal-insulator transition pointnc is indicated. Also shown is the data for the effective mass m∗ obtained in transportmeasurements on the same samples [28]. The dashed line is a linear fit.

39

2.4. COMPARISON WITH THERMOPOWER IN HIGH DISORDER SYSTEMS

2.4 Comparison with thermopower in high disorder systems

We now compare the results obtained in the current experiment on low-disorder silicon

samples with those obtained by Fletcher etal. [38] in a silicon sample with high level of

disorder, as indicated by the appreciably higher density nc for the (Anderson) metal-insulator

transition (cf. Fig. 2.7). A replot of the thermopower data taken from Ref. [38] shown in

the inset of Fig. 2.8 demonstrates that (−T/S) exhibits very similar behavior in the critical

region, vanishing at the same density nt. This indicates that the thermopower divergence

is not related to the degree of disorder [39] and reflects the divergence of the effective mass

m at a disorder-independent density nt — behavior that is typical in the vicinity of an

interaction-induced phase transition.

In the main panel of Fig. 2.8, we show the factor (−Sσ), which determines the thermoelectric

current j = −Sσ∇T , as a function of electron density at different temperatures. (−Sσ)

stays approximately constant in the critical region, i.e., 1/S is proportional to σ in the

low-disordered 2D electron system. In contrast, for the highly-disordered silicon samples of

Ref. [38], (−Sσ) tends to zero at the (higher-density) Anderson transition point nc, and is

caused by a rapidly decreasing conductivity σ for ns < nc. We note that this signals that

the transitions in low- and high-disordered silicon derive from different sources: whereas

in highly-disordered 2D electron systems the conductivity tends to zero at the Anderson

transition because of disorder, in the clean 2D electron system the drop of the conductivity

at the phase transition is controlled by the increasing mass [28].

40

2.4. COMPARISON WITH THERMOPOWER IN HIGH DISORDER SYSTEMS

0

1

2

3

4

5

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

400 mK700 mK

-Sσ

(nΑ

/Κ)

ns (1011 cm-2)

0

0.2

0.4

0.6

0.8

0.5 1 1.5 2-(

k B

2 /e)

T/S

(m

eV)

ns (1011 cm-2)

nc

nt

Figure 2.8: The factor −Sσ, which determines the thermoelectric current, versus electrondensity at different temperatures. Inset: the value −T/S versus electron density at T =0.3 K in a highly-disordered 2D electron system in silicon [38]. The linear fit (solid line)extrapolates to zero at the same density nt as in Figs. 2.6 and 2.7. The density nc for themetal-insulator transition is indicated.

41

2.5. ENHANCED EFFECTIVE MASS

2.5 Enhanced effective mass

We now examine the results obtained for the enhanced effective mass. Since both the

thermopower and the conductivity are related to electron transfer at the Fermi level, our

experiment yields the effective mass of the electrons at the Fermi level, renormalized by

electron-electron interactions. The value of m can be extracted from the thermopower data

by requiring that the two data sets in Fig. 2.7 in the range of electron densities where they

overlap should correspond to the same value of mass. The coefficient α is determined from

the ratio of the slopes and is equal to α ≈ 0.18. The corresponding mass enhancement in

the critical region reaches m/mb ≈ 25, where the band mass mb = 0.19me and me is the free

electron mass. This exceeds by far the mass values obtained from previous experiments on

the 2D electron system in silicon and other 2D electron systems.

It is worth noting that the effective mass determined previously by parallel-field magneti-

zation measurements [40] should be related to the band width which is the Fermi energy

counted from the band bottom [32]. For ns ≥ 1011 cm−2, this mass value was found to

be practically the same as the effective mass measured in transport [18, 19]. However, the

behavior is different the densities reached in our experiment in the close vicinity of the

critical point nt (ns < 1011 cm−2). We argue that the band-width-related mass does not

increase strongly as the density nt is approached. Indeed, if it did, the ratio of the spin

and cyclotron splittings in perpendicular magnetic fields would increase considerably with

decreasing electron density. In this case one would observe with decreasing electron density a

Shubnikov-de Haas oscillation beating pattern, including several switches between cyclotron

and spin minima in weak magnetic fields. Instead, the Shubnikov-de Haas oscillations in a

dilute 2D electron system in silicon reveal one switch from cyclotron to spin minima as the

electron density is decreased [41], the spin minima surviving down to ns ≈ nc and even be-

low [42]. Another argument is that the dependence of −T/S on electron density stays linear

42

2.6. EFFECT OF PARALLEL MAGNETIC FIELDS ON THERMOPOWER SIGNAL

down to the lowest ns achieved, indicating that the 2D electron system remains degenerate.

Thus, while the effective electron mass at the Fermi level tends to diverge, the band width

does not decrease appreciably in the close vicinity of the critical point nt.

The observed scenario is in principle consistent with the Fermi-liquid-based model of Ref. [23]

in which a flattening at the Fermi energy in the spectrum that leads to a diverging effective

mass has been predicted. The prediction that the mass at the Fermi level and the band-

width-related mass are different is in contrast to the majority of the theories that presume

a parabolic spectrum. The experimental results are likely to favor the model [23], which

implies the existence of an intermediate phase that precedes Wigner crystallization. Still,

the origin of the low-density phase in the samples that are currently available is masked by

residual disorder.

2.6 Effect of Parallel Magnetic Fields on Thermopower signal

Fig. 2.9 shows the increase in the thermopower signal (−S∇T ) as a function of parallel

magnetic field for a particular carrier density, ns = 1.275 × 1011cm−2 at 300mK mixing

chamber temperature. The signal is seen to saturate and then decrease beyond the critical

field B∗ = 3.25T corresponding to full spin polarization for this value of electron density.

The increase in the thermopower is not due to any first harmonic magnetoresistance, as we

have verified that the first harmonic stays very low and the lock-in is capable of suppressing

first order influencing the second harmonic thermopower signal up to 103. The first harmonic

is observed to increase as we increase the field beyond B∗. Further systematic investigations

are required to elucidate the effects of parallel magnetic fields.

43

2.6. EFFECT OF PARALLEL MAGNETIC FIELDS ON THERMOPOWER SIGNAL

Figure 2.9: The effect of in-plane magnetic field on the thermopower signal for electrondensity, ns = 1.275 × 1011cm−2 at 300mK mixing chamber temperature. The arrow pointsto the critical field B∗ = 3.25T corresponding to complete spin polarization - this value isobtained from the previous experiments. Refer to Fig. 1.6

44

2.7. CONCLUSION

2.7 Conclusion

In summary, we have found that the diffusion thermopower in a low-disordered strongly-

interacting 2D electron system in silicon tends to diverge at a density nt as the electron

density is decreased. The density nt is close to the critical density for the metal-insulator

transition in this electron system but, unlike the latter, it is independent of disorder. The

thermopower data indicates a diverging effective mass in the vicinity of a phase transition.

45

Chapter 3

Extension of Scaling Theory to include Valley

Effects

This chapter is adapted from: A. Punnoose, A. M. Finkel’stein, A. Mokashi and S. V.

Kravchenko, 2010, Test of the scaling theory in two dimensions in the presence of valley

splitting and intervalley scattering. Phys. Rev. B. 82, 201308(R).

Previous experiments [43] on simultaneous measurements of parallel field magnetoconduc-

tance and resistivity in the 2DES of Si-MOSFETs and extracting the disorder (ρ) and inter-

action parameters (Cee and γ2) made possible the description of the MIT in terms of a flow

diagram confirming the predicted Quantum Critical Point [25]. The two-parameter (disor-

der and interactions) scaling theory by Punnoose and Finkel’stein has been successful in

describing the metallic side of the MIT without any fitting parameters. Our recent work has

enabled extending these results to even lower temperatures by accounting for valley splitting

and intervalley scattering.

46

3.1. TWO VALLEYS IN THE [100] DIRECTION IN SI CONDUCTION BAND

Figure 3.1: The equal energy surfaces of conduction band of Si in k-space. Ref. [45]

3.1 Two valleys in the [100] direction in Si conduction band

The equal energy surfaces of the conduction band of Si are made of equivalent ellipsoids

in the six crystal directions (Fig 3.1). The ground state of the electrons in the inversion

layer of [100] Si MOSFET devices is shown as two circles that overlap, obtained by taking

a projection of the two ellipsoids onto the [100] plane. These overlapping circles represent

the two valleys of the conduction band of electrons. These valleys are fully degenerate above

Tv ≈ 0.5K which is the temperature associated with splitting of the valleys.

The sharpness of the interface of the inversion layer leads to the splitting, ∆v, of the two val-

ley bands as a result, for temperatures lower than Tv = ∆v/kB, the two valleys are no longer

degenerate. The atomic scale irregularities found at the interface give rise to a finite inter-

valley scattering rate, ~/τ⊥ [51], as a result the valleys are mixed below the corresponding

temperature T⊥ ≈ 0.2K.

We show that once the effects of valley splitting and intervalley scattering are incorporated,

47

3.2. PREDICTIONS OF THE TWO-PARAMETER SCALING THEORY

renormalization group theory consistently describes the metallic phase in silicon metal-oxide-

semiconductor field-effect transistors down to the lowest accessible temperatures.

3.2 Predictions of the Two-parameter Scaling Theory

3.2.1 Main results from theory and experimental verification

The two-parameter scaling theory of quantum diffusion in two dimensions [44, 25] has been re-

markably successful in describing the properties around the metal-insulator transition (MIT)

in electron systems confined to silicon inversion layers (MOSFETs) [43, 46, 47]. The theory is

based on the scaling hypothesis that both the resistivity and the electron-electron scattering

amplitudes become scale dependent in a diffusive system due to the singular long ranged

nature of the diffusive propagators, D(q, ω) = 1/(Dq2 + ω), in a disordered medium [48, 49].

In their recent results [43], Anissimova et al reported the first ever measurements of the

temperature dependence of the interaction parameter Cee using the two-parameter scaling

theory to extract the parameters from the experimental data. Like the resistivity, the in-

teraction parameter was also observed to show a fan-like spread around the Metal-Insulator

Transition (Fig. 3.2). It is seen that on the metallic side, it increases as temperature is

reduced, with the resistivity decreasing simultaneously - thus confirming the importance of

electron electron interactions for the existence of the metallic state in 2D.

By simultaneously plotting the two parameters ρ and Cee that stand for disorder and inter-

action strength respectively, they were able to draw a disorder-interaction flow diagram that

clearly indicates the existence of a Quantum Critical Point (Fig. 3.3).

48


Figure 3.2: Fan like spread for both parameters ρ and Cee around the MIT. Adapted from[43].

Figure 3.3: Two-parameter flow diagram around the QCP. Adapted from [43].

49


3.2.2 Extention to lower temperatures

The predicted scale dependencies calculated using renormalization group (RG) theory [44]

were recently verified experimentally in Ref. [43] without any fitting parameters. Since the

theory considered the valleys to be degenerate and distinct, the experiments were limited

to temperatures larger than the characteristic valley splitting and intervalley scattering rate

(T & 500 mK). The effects of scaling are, however, significant at low temperatures and it is

therefore important to test the scaling hypothesis at much lower temperatures. We show that

when the RG theory is extended to include valley splitting and intervalley scattering [50] the

scaling properties in the metallic phase can be described quantitatively down to the lowest

reliably accessible temperatures, T ≈ 200 mK.

The evolution with scale (temperature) of the two-parameters, namely, the resistance, ρ,

and the electron-electron interaction strength, γ2, in the spin-triplet channel were discussed

in detail for ρ . 1 (in units of πh/e2) in terms of RG theory in Ref. [44]. (In Fermi-liquid

notation, γ2 is related to the amplitude F a0 as γ2 = −F a

0 /(1+F a0 ).) The theory predicts that,

while γ2 increases monotonically as the temperature is reduced, ρ behaves non-monotonically,

changing from insulating behavior (dρ/dT < 0) at high temperatures to metallic behavior

(dρ/dT > 0) at low temperatures, with the crossover occurring when γ2 attains the value

γ∗2 = 0.45. Although the maximum value ρmax occurs at a crossover temperature T = Tmax,

both of which are sample specific and hence non-universal, the two-parameter scaling theory

predicts that the behaviors of ρ(T )/ρmax and γ2(T ) are universal when plotted as functions

of ξ = ρmax ln(Tmax/T ). The above predictions, including the value of γ∗2 , were verified exper-

imentally in Refs. [44, 43] in the temperature range where the two valleys may be considered

to be degenerate and distinct.

50


For n-(001) silicon inversion layer the conduction band has two almost degenerate valleys

located close to the X-points in the Brillouin zone. While the sharpness of the interface

of the inversion layer leads to the splitting, ∆v, of the two valley bands, the atomic scale

irregularities found at the interface gives rise to a finite intervalley scattering rate, ~/τ⊥ [51].

The singularity of the diffusion modes, especially those in the valley-triplet sector, are cut-off

at low frequencies as a result [52, 50]. Hence, the specific form of the RG equations, which

is sensitive only to the number of singular modes, depends on if kBT is greater than or less

than the scales ∆v or/and ~/τ⊥.

The relevant RG equations for the different temperature ranges may be combined as fol-

lows [50]:

dρ

dξ= ρ2

[1− (4K − 1)

(γ2 + 1

γ2

log(1 + γ2)− 1

)](3.1a)

dγ2

dξ= ρ

(1 + γ2)2

2(3.1b)

The parameter K accounts for the number of singular diffusion modes in each temperature

range. For temperatures T & Tv and T⊥, where kBTv = ∆v and kBT⊥ = ~/τ⊥, the two

bands are effectively degenerate and distinct; the constant K in this case is proportional

to the square of the number of valleys, nv, i.e., K = n2v = 4 (nv = 2 for silicon). In the

temperature range T⊥ . T . Tv, the two bands remain distinct but are split and hence each

valley contributes independently to ∆σ(b), i.e., K = nv = 2. At still lower temperatures

T . T⊥, intervalley scattering mixes the two valleys to effectively produce a single valley so

that K = 1.

A few important clarifications regarding the use of Eq. (3.1) are discussed below. First, for

the case K = 2, when the bands are split but distinct, it has been shown that using a single

51


amplitude γ2 to describe the interaction in all the seven (4K−1) modes is an approximation

that is valid only if the temperature range T⊥ . T . Tv is not too wide [50]. In general,

when the bands are split certain amplitudes evolve differently from γ2, thereby necessitating

the need to go beyond the two-parameter scaling description [53, 50]. The deviation is large

when the RG evolution is allowed to proceed to exponentially large scales or T Tv. In

our case, however, since T⊥, which effectively mixes the two bands, is only a fraction smaller

than Tv, the deviation of the amplitudes is quickly limited by T⊥. We therefore assume that

all the amplitudes remain degenerate and contribute equally to ρ, which amounts to taking

K = 2 in Eq. (3.1).

The second point concerns the weak-localization (WL) contribution [54] to Eq. (3.1). It

is seen experimentally that the phase breaking rate, ~/τϕ, saturates at low electron densities

(n . 1011 cm−2) for T . 500 K. Correspondingly, a strong suppression of the WL correction

is also observed in this regime [55]. These observations are consistent with our results, as is

discussed later. We have therefore neglected the weak-localization contribution in Eq. (3.1)

when analyzing the cases K = 2 and 1 (these are the relevant cases at low temperatures).

In Ref. [43] it was shown that γ2 may be determined experimentally by exploiting the b2

dependence of the magnetoconductance ∆σ(b) ≡ ∆σ(B, T ) = σ(B, T ) − σ(0, T ) in a weak

parallel magnetic field b = gµBB/kBT . 1. In the weak field limit ∆σ(b) is given as [56, 57]

∆σ(b) = −0.091e2

πhKγ2 (γ2 + 1) b2 (3.2)

Hence the slope of ∆σ(b2) provides a direct measure of γ2 (Fig 3.4), given of course that K

is known.

52

3.3. EXACT AGREEMENT WITH PREDICTED CROSSOVER WITHOUT FITTINGPARAMETERS

Figure 3.4: Procedure for extracting interaction parameter γ2 or Cee from the slopes of∆σ(B, T ) vs b2 (From the magnetoresistivity data). Adapted from [43]. Data is shown forelectron density ns = 9.14 × 1010cm−2 measured at different temperatures. a, Resistivityvs parallel magnetic field. b, Magnetoconductivity σ(B, T ) ≡ σ(B, T )− σ(0, T ) (in units ofe2/h) vs b2. It is clear that the slopes that signify the interaction parameter decrease withtemperature.

3.3 Exact agreement with predicted crossover without fitting param-

eters

In the upper panels in Fig. 3.5, we plot ρ(T ) at zero magnetic field for three different electron

densities. They show a characteristic non-monotonic behavior as predicted in (3.1). In the

lower panels in Fig. 3.5 we plot the extracted values of γ2 using Eq. (3.2) with K = 4, i.e.,

assuming that the valleys are degenerate and distinct. The dashed horizontal line marks

the point γ2 ≈ 0.45 approximately where ρ(T ) attains its maximum value in remarkable

agreement with Eq. (3.1). (At these temperatures quantum coherence is relevant and its

contribution to weak localization, dρ/dξ = nvρ2, is to be added to Eq. (3.1a).)

53

3.4. COMPARISON OF THEORY AND EXPERIMENT CONSIDERING THECHANGING VALLEY DEGENERACY

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a)

ρ (

πh/e

2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b)

ρ (

πh/e

2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(c)

ρ (

πh/e

2)

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5

γ2

T (K)

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5

γ2

T (K)

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5

γ2

T (K)

ns = 8.4x10

10 cm

-2n

s = 8.7x10

10 cm

-2n

s = 9.1x10

10 cm

-2

Figure 3.5: Upper panels: ρ(T ) traces (in units of πh/e2) for three different electron densities,ns = 9.87, 9.58 and 9.14× 1010 cm−2. Lower panels: Extracted values of γ2(T ) using Eq. 3.2using K = 4, for the same electron densities. (See Ref. [43] for further details.) The dashedlines are positioned at the critical value γ∗2 = 0.45. Note that the maximum in ρ(T ) occurswhen γ2 attains approximately the value γ∗2 .

3.4 Comparison of theory and experiment considering the changing

valley degeneracy

The results of the comparison between theory and experiment are presented in Fig. 3.6. The

solid squares () are the experimental data points for ns = 9.1 × 1010 cm−2, reproduced

here from Fig. 3.5(a). The solid lines are the predicted theoretical curves for ρ(T ) and γ2(T )

with the parameters K = 4, ρmax = 0.4 and Tmax = 2.3 K. (Here, Tmax is the temperature

at which ρ(T ) attains its maximum value, ρ(Tmax) = ρmax.) The remarkable agreement

between theory and experiment is especially striking given that the theory has no adjustable

parameters.

54

3.4. COMPARISON OF THEORY AND EXPERIMENT CONSIDERING THECHANGING VALLEY DEGENERACY

At temperatures below 0.5 K, the experimentally extracted values of γ2(T ) in Fig. 3.6(b)

seem to saturate with further decrease in T . We believe that the saturation is an artifact

of the analysis related to our assumptions that both the valley splitting and the intervalley

scattering are negligible at the lowest temperatures. As noted earlier, the large number of

valley modes K = n2v reduces to just K = nv for temperature T⊥ . T . Tv and to just

K = 1 for T . T⊥. In the following, we recalculate γ2(T ) taking these considerations into

account.

The experimentally extracted values of γ2, using K = 2 and K = 1, are shown in Fig. 3.6(b)

as diamonds (red ) and stars (blue ), respectively. The procedure used to extract these

values are the same as that used for K = 4, namely, by fitting the σ(b2) traces in Fig. 3.5(b)

to Eq. (3.2) using the appropriate K values. We find very favorable agreement with theory

(solid line) if the crossover scales are chosen such that Tv ≈ 0.5 K and T⊥ ≈ 0.2 K. (Note

that for these temperatures the WL corrections have not been included in Eq. (3.1) for the

reasons discussed earlier.) These values are in good agreement with earlier estimates of

Tv [58] and T⊥ [59] obtained at higher densities employing different methods. We checked

by direct calculation using Eq. (3.1) that the theoretical values of ρ and γ2 are not affected

significantly when crossing these scales, provided that the WL corrections are not included

below T . 500K. Deviations from the solution for K = 4 taking K = 2 and K = 1 are

shown in Fig. 3.6 as long (red) and short (blue) dashed lines, respectively. As can be seen,

the deviations are insignificant (almost indiscernible) down to T = 0.2 K.

By comparing with experiments we have extended the test of the scaling equations (3.1)

down to the lowest reliably measurable temperatures T ≈ 0.2 K. Concerning still lower tem-

peratures, i.e., lower than T = 0.2 K, the theory predicts (not shown here) that while ρ(T )

saturates and then begins to drop again at ultra low temperatures (T . 50 mK), γ2(T ) rises

55

3.5. SUMMARY

fast monotonically for K = 1. Further tests of these predictions are in progress.

3.5 Summary

To conclude, we have shown that if valley splitting and intervalley scattering are incorporated

into the RG theory, the latter quantitatively describes the metallic phase down to the lowest

readily accessible temperatures. The extracted values of intervalley scattering time and

valley splitting are in good agreement with those previously obtained at higher densities

using different methods.

56

3.5. SUMMARY

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

(a)

T (K)

Theory K=4 K=2 K=1

Experiment

(h/

e2 )

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0(b)

Experiment K=4 K=2 K=1

2

T (K)

Theory K=4 K=2 K=1

Figure 3.6: The result of the comparison between theory (lines) and experiment (symbols)for ρ and γ2 are presented in (a) and (b), respectively. The parameter K = 4 correspondsto the case when the two valleys (nv = 2) are degenerate, i.e., T > Tv, where Tv ≈ 0.5 K isthe estimated valley splitting. K = 2 corresponds to the temperature range T . Tv, andK = 1 corresponds to the region T . T⊥ ≈ 0.2 K where the intervalley scattering mixes thetwo valleys to give one valley.

57

Chapter 4

Transport in Strongly Correlated Regime at

intermediate temperatures

In this chapter, we discuss our recent results on transport measurements in strongly inter-

acting two-dimensional electron systems in high mobility Si-MOSFETs in a range of tem-

peratures that has not been probed in detail. We present our findings and compare them

with the predictions available for such systems drawing an analogy of the resistivity in 2DES

with a hydrodynamic description of transport in correlated electron systems.

4.1 Broad range of temperatures unexplored

Two-dimensional electron systems in clean, high-mobility samples that show the Metal-

Insulator Transition in zero magnetic field - considered to be a Quantum Phase Transition

at 0K (in Si-MOSFETs, GaAs heterojunctions and other systems - which show similar data

despite having dissimilar electronic structure) display distinct non-Fermi liquid character-

istics in the strongly correlated regime (e.g. metal-insulator transition, magnetoresistance

58

4.1. BROAD RANGE OF TEMPERATURES UNEXPLORED

in parallel and perpendicular fields, etc.). These novel properties are dramatically different

from those seen in weakly-interacting systems and signify completely new physics[60]. Al-

though there has been significant progress towards finding theoretical explanations of these

phenomena, a comprehensive theory is yet to be developed. What are the implications of this

new physics at higher temperatures away from the MIT where the system is still strongly

correlated and quantum mechanical (and two-dimensional) though non-degenerate? Are

there any unexpected findings in this relatively less explored regime? Do these anomalous

properties survive or get modified at significantly higher temperatures? We have tried to

answer some of these questions in the experimental work described below and we compare

our experimental findings with theoretical predictions.

The resistivity of 2DES in zero magnetic field near the MIT has been extensively stud-

ied in various clean samples and for Si-MOSFETs in particular, data are available from

∼ 50mK to ∼ 8K. The longitudinal and Hall resistances in perpendicular fields and the

strongly enhanced magnetoresistance in parallel fields have also been investigated in detail

at these temperatures. However, there are no systematic measurements above typical Fermi

temperatures (e.g. above TF ∼ 7.5K for ns = 1011cm−2).

59

4.2. ACCESSING THE REGIME OF STRONG CORRELATIONS

Gas Strongly correlated liquid Wigner Crystal

𝑟𝑠~1 𝑟𝑠~35

Figure 4.1: Strength of interactions signified by parameter rs

4.2 Accessing the regime of strong correlations

The interaction parameter signifying the correlations is characterized by the ratio of the

Coulomb energy and the Fermi energy. In the quantum mechanical regime it is given by,

rs =Coulomb Energy

Fermi Energy=

1√πns(a∗B)2

(4.1)

where ns is the electron density and a∗B = ~2ε/m∗e2, the effective Bohr radius (m∗ being

the effective electron mass which can be assumed to be the band mass of electrons). As

the interaction parameter is inversely proportional to the square root of the electron density

ns, we can access the strongly correlated region in clean high-mobility (i.e. low-disordered)

samples by going to lower densities which is achieved in Si-MOSFETs by changing the gate

voltage VG.

60

4.3. SYSTEM TWO-DIMENSIONAL EVEN AT HIGH TEMPERATURES

4.3 System two-dimensional even at high temperatures

The inversion layer of electrons formed at the interface of the oxide and semiconductor

in the MOSFET in which the electrons are constrained to move only in two dimensions,

remains two-dimensional even at the highest temperatures (70K) we have investigated in

these experiments. This is demonstrated in the following lines. The energy of the electrons

is quantized in the third dimension with the energy eigenvalues of Airy functions obtained

using the triangular potential approximation given by,

En = cn

[~2

2mb

(e2nsε0εr

)2] 1

3

(4.2)

where c1 = 2.338, c2 = 4.088 and so on. The band mass of electrons in the (100) direction in

the Si conduction band is given by mb = 0.19me and the relative permittivity is the average

of that for SiO2 and Si, εr = 7.7.

The above equation leads us to the expression for the energy gap between the first two levels

expressed as the temperature for lifting of quantum degeneracy in the third dimension,

E2 − E1

kB= 153

( ns1011cm−2

) 23K (4.3)

The system thus remains two-dimensional for all the electron densities up to the highest

temperatures accessed in our experiments.

61

4.4. TEMPERATURE DEPENDENCE OF RESISTIVITY FOR T > TF

4.4 Temperature dependence of resistivity for T > TF

The resistivity as a function of temperature ρ(T ) is found to be non-monotonic in different

clean high mobility ”metallic samples” samples with large rs and for ρ < he2

. It is observed to

increase first, reach a peak near the Fermi temperature TF (which some authors consider as

TM , the melting temperature of the ”Wigner Crystal”) and then decrease till a temperature

Tph beyond which it increases as the electron-phonon interactions become significant [Fig 4.2].

At smaller rs deeper in the metallic regime the resistivity is found to increase monotonically

with temperature.

Let us consider the temperature regime that lies between the Fermi temperature TF and

the temperature at which electron-phonon interactions take over Tph. In this regime, the

transport properties of the correlated electron liquid in low disorder samples are better

described by hydrodynamics rather than by the Boltzmann equation. This assumption is

valid only for low disorder, for no Umklapp processes and no electron-phonon scattering -

which are all true for the temperature range we have just mentioned in the samples we use.

In this case, the viscosity of the fluid η becomes an important quantity to consider.[61]

4.4.1 Disorder as an effective medium for hydrodynamic flow of the electron

liquid

For low-disordered systems, the electron liquid can be considered to be flowing through an

effective medium of slight disorders. In other words, if the electron-electron mean free path

(le-e)is smaller than the spacing between the impurities/the length across which the disorder

potential changes, the resistivity is assumed to be proportional to the viscosity of the electron

liquid. i.e.

ρ ∝ η (4.4)

62


(a)5000

1 104

1.5 104

2 104

2.5 104

0.2

0.4

0.6

0.8

0 10 20 30 40 50

ρ (Ω

/sq

ua

re)

T (K)

ρ (

h/e

2)

(b)

0 2 4 6 8 10 120

1k

2k

3k

4k

5k

[h/e

2 ]

[/

]

Temperature T [K]

Holes in a 10nm GaAs quantum wellresistivity vs. T at B=0 hole density p = 1.3*1010/cm2

calculated phonon scattering resistivity for deformation constant D=6eV

0.00

0.05

0.10

0.15

(c)

4

6

8

10

0 5 10 15 20 25 30

0.2

0.3

0.4

ρ (

kO

hm

/sq

uare

)

T (K)

2D holes in p-SiGe

p = 1.2 x 1011

cm-2

ρ (

h/e

2)

(d)

1

2

3

4

0 5 10 15 20 25 30 35

0.05

0.1

0.15

ρ (

kO

hm

/sq

uare

)

T (K)

2D electrons in (111) Si MOSFET

n = 1.9 x 1011

cm-2

ρ (

h/e

2)

Figure 4.2: Non-monotonic temperature dependence of the resistivity in a (100) Si MOSFET(a), p-GaAs quantum well (b), p-SiGe quantum well (c), and (111) Si MOSFET (d) deep inthe metallic regime over an extended temperature range. The Fermi temperatures are 7.5 K(a), 0.75 K (b), and 7 K (c). From Spivak et. al, 2010[60].

63


In their latest result for two dimensions (2011), Andreev, Kivelson and Spivak [62] arrive at

an expression that directly relates ρ2D and η through a set of equilibrium coefficients, the

thermal conductivity κ and the correlation length of the disorder potential ξ.

4.4.2 Two temperature regimes for strong correlations above TF

Two distinct regimes are identified in the interval T ∈ EF , V = rsEF [61]. By V , here we

mean the temperature associated with the interaction energy VkB

and similarly for the Fermi

Energy. The first is the semiquantum regime for EF < T < ΩP =√V EF =

√rsEF . Here

the electron liquid is expected to be strongly correlated and non-degenerate yet quantum

mechanical. The viscosity of a strongly interacting liquid in this regime (and consequently

the resistivity) is expected to follow [63]

ρ ∝ η(T ) ∝ 1/T (4.5)

The second regime is in the temperature range√rsEF < T < rsEF . This is the regime in

which it is a highly correlated classical electron liquid where the expected behavior observed

in a large number of liquids [64][65] is,

ρ ∝ η(T ) ∝ eAVkBT (4.6)

where A is a constant of order 1.

These predictions are consistent with the measured values of resistivity between TF and

Tph in the 2DES of different samples as shown in Fig 4.2 which clearly decreases with tem-

perature. Also, the viscosity of bulk liquid 4He in a similar semiquantum region is observed

to be inversely proportional to the temperature[63].

64


𝜌 (ℎ 𝑒2 )

T (K) 𝐸𝐹 𝑇𝑝ℎ Ω𝑃? 𝑟𝑠𝐸𝐹?

Figure 4.3: The different energy regimes for the correlated electron liquid.

65


4.4.3 Resistivity beyond Tph

Once the electron-phonon scattering becomes dominant at even higher temperatures, the

resistivity is expected to increase. From Matthiessen’s Rule, the different contributions to

the inverse scattering times/mobilities add up.

1

τ=

1

τdisorder+

1

τinteraction+

1

τphonon1

µ=

1

µdisorder+

1

µinteraction+

1

µphonon

ρ = ρdisorder + ρinteraction + ρphonon (4.7)

The resistivity due to phonons is expected to follow a power-law dependence. Thus, the effect

of the onset of electron-phonon scattering would be to overshadow the other contributions

to the resistivity.

4.5 Experimental setup

These experiments were carried out in Prof. Don Heiman’s lab. To access the temperatures

mentioned (T > TF ), we used a sample-in-vacuum cryo-free variable temperature cryostat

from Cryo Industries Limited with a base temperature of ∼ 8K to 10K. We also utilized a

cryo-free superconducting magnet made by Cryogenics Ltd. which can be swept from −14T

to 14T .

The sample is loaded on a 16-pin chip holder at the end of a copper ’cold finger’ in thermal

contact with the ’cold head’ of the cryostat. Thermal contact of the sample with the cold

finger is established by wrapping few turns of the (electrically) insulated copper wires that

go to the sample terminals around the cold finger at a couple of places. A radiation shield

66


and a vacuum chamber cover are clamped in place and the air in the vacuum chamber is

evacuated to around 2×10−5mbar. The high purity 4He gas in the closed system of the cold

head is pumped on and the system cools down to base temperature in about three hours.

The superconducting magnet needs to be cooled down in stages for 24 hours before it is

ready to be used.

There are two thermometers on the cold finger - one near the top (close to the cold head)

and the other at the end (near the sample). The temperature of the cold finger can be varied

by means of a heater located near the top thermometer. Since there is a slight leak in the

cryostat near the temperature controller, the system has to be evacuated every day. A glass

window at the end of the vacuum chamber cover that is used for optics experiments has to

be covered with thick copper tape to prevent the sample from being exposed to light. There

are two orientations - perpendicular to the field (and to the length of the cold finger) and

parallel to the field. A circular copper frame at the end of the cold finger has to be carefully

removed and reoriented to achieve this. The cryostat/insert has to be placed vertically inside

the magnet cavity whenever we need a magnetic field.

Standard 4-terminal resistance measurements with low frequency AC are done using a

Stanford Instruments SRS630 lock-in amplifier and a low-noise pre-amplifier. A Voltage

to Current Convertor is used to convert the input/reference AC Voltage from the lock-in

into a current. Coaxial cables are used and they enter the cryostat through vacuum-sealed

”spacecraft-grade” connectors. The data is recorded through HP digital multimeters read

by LabView programs. The gate voltage is applied using a Yokogawa source and some times

a Keithley Voltage source.

67


SR830 Lock-in VI Convertor

Cryostat SR560

Pre-Amplifier

HP 34401A Multimeter

Desktop Computer

Input AC Signal

Data recorded

Signal from sample

Yokogawa 7651 DC source

T

VG

B

Superconducting Magnet

Figure 4.4: Setup diagram. Typically we have used frequencies ∼ 1Hz and currents ∼ 10nAin our measurements checking for linearity every time to make sure there are no overheatedelectrons.

68

4.6. EXPERIMENTAL RESULTS

4.6 Experimental Results

4.6.1 ρxx versus T at B = 0

We observed that the shapes of the ρ(T ) plots in the range of temperatures accessed in our

experiments depend significantly on the electron density ns (and on the interaction parameter

rs). To probe the strongly interacting regime, we investigated the resistivities at densities

corresponding to rs values going from about 35 to 15. The shapes and characteristics of the

plots change gradually as we tune the interactions. We have included some representative

plots and discuss their properties below.

Analysis for ρxx(T ) of a particular carrier density ns = 0.375× 1011cm−2

Among various linear functions of various functional dependencies plausible for temperature

dependence of viscosity, viz. exp(1/T ), 1/T, ln(1/T ), we found that the experimental results

for ρxx(T ) before the onset of phonons fit best to a linear combination of exp(1/T ) and

ln(1/T ) (Fig 4.5). Using an exp(1/T ) term is justified here because for this ns, ΩP = 14.7K

and rsEF = 76.6K.

The logarithmic temperature dependence term, ln(1/T ) in the fit of the resistivity ρxx(T )

could be interpreted as a weak localization term (in the Mattheissen Rule) arising from the

Scaling Theory of Localization for disordered electron systems. [4][66]

δσ2D(T ) =p

2

e2

~π2ln

(T

T0

)(4.8)

where p (from τ ∝ T−p where τ is the inelastic scattering time that increases as temperature

decreases) is an index that depends on the scattering mechanism.

69


However the exp(1/T ) term which could have been from the viscosity-like term has a nega-

tive coefficient - which cannot be interpreted physically. Fitting it just to exp(1/T ) assuming

the classical correlated regime gives an unacceptable fit.

Since it does not fit to a sum of linear terms as suggested by Matthiessen’s Rule, it is possi-

ble that non-linear terms (arising due to non-equilibrium phenomena analogous to turbulent

flow, convection, eddies in the correlated two-dimensional liquid?) are present.

Trend for ρ(T ) as a function of ns

On the other hand, it can be argued that this is a very limited temperature range to find a

fit to the data. The expected behavior of ρ(T ) between TF and Tph, viz., a drop in resistivity

with increasing temperatures is found to be true for a wide range of rs. Comparing the

plots for rs = 30.4 and rs = 35.2 shows one limit of this qualitative trend (Fig 4.6). As the

interaction parameter gets smaller, we reach the other limit in which the drop in resistivity

with temperature gets smaller and smaller till the point where it is not possible to observe

it (Fig 4.8). The plot of −∆ρ∆T

vs ns clearly shows how this quantity drops as the system is

brought from high rs values to lower ones (Fig 4.9).

Temperature at which phonon excitations become significant, Tph for different ns

From the plots of ρ(T ) vs ns, we could find the dependence of the temperatures above which

electron-phonon interactions become significant (Tph) with the electron densities. We found

that Tph is inversely proportional to the square root of ns (Fig 4.10). The electron-phonon

scattering times can be obtained from Tph values using ~τe−ph

= kBTph (Fig 4.11).

70


20 30 40 50 60

0.89

0.90

0.91

0.92

0.93

0.94

0.95

THKL

Ρ xx

Hhe2

Lns=0.375´1011cm-2,rs=27.2,8EF ,WP,rsEF ,Tph<=82.8,14.7,76.6,45.2HKL<

25 30 35 40 45

0.890

0.895

0.900

0.905

0.910

THKL

Ρ xx

Hhe2

L

Fit data to 1, expH1TL, logH1TL

Figure 4.5: ρxx versus T @ B = 0 for VG = 0.7V, ns = 0.375× 1011cm−2

71


20 30 40 50 60 70

1.26

1.28

1.30

1.32

1.34

THKL

Ρ xx

Hhe2

Lns=0.225´1011cm-2,rs=35.2,8EF ,WP,rsEF ,Tph<=81.7,10,59.3,57HKL<

30 40 50 60 70

1.07

1.08

1.09

1.10

1.11

1.12

THKL

Ρ xx

Hhe2

L

ns=0.3´1011cm-2,rs=30.4,8EF ,WP,rsEF ,Tph<=82.25,12.4,68.5,49HKL<

Figure 4.6: ρxx versus T @ B = 0 for VG = 0.6V, ns = 0.225 × 1011cm−2 and for VG =0.65V, ns = 0.3× 1011cm−2

72


32 34 36 38 40 42 44 460.763

0.764

0.765

0.766

0.767

0.768

0.769

0.770

THKL

Ρ xx

Hhe2


30 35 40 45

0.660

0.662

0.664

0.666

0.668

THKL

Ρ xx

Hhe2

L

ns=0.525´1011cm-2,rs=23,8EF ,WP,rsEF ,Tph<=83.9,18.9,90.6,37.5HKL<

Figure 4.7: ρxx versus T at B = 0 for VG = 0.75V and VG = 0.8V

73


20 25 30 35 40 450.530

0.535

0.540

0.545

0.550

THKL

Ρ xx

Hhe2


20 25 30 35 40

0.445

0.446

0.447

0.448

0.449

0.450

0.451

THKL

Ρ xx

Hhe2

L

ns=0.825´1011cm-2,rs=18.4,8EF ,WP,rsEF ,Tph<=86.2,26.5,113.6,29.6HKL<

Figure 4.8: ρxx versus T at B = 0 for VG = 0.9V and VG = 1V

74


DΡ

DT

0.4 0.6 0.8 1.0 1.2 1.40.0000

0.0005

0.0010

0.0015

nsH´1011cm-2L

DΡD

THh

e2-

KL

DΡ

DT=

Ρ0 - Ρph

Tph - T0

vs electron density ns

THKL

ΡxxHhe2L

Figure 4.9: The average change in resistivity with temperature ρ as a function of ns

75


Tph

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.925

30

35

40

45

50

55

60

nsH´1011cm-2L

Tph

HKL

Tph Hµ ns-12L vs electron density ns

THKL

ΡxxHhe2L

Figure 4.10: The temperatures at which phonon scattering starts Tph as a function of carrierdensity ns

76


Ñ

Τe-ph~ kBTph

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08

0.10

0.12

0.14

0.16

nsH´1011cm-2L

Τ e-

phHx1

0-3se

cL

Τe-ph Hµ ns12L vs electron density ns

Figure 4.11: The extracted minimum electron-phonon scattering times τ versus ns

77


Mobility

The mobility values are extracted from the resistivity data at different temperatures by

sweeping the gate voltage and recording the resistivity at different carrier densities. The

mobility of electrons in two-dimensions is given by,

µ =1

nseρ(4.9)

where ρ = Rl/b

is the resistivity per square in Ohms. The plot of µ versus ns (Fig 4.12)clearly

shows that mobility reaches values from 0.5 up to around 0.73 m2

V−sec at lowest densities and

temperatures. These values are high for Si MOSFETs considering the temperatures we have

accessed - so the system still qualifies to be called as ”high-mobility” and ”low-disordered”.

Electron mean free times

The experimental values of electron mobility obtained from the resistivities at different ns and

temperatures are used to extract the electron mean free times (momentum relaxation times)

using the expression from Drude theory, µ = eτmb

. e.g. for µ = 0.7 m2

V−sec , τ = 0.756×10−12sec.

We convert the τ ’s into temperature units 1kB

~τ

to compare them with the Fermi temperature

and actual experimental temperatures. Looking at the τ values for T = 10.6K data, we find

that there are four temperature regimes (Fig 4.13).

For ns . 0.9× 1011cm−2,

EF <~τ< kBT

For 0.9× 1011cm−2 . ns . 1.4× 1011cm−2,

EF < kBT <~τ

78


For 1.4× 1011cm−2 . ns . 1.6× 1011cm−2,

kBT < EF <~τ

And for ns & 1.6× 1011cm−2, the diffusive regime,

kBT <~τ< EF

79


=

·

=

·

Figure 4.12: Mobility µ as a function of ns at different temperatures

80


=

=

= =

h

h

h

=

Figure 4.13: TF and ~/τ vs ns for T = 10.6K

81


4.6.2 Magnetoresistance in parallel field

The resistivity as a function of temperature ρ(T ) in finite parallel magnetic fields showed

the same behavior as that for B = 0 in the range of temperatures we explored. On the other

hand, the magnetoresistance ρxx(B‖) was observed to be weakly enhanced in the parallel

fields applied, e.g. for ns = 0.975 × 1011cm−2, ρ(B) went from 0.415h/e2 to 0.44h/e2 when

the field was ramped from 0T to 14T at 10K (Fig 4.14). Though the magnetoresistance

plots look different at different temperatures (Fig 4.15), the plots of the ratio, ρ(B)/ρ(0)

exactly overlap (Fig 4.16). We have plotted first the magnetoresistance and then the ratio

for temperatures 10K, 16K, 25K and 60K on the same graph for comparison. This strongly

indicates that this is an orbital effect and it is not related to spin properties.

82


¹

¹

¹

¹

¹

¹

¹

¹

¹ ¹

Ã

Ã

¹¹

¹

Figure 4.14: Magnetoresistance ρ(B‖) versus B‖ at T = 10K for VG = 1.1V

83


¹

¹

¹

¹

¹

¹

¹

¹

¹ ¹

¹

¹

¹¹

Figure 4.15: Magnetoresistance ρ(B‖) versus B‖ at different T

84


Figure 4.16: Magnetoresistance ratio ρ(B‖)/ρ(0) versus B‖ at different T

85


4.6.3 Results in perpendicular field

ρxy v/s B⊥

Fig 4.17 shows the Hall effect observed at 10.2K for different carrier densities with the plot

of transverse resistivity ρxy versus B⊥ displaying the expected negative slope (for negatively

charged carriers) and inverse proportionality with the carrier density (∝ 1/ns). The Quan-

tum Hall effect is not seen i.e. no plateaus are observed which means that the electrons are

not degenerate.

ρxx v/s B⊥

Despite the fact that the electrons are non-degenerate, a distinct minimum is observed at

various electron densities in the perpendicular field configuration when we measure the lon-

gitudinal resistivity as a function of magnetic field (Fig 4.19, 4.20, 4.21, 4.22). These can be

interpreted as a Quantum Hall Effect resistivity minimum [67] that has survived up to the

relatively high temperatures reached in our experiments. The position of the minimum is

very sensitive to temperature changes. e.g. at ns = 2.325×1011cm−2 (Fig 4.19), the position

of the resistivity minimum shifts from 4.9T to 5.3T for a small temperature change from

7.1K to 8K.

We observe a broadening in the resistivity minimum for higher temperature until around

the Fermi temperature where the feature disappears.

It is possible to find the filling factor corresponding to these observed features from the

values of the position of the resistivity minima for different electron densities measured at

86


Figure 4.17: Transverse resistivity ρxy versus B⊥

87


nsH´1011cm-2L BminHTL Filling factor HΝL1.575 3.3 1.9763

2.325 5.23 1.8408

3.825 5.9 2.6845

5.325 8.2 2.689

6.825 9.8 2.8838

Figure 4.18: The filling factor ν calculated for the observed QHE resistivity minimum Bmin

at various densities ns

8K (Table 4.18).

ν =ns

eB/h(4.10)

In their experiments on similar Si MOSFET samples at 40mK, Kravchenko et al [67] observed

QHE resistivity minima at fields corresponding to filling factors that correspond to spin

splitting between Landau levels. From the table of values of ν’s extracted at 8K, is reasonable

to infer that the filling factors are actually all ν = 2 that have been shifted due to the effect

of temperature (and broadening). This interpretation becomes plausible because of the

presence of a second resistivity minimum at exactly half of the value of the other Bmin

observed for VG = 5V (ns = 6.825 × 1011cm−2) (Fig 4.22)that could correspond to filling

factor ν = 4.

88


¹

¹

¹

¹

¹

¹

¹ ¹

¹

Figure 4.19: Longitudinal resistivity ρxx versus B⊥ at VG = 2V

89



90



91


¹¹

¹¹

¹¹

¹¹

¹¹

¹¹

¹¹

¹¹

¹ ¹

¹


92

4.7. SUMMARY

4.7 Summary

Based on our experiments to elucidate transport properties in the strongly correlated regime

in 2DES of high mobility Si-MOSFETs, we could establish that the resistivity in the in-

terval, [TF , rsTF ] qualitatively follows the description based on an analogy with viscosity

of liquid 3He and 4He [60]. The temperature at which the phonon-component of resistiv-

ity starts getting significant (due to increasing electron-phonon interactions) is observed to

be proportional to the inverse square root of the carrier density. The electron mean free

times extracted from the mobilities indicate that the electrons are in the diffusive regime for

carrier densities ns & 1.6 × 1011cm−2. The magnetoresistance in parallel magnetic fields is

observed to increase weakly with field and the plots for the ratio with zero field resistivity,

ρ(B)/ρ(0) at different temperatures are seen to exactly overlap which indicates that this is

an orbital effect and not a spin-related one. In perpendicular magnetic fields, the transverse

Hall resistivity does not show plateaus - i.e. the electrons are not degenerate. However, the

longitudinal resistivity, ρxx showed evidence of Landau quantization even at temperatures

∼ TF . This could possibly be due to the fact that our measurements are taken in the strongly

interacting regime.

93

Bibliography

[1] D. K. C. MacDonald, Near Zero: An Introduction to Low Temperature Physics, Anchor

Books, Garden City, New York, 1961.

[2] Subir Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge,

UK, 1999.

[3] Edwards, P. P., Johnston, R. L., Rao, C. N. R., Tunstall, D. P. and Hensel, F., 1998,

The Metal-Insulator Transition: A Perspective, Philos. Trans. R. Soc., A 356, 5.

[4] Abrahams, E., P. W. Anderson, P. A. Lee, and T. V. Ramakrishnan, 1979, Scaling Theory

of Localization: Absence of Quantum Diffusion in Two Dimensions , Phys. Rev. Lett. 42,

673.

[5] Zavaritskaya, T. N. and Zavaritskaya, E. I., 1987, Metal-insulator transition in inversion

channels of silicon MOS structures, JETP Lett. 45, 609.

[6] Kravchenko, S. V., Kravchenko, G. V., Furneaux, J. E., Pudalov, V. M. and D’Iorio, M.,

1994, Possible metal-insulator-transition at B = 0 in 2 dimensions, Phys. Rev. B 50,

8039.

[7] Kravchenko, S. V., Mason, W. E., Bowker, G. E., Furneaux, J. E., Pudalov, V. M. and

D’Iorio, M., 1995, Scaling of an anomalous metal-insulator-transition in a 2-dimensional

system in silicon at B = 0, Phys. Rev. B. 51, 7038.

94

BIBLIOGRAPHY

[8] Pudalov, V. M., Brunthaler, G., Prinz, A. and Bauer, G., 1997, Instability of the twodi-

mensional metallic phase to a parallel magnetic field, JETP Lett. 65, 932.

[9] Shashkin, A. A., Kravchenko, S. V., Dolgopolov, V. T. and Klapwijk T. M., 2001, In-

dication of the Ferromagnetic Instability in a Dilute Two-Dimensional Electron System,

Phys. Rev. Lett. 87, 086801.

[10] A A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev.

Lett. 87, 086801 (2001).

[11] Kravchenko, S.V., Shashkin, A.A., Anissimova, S., Venkatesan, A., Sakr, M.R., Dol-

gopolov, V.T. and Klapwijk, T.M., 2006, Thermodynamic magnetization of a strongly

correlated two-dimensional electron system, Ann. Phys. 321, 1588.

[12] Heemskerk, R., 1998, Transport at Low Electron Density in the Two-Dimensional Elec-

tron Gas of Silicon MOSFETs, Ph. D. thesis, University of Groningen.

[13] Oxford Instruments Ltd., 1996, Kelvinox Dilution Refrigerator and Superconducting

Magnet system: Operater’s Handbook, Instrument Documentation.

[14] Wigner, E., 1934, On the Interaction of Electrons in Metals, Phys. Rev. 46, 1002.

[15] Tanatar, B. and Ceperley, D. M., 1989, Ground state of the two-dimensional electron

gas, Phys. Rev. B 39, 5005.

[16] Landau, L. D., 1957, Kinetic Equation Approach, Sov. Phys. JETP 3, 920.

[17] Vitkalov, S. A., Zheng, H., Mertes, K. M., Sarachik, M. P. and Klapwijk, T. M., 2001,

Scaling of the magnetoconductivity of silicon MOSFET’s: evidence for a quantum phase

transition in two dimensions, Phys. Rev. Lett. 87, 086401.

[18] Kravchenko, S. V. and Sarachik, M. P., 2004, Metal-Insulator Transition in Two-

Dimensional Electron Systems, Rep. Prog. Phys. 67, 1.

95

BIBLIOGRAPHY

[19] Shashkin, A. A., 2005, Metal – insulator transitions and the effects of electron – electron

interactions in two-dimensional electron systems , Phys. Usp. 48, 129.

[20] Attaccalite, C. , Moroni, S., Gori-Giorgi, P. and Bachelet, G. B., 2002, Correlation

Energy and Spin Polarization in the 2D Electron Gas, Phys. Rev. Lett. 88, 256601.

[21] Spivak, B. 2003, Phase separation in the two-dimensional electron liquid in MOSFET’s,

Phys. Rev. B 67, 125205.

[22] Spivak, B. and Kivelson, S. A., 2004, Phases intermediate between a two-dimensional

electron liquid and Wigner crystal, Phys. Rev. B 70, 155114.

[23] Khodel, V. A., Clark, J. W. and Zverev, M. V. , 2005, Thermodynamic properties of

Fermi systems with flat single-particle spectra , Europhys. Lett. 72, 256.

[24] S. Pankov and V. Dobrosavljevic, 2008, Problems with reconciling density functional

theory calculations with experiment in ferropnictides, Phys. Rev. B 77, 085104.

[25] Punnoose, A. and Finkel’stein, A. M., 2005, Metal-Insulator transition in disordered

two-dimensional electron systems, Science 310, 289.

[26] Marchi, M. , De Palo, S., Moroni, S. and Senatore, G., 2009, Correlation energy and

spin susceptibility of a two-valley two-dimensional electron gas, Phys. Rev. B 80, 035103.

[27] Fleury, G. and Waintal, X., 2010, Energy scale behind the metallic behaviors in low-

density Si MOSFETs, Phys. Rev. B 81, 165117.

[28] Shashkin, A. A., Kravchenko, S. V. Dolgopolov, V. T. and Klapwijk, T. M., 2002, Sharp

increase of the effective mass near the critical density in a metallic two-dimensional

electron system, Phys. Rev. B 66, 073303.

[29] Pudalov, V. M., Gershenson, M. E., Kojima, H. , Butch, N., Dizhur, E. M., Brunthaler,

G. , Prinz, A. and Bauer, G., 2002, Low-Density Spin Susceptibility and Effective Mass

96

BIBLIOGRAPHY

of Mobile Electrons in Si Inversion Layers, Phys. Rev. Lett. 88, 196404.

[30] Shashkin, A. A., Rahimi, M. , Anissimova, S., Kravchenko, S. V., Dolgopolov, V. T.,

and Klapwijk, T. M., 2003, Spin-Independent Origin of the Strongly Enhanced Effective

Mass in a Dilute 2D Electron System, Phys. Rev. Lett. 91, 046403.

[31] Khrapai, V. S., Shashkin, A. A., and Dolgopolov, V. T., 2003, Direct Measurements of

the Spin and the Cyclotron Gaps in a 2D Electron System in Silicon, Phys. Rev. Lett.

91, 126404.

[32] Shashkin, A. A. , Anissimova, S., Sakr, M. R., Kravchenko, S. V., Dolgopolov, V. T.

and Klapwijk, T. M., 2006, Pauli Spin Susceptibility of a Strongly Correlated Two-

Dimensional Electron Liquid, Phys. Rev. Lett. 96, 036403; Anissimova, S., Venkatesan,

A. , Shashkin, A. A., Sakr, M. R., Kravchenko, S. V. and Klapwijk, T. M., 2006, Mag-

netization of a Strongly Interacting Two-Dimensional Electron System in Perpendicular

Magnetic Fields, Phys. Rev. Lett. 96, 046409.

[33] Heemskerk, R. and Klapwijk, T. M., 1998, Nonlinear resistivity at the metal-insulator

transition in a two-dimensional electron gas, Phys. Rev. B 58, R1754 (1998).

[34] Fletcher, R.,Pudalov, V. M., Feng, Y., Tsaousidou, M. and Butcher, P. N., 1997, Ther-

moelectric and hot-electron properties of a silicon inversion layer, Phys. Rev. B 56, 12422.

[35] Faniel, S., Moldovan, L. , Vlad, A., Tutuc, E., Bishop, N., Melinte, S., Shayegan, M.

and Bayot, V., 2007, In-plane magnetic-field-induced metal-insulator transition in (311)A

GaAs two-dimensional hole systems probed by thermopower, Phys. Rev. B 76, 161307(R).

[36] Goswami, S. , Siegert, C., Baenninger, M., Pepper, M., Farrer, I., Ritchie, D. A. and

Ghosh, A., 2009, Highly Enhanced Thermopower in Two-Dimensional Electron Systems

at Millikelvin Temperatures, Phys. Rev. Lett. 103, 026602.

97

BIBLIOGRAPHY

[37] Dolgopolov, V. T. and Gold, A., 2011, Thermoelectric properties of the interacting two

dimensional electron gas in the diffusion regime? , JETP Lett. 94, 446; Gold, A. and

Dolgopolov, V. T., 2011, Temperature dependence of the diffusive thermopower in the

two-dimensional interacting electron gas, EPL 96, 27007.

[38] R. Fletcher, V. M. Pudalov, A. D. B. Radcliffe, and C. Possanzini, Semicond. Sci.

Technol. 16, 386 (2001).

[39] Strictly speaking, a theory of the thermopower near the Anderson transition is still not

developed. Therefore, the obtained experimental results can serve as a confirmation that

the thermopower behavior near the Anderson transition (but not the absolute value) is

insensitive to the level of disorder.

[40] In the parallel-field configuration, the spin susceptibility χ ∝ gm is measured. Its en-

hancement is caused by the mass m while the g factor enhancement is weak [32].

[41] S. V. Kravchenko, A. A. Shashkin, D. A. Bloore, and T. M. Klapwijk, Solid State

Commun. 116, 495 (2000).

[42] M. D’Iorio, V. M. Pudalov, and S. G. Semenchinsky, Phys. Lett. A 150, 422 (1990).

[43] Anissimova, S., Kravchenko, S. V., Punnoose, A., Finkel’stein, A. M. and Klapwijk, T.

M., 2007, Flow diagram of the metal-insulator transition in two dimensions, Nat. Phys.

3, 707.

[44] Punnoose, A. and Finkel’stein, A. M., 2002, Dilute Electron Gas near the Metal-

Insulator Transition: Role of Valleys in Silicon Inversion Layers, Phys. Rev. Lett. 88,

016802.

[45] Akira Isihara, Condensed Matter Physics, Oxford University Press, New York, 1991.

98

BIBLIOGRAPHY

[46] Knyazev, D. A., Omel’yanovskii, O. E., Pudalov, V. M. and Burmistrov, I. S., 2006,

Critical behavior of transport and magnetotransport in 2D electron system in Si in the

vicinity of the metal-insulator transition, JETP Lett. 84, 662.

[47] Knyazev, D. A., Omel’yanovskii, O. E., Pudalov, V. M. and Burmistrov, I. S., 2008,

Metal-Insulator Transition in Two Dimensions: Experimental Test of the Two-Parameter

Scaling , Phys. Rev. Lett. 100, 046405.

[48] Finkel’stein, A. M., 1990, Sov. Sci. Rev. A, Phys. Rev. 14,1.

[49] Castellani, C., Castro, C. D., Lee, P. A. and Ma, M., 1984, Interaction-driven metal-

insulator transitions in disordered fermion systems, Phys. Rev. B 30, 527.

[50] Punnoose, A. arXiv:0910.0037.

[51] Ando, T., Fowler, A. B., Stern, F., 1982, Electronic Properties of Two-Dimensional

Systems, Rev. Mod. Phys.54, 437.

[52] Fukuyama, H., 1980, Effects of Intervalley Impurity Scattering on the Non-Metallic

Behavior in Two-Dimensional Systems, J. Phys. Soc. Jpn. 49, 649.

[53] Burmistrov, I. S. and Chtchelkatchev, N. M., 2008, Electronic properties in a two-

dimensional disordered electron liquid: Spin-valley interplay, Phys. Rev. B 77, 195319.

[54] Altshuler, B. L. and Aronov, A. G., chap. Electron-Electron Interactions in Disordered

Systems in Modern Problems in Condensed Matter Physics, Elsevier, North Holland,

1985.

[55] Rahimi, M., Anissimova, S., Sakr, M. R. and Kravchenko, S. V., 2003, Coherent

back-scattering near the two-dimensional metal-insulator transition, Phys. Rev. Lett. 91,

116402.

[56] P. A. Lee and T. V. Ramakrishnan, Phys. Rev. B 26, 4009 (1982).

99

BIBLIOGRAPHY

[57] C. Castellani, C. D. Castro, and P. A. Lee, Phys. Rev. B 57, R9381 (1998).

[58] S. A. Vitkalov, K. James, B. N. Narozhny, M. P. Sarachik, and T. M. Klapwijk, Phys.

Rev. B 67, 113310 (2003).

[59] A. Y. Kuntsevich, N. N. Klimov, S. A. Tarasenko, N. S. Averkiev, V. M. Pudalov,

H. Kojima, and M. E. Gershenson, Phys. Rev. B 75, 195330 (2007).

[60] Spivak, B. and Kravchenko, S. V. and Kivelson, S. A. and Gao, X. P. A., 2010, Collo-

quium: Transport in strongly correlated two dimensional electron fluids , Rev. Mod. Phys.

82, 1743.

[61] Spivak, B. and S. A. Kivelson, 2006, Transport in two dimensional electronic micro-

emulsions , Ann. Phys. (Paris) 321, 2071.

[62] Andreev, A. V. and Kivelson, S. A. and Spivak, B., 2011, Hydrodynamic Description of

Transport in Strongly Correlated Electron Systems , Phys. Rev. Lett. 106, 256804.

[63] Andreev, A. F., 1979, Kinetic Phenomena in Semiquantum Liquids , Sov. Phys. - JETP

50, 1218.

[64] Frenkel, J., Kinetic Theory of Liquids , Oxford, 1946.

[65] Ferrer, M. L. and Kivelson, D., 1999, Constant density activation energies and the role

of activated dynamics in liquids above their melting points , J. Chem. Phys. 110, 10963.

[66] Lee, P. A. and Ramakrishnan, T. V., 1985, Disordered electronic systems , Rev. Mod.

Phys. 57, 287

[67] Kravchenko S. V., Shashkin A. A., Bloore D. A. and Klapwijk T. M., 2000, Shubnikov-

de Haas oscillations near the metal–insulator transition in a two-dimensional electron

system in silicon, Solid State Commun. 116, 495

100

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