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
1.3. METAL-INSULATOR TRANSITION IN TWO DIMENSIONS
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
1.3. METAL-INSULATOR TRANSITION IN TWO DIMENSIONS
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
1.5. DIVERGING SPIN SUSCEPTIBILITY
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
1.5. DIVERGING SPIN SUSCEPTIBILITY
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
1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET
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
1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET
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
1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET
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
1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET
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
1.6. DILUTION REFRIGERATOR AND SUPERCONDUCTING MAGNET
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
1.8. OUTLINE OF THESIS
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
1.8. OUTLINE OF THESIS
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
2.2. EXPERIMENTAL SETUP
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
2.2. EXPERIMENTAL SETUP
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
2.2. EXPERIMENTAL SETUP
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
2.3. THERMOPOWER RESULTS
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
2.3. THERMOPOWER RESULTS
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
2.3. THERMOPOWER RESULTS
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
2.3. THERMOPOWER RESULTS
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
3.2. PREDICTIONS OF THE TWO-PARAMETER SCALING THEORY
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. PREDICTIONS OF THE TWO-PARAMETER SCALING THEORY
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
3.2. PREDICTIONS OF THE TWO-PARAMETER SCALING THEORY
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
3.2. PREDICTIONS OF THE TWO-PARAMETER SCALING THEORY
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
4.4. TEMPERATURE DEPENDENCE OF RESISTIVITY FOR T > TF
(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
4.4. TEMPERATURE DEPENDENCE OF RESISTIVITY FOR T > TF
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
4.4. TEMPERATURE DEPENDENCE OF RESISTIVITY FOR T > TF
𝜌 (ℎ 𝑒2 )
T (K) 𝐸𝐹 𝑇𝑝ℎ Ω𝑃? 𝑟𝑠𝐸𝐹?
Figure 4.3: The different energy regimes for the correlated electron liquid.
65
4.5. EXPERIMENTAL SETUP
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
4.5. EXPERIMENTAL SETUP
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
4.5. EXPERIMENTAL SETUP
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
Lns=0.45´1011cm-2,rs=24.9,8EF ,WP,rsEF ,Tph<=83.4,16.8,83.9,40.5HKL<
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
4.6. EXPERIMENTAL RESULTS
20 25 30 35 40 450.530
0.535
0.540
0.545
0.550
THKL
Ρ xx
Hhe2
Lns=0.675´1011cm-2,rs=20.3,8EF ,WP,rsEF ,Tph<=85.1,22.8,102.7,33.7HKL<
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
=
·
=
·
Figure 4.12: Mobility µ as a function of ns at different temperatures
80
4.6. EXPERIMENTAL RESULTS
=
=
= =
h
h
h
=
Figure 4.13: TF and ~/τ vs ns for T = 10.6K
81
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
Figure 4.16: Magnetoresistance ratio ρ(B‖)/ρ(0) versus B‖ at different T
85
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
Figure 4.17: Transverse resistivity ρxy versus B⊥
87
4.6. EXPERIMENTAL RESULTS
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
4.6. EXPERIMENTAL RESULTS
¹
¹
¹
¹
¹
¹
¹ ¹
¹
Figure 4.19: Longitudinal resistivity ρxx versus B⊥ at VG = 2V
89
4.6. EXPERIMENTAL RESULTS
Figure 4.20: Longitudinal resistivity ρxx versus B⊥ at VG = 3V
90
4.6. EXPERIMENTAL RESULTS
Figure 4.21: Longitudinal resistivity ρxx versus B⊥ at VG = 4V
91
4.6. EXPERIMENTAL RESULTS
¹¹
¹¹
¹¹
¹¹
¹¹
¹¹
¹¹
¹¹
¹ ¹
¹
Figure 4.22: Longitudinal resistivity ρxx versus B⊥ at VG = 5V
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
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