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The microwave response of ultra thin microcavity arrays

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BROADBAND MICROWAVE MEASUREMENTS OF
TWO DIMENSIONAL QUANTUM MATTER
by
Wei Liu
A dissertation submitted to The Johns Hopkins University in conformity with the
requirements for the degree of Doctor of Philosophy.
Baltimore, Maryland
January, 2013
© Wei Liu 2013
All rights reserved
UMI Number: 3572735
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Di!ss0?t&iori P iiblist’Mlg
UMI 3572735
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Abstract
Employing broadband microwave spectroscopy, we study a field tuned 2D superconductor-metal quantum phase transition in a low-disorder InOx film. We measure the
complex conductance as a function of temperature, frequency and magnetic field. The
zero field transition temperature
Tc = T
k tb
is determined as the temperature where
the phase stiffness starts to acquire frequency dependence. The thermal phase tran­
sition is consistent with the Kosterlitz-Thouless-Berezinsky formalism. The AC data
demonstrate the critical slowing down close to
ity of a vortex plasma model above
T
k tb
T
k tb
and in general the applicabil­
• For finite field measurements, we find
the strong suppression in the superfluid stiffness above the nominal quantum critical
point Bcross where different isotherms of resistance as a function of magnetic field
cross each other. The critical slowing down of the fluctuation rate near a continuous
quantum phase transition supports a possible scenario that the quantum critical point
is at a much lower field B sm above which the film transits into an anomalous metallic
state with superconducting correlations. A phase diagram is established that includes
the magnetic field dependence of the superconducting transition temperatures, phase
ABSTRACT
stiffness at the lowest and highest accessed frequencies, and the fluctuation rates at
the base temperature.
Primary Reader: N. Peter Armitage
Secondary Reader: Predrag Nikolic
Acknowledgments
My interests in science were inspired actually by anecdotes of scientists I read
when I was a kid; how they would do something ridiculous cause they were so focused
on their research and thinking. From then on, for some reason, I was always imagining
being one of those nerds.
So now here I am. After 13 years of majoring in physics and doing research in
physics, I am much closer to the type of people I always want to be. Of course, there
is no end in the pursuing. I have grown a lot in the past few years in Hopkins and
this in turn makes my stay and PhD life the most valuable years in my life. W hat I
learned here is not just knowledge, but the way of thinking and solving problems.
So many people here I would like to acknowledge for their help, support, love
and companionship. I could not have carried on in the journey of pursuing my PhD
without them.
First I would like to acknowledge my advisor Prof. N. Peter Armitage. He has set
a great example to me about how to be a great scientist by his passion and consistent
pursuit in physics. I appreciate all his support and trust along the way. He always
ACKNOWLEDGMENTS
has the patience for my questions even though most of time my questions may be
simple and naive and may not be of physics nature. There have been countless times
that I call from the lab about any possible situations happening in the lab and Peter
always has time to discuss. I also feel grateful for all the travel opportunities that I
have throughout these years to present my work to colleagues outside the university.
I feel really lucky to have Peter as my PhD advisor.
Natalia Drichko has been a great mentor in life for me. She always has time when
I want to talk to her. Thank you for being so generous of your time. My lab mate
Luke Bilbro, who joined the lab half a year earlier, has been a great accompany for
more than five years of working together in the lab. I would like to thank him for
his support and friendship throughout almost my whole graduate school. Rolando
Valdes Aguilar has taught me so much about physics, ways of thinking and countless
useful techniques and skills in writing and giving presentation. I thank him for his
inspiring and useful conversations. I also thank all the rest lab members: Andreas,
Chris, LiDong, Mohammad, Yuval, Grace, Liang, Nick and Ji, thank you all for the
great time together and friendship. I am glad to have this great opportunity to work
with you all.
I acknowledge my long time collaborator Prof. Sambandamurthy at University of
Buffalo for his efforts and supports in many ways. I enjoyed all the conversations and
discussions with him. I also thank Yufeng from Prof. Ruoff’s group at UT Austin for
the high quality graphene samples.
ACKNOWLEDGMENTS
I would like to acknowledge all the professors who taught me during my graduate
school. I have learned so much from all the lectures and valuable discussions. I
also have been fortunate to meet all my dear friends through the department, JHU
Taekwondo club and the school.
I thank my family for always being supportive, especially my husband Brian.
Without their love and understanding, this thesis work would have been impossible.
D edication
Dedicated to my family
Contents
A bstract
ii
Acknowledgments
iv
List of Figures
xi
1 Introduction
1
1.1
Ginzburg-Landau th e o r y ..........................................................................
2
Enhanced conductivity above Tcby amplitude fluctuations . .
5
1.1.1
1.2
Kosterlitz-Thouless-Berezinsky phase tr a n s itio n ..................................
8
1.2.1
2D XY m o d e l................................................................................
9
1.2.2
Superfluid stiffness and the universal j u m p ................................
14
1.2.3
Nonlinear IV characteristic...........................................................
16
1.3
2D superconductor-insulator quantum phase tr a n s itio n .......................
17
1.4
AC response of a su p erco n d u cto r..........................................................
23
1.5
Thesis overview .........................................................................................
26
viii
CONTENTS
2
Broadband Corbino microwave spectrom eter
29
2.1
Experimental setup overview...................................................................
31
2.2
Data a n a ly s is ............................................................................................
39
2.3
C alibrations...............................................................................................
42
2.3.1
Room temperature c a lib ra tio n s..................................................
43
2.3.2
Low temperature c a lib ra tio n s.....................................................
52
2.3.2.1
Effects of the superconducting c a b l e .........................
53
2.3.2.2
Error coefficients...........................................................
60
2.3.3
2.4
3
4
5
Microwave measurements in a perpendicular magnetic field. .
Review of the Corbino spectrometers from other g ro u p s......................
62
67
AC studies o f th e zero field phase transition
69
3.1
Superconducting fluctuations...................................................................
69
3.2
Dynamics of KTB phase tr a n s itio n .......................................................
73
3.3
Sample Details
.........................................................................................
76
3.4
R esults........................................................................................................
78
3.5
Conclusion..................................................................................................
89
2D field tuned superconductor-m etal quantum phase transition
91
4.1
Dynamics of 2D quantum phase tr a n s itio n ..........................................
93
4.2
Conclusion..................................................................................................
107
Summary
108
ix
CONTENTS
A
Transmission line m odel
111
A.0.1 Impedance m a tc h in g ....................................................................
B
113
Experim ental procedures
116
B.l
Measurements at zero fie ld ......................................................................
116
B.1.1 Operating sequence.......................................................................
118
Measurements at finite field s...................................................................
125
B.2.1 Operating sequence.......................................................................
126
B.2
C
Reflection coefflcients for zero-field m easurem ents
130
D
M agnetic field distributions
134
E
Microwave conductance o f InO* at finite fields
140
F
AC conductance o f CVD grown graphene
148
F .l Electrodynamics of single layer
graphene......................................................................................................
149
F.2 Conclusion...................................................................................................
158
Bibliography
160
V ita
178
x
List of Figures
1.1
1.2
1.3
GL free energy density function for a > 0 and a < 0.......................
The spin configuration of a vortex.......................................................
A proposed phase diagram for the dirty boson m o d e l ...................
2.1
2.2
2.3
Scheme of the overall experimental s e t u p .........................................
31
Pictures of the Corbino p r o b e ...........................................................
33
Scheme and a picture of the Au pattern for the sampleprepared for
microwave m easurem ents.....................................................................
34
Scheme plot of the sample s ta g e ........................................................
35
Pictures of the setup in our l a b ........................................................
36
Magnitude of S ll for different open su b stra te s...............................
46
Comparison of calibrated impedance for 40 nm NiCr on Siusing dif­
ferent open stan d ard s
47
Effects of the position of the set s c r e w ............................................
48
Impedance of a 10 nm A1 film with different spring configurations . .
50
Impedance of InOj at room temperature showing the importance of
substrate correction..............................................................................
50
Magnitude of the substrate im p e d a n c e ............................................
52
Four temperature scans for the bulk copper at 7 GHz with a 20 cm
superconducting cable in the transmission l i n e ................................
55
Magnitude and phase of S™ of Si as a function of temperature at 7
GHz with a 10 cm superconducting cable in the transmission line . . 56
Magnitude and phase of S'™ of InOx as a function of temperature at 7
G H z .......................................................................................................
59
Temperature dependence of the magnitude and phase of the three error
terms at 9 G H z ....................................................................................
61
The hysteresis of the magnet in our sy stem ...................................... 64
Magnitude and phase of S™ of a Si standard as a function of field at
300 mK for 4 fixed frequencies............................................................
65
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
xi
4
13
19
LIST OF FIGURES
2.18 Ratio of real conductance of InOx measured at 3.5 Tesla but calibrated
by standards measured at 2 Tesla and 3.5 T e s l a ..................................
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
4.1
4.2
4.3
4.4
4.5
4.6
A .l
B.l
66
Rq as a function of temperature for one of the InOx films...................
Measured R q as a function of temperature along with two fitted effec­
tive normal sheet resistances....................................................................
The AFM image, TEM diffraction pattern and TEM image of a co­
deposited insulating InOx f i lm .................................................................
Sheet resistance of a granular InOx film ................................................
Temperature and frequency dependence of the complex conductance
at zero f i e l d ................................................................................................
The phase stiffness at different frequencies as a function of temperature
The phase stiffness on a linear scale near Tc .......................................
The phase and magnitude of the complex c o n d u c ta n c e ....................
The phase and magnitude of the scaling function ..............................
Comparison of the scaling forms with the 2D AL formalism..............
Different fittings to the temperature dependence of the characteristic
fluctuation frequen cy................................................................................
71
72
77
77
79
81
83
84
85
86
90
Co-measured sheet resistance as a function of temperature in different
fields..............................................................................................................
95
Rq measured in a separate dilution fridge e n v iro n m en t.....................
98
Frequency dependence of the real and imaginary conductance at dif­
ferent f i e l d s ................................................................................................
99
Frequency dependence of the phase stiffness at 300 mK at different fields 100
Experimentally measured real and imaginary conductance and fitted
data to a Lorentzian lineshape model....................................................... 103
Fluctuation rates at different magnetic fields and a proposed phase
d ia g ra m ...................................................................................................... 105
A schematic plot of the transmission line m o d e l.................................
Ill
Examples of temperature profiles for the standardized zero field mea­
surements...................................................................................................... 117
C .l
C.2
C.3
Magnitude and phase of S'™ of one Nb film at 7 G H z ........................
Magnitude and phase of S™ of one NiCr film at 7 G H z .....................
Magnitude and phase of S™ of a glass substrate at 7 G H z..................
131
132
133
D .l
D.2
D.3
D.4
The design of our cryostat from Janis company .................................
Contour plot of B z ..................................................................................
Contour plot of B r ....................................................................................
Contour plot of the magnitude of the magnetic field.............................
136
137
138
139
LIST OF FIGURES
E .l
E.2
E.3
E.4
E.5
E.6
E.T
Complex
Complex
Complex
Complex
Complex
Complex
Complex
conductance
conductance
conductance
conductance
conductance
conductance
conductance
and
and
and
and
and
and
and
phase
phase
phase
phase
phase
phase
phase
stiffness
stiffness
stiffness
stiffness
stiffness
stiffness
stiffness
F .l
F.2
at
at
at
at
at
at
at
B
B
B
B
B
B
B
=
=
=
=
=
=
=
0
T e s l a ........
1T e s l a ....................
2T e s l a ....................
3.5 T e s l a ...............
4
T e s l a ........
5T e s l a ....................
6T e s l a ....................
141
142
143
144
145
146
147
Raman spectra and DC values of graphene on S i ......................... 151
Calibrated impedance and conductance of graphene in the microwave
ra n g e ................................................................................................... 153
F.3 Ratio of the real and imaginary conductance of graphene in the mi­
crowave r a n g e
155
F.4 Complex normalized conductance of graphene in terahertz range . . . 157
xiii
Chapter 1
Introduction
Superconductivity was first discovered by Heike Kamerlingh Onnes in 1911. The
three hallmarks of superconductivity are zero electrical resistance, Meissner effect
(perfect diamagnetism) and flux quantization. One of the early phenomenological
descriptions of superconductivity that remains useful nowadays was proposed by the
Londons [1]. The two famous London equations
47rA2
C^
t
V x Js =
c2
a3dt
-B
- E.
(1.1)
(1.2)
govern the microscopic electronic and magnetic fields. J s is the current density. A
is the penetration length which is defined as 47rA2/c2 = m /( n se2) where n8 is the
number density of superconducting electrons. These equations were followed by the
phenomenological mean-field Ginzburg and Landau (GL) theory [2], and then the
fundamental microscopic BCS theory [3].
1
CHAPTER 1. INTRODUCTION
The well accepted microscopic BCS theory, was developed by Bardeen, Cooper and
Schrieffer to explain the superconducting state. According to this theory, even a weak
attractive interaction between electrons causes an instability of the Fermi-sea to the
formation of bound pairs of electrons with equal and opposite momentum and spin.
These bound pairs of electrons, so called Cooper pairs, comprise the superconducting
charge carriers. Therefore, superconductivity can be understood as an effect caused
by a condensation of Cooper pairs into a macroscopic quantum state. In conventional
superconductors, the attraction is caused by the electron phonon interaction. One of
the key predictions of the BCS theory is the existent of an energy gap A, of order
kBTc. A minimum energy Eg — 2A is required to break a Cooper pair and create two
quasi-particle excitations. In the weak coupling limit, near the critical temperature
Tc, Eg depends on temperature as Eg = 3.52ksTcy / l —(T /T c). This expression is
independent of materials and has been confirmed in numerous experiments.
1.1
Ginzburg-Landau theory
Despite the triumphal success of BCS theory, the full microscopic theory becomes
very difficult in some situations, such as systems with the presence of inhomogeneous
ns. Another powerful tool that has been widely used to describe superconductivity
is the Ginzburg-Landau (GL) theory, which was postulated as a phenomenological
model. GL theory is also very useful in describing thermal and quantum supercon­
2
CHAPTER 1. INTRODUCTION
ducting fluctuations, which are the main focuses of this thesis. In this section, I
will mainly discuss the GL mean field theory and the Aslamazov-Larkin conductiv­
ity. The treatment given here loosely follows the more explicit treatments given in
references [4,5].
In the GL theory, a pseudowavefunction tp(r) = A eltp is introduced as a complex
order parameter, which describes the center-of-mass motion of the electron pairs.
|^ (r)|2 is the local number density of superconducting electrons ns(r), which goes to
zero at Tc when there are no thermal fluctuations present. Assuming ip is small and
varies slowly in space, the free energy of a superconductor consists of three main parts
Fs = Fn + Condensation energy + Kinetic energy + Field energy
Fn is the free energy in the normal phase. Therefore, the free energy density can be
written as
(1.3)
If ip — 0, fs =
+
Here, because of the pairing, e* = 2e and m* = 2m e, where
m e is the electron mass, a and j3 (fi > 0) are both material dependent parameters.
In the absence of fields and gradients, equation 1.3 becomes
(1.4)
The minimum of f s - f n occurs at |-0|2 = 0, f s - /„ = 0 when a > 0, and \ip\2 —
| = |^oo|2, fs — fn = —|g =
when a < 0 as shown in 1.1. Thus, or must
change sign at the GL transition temperature Tc. Following Ginzburg and Landau,
3
CHAPTER 1. INTRODUCTION
fs - fn
fs - fn
A
A
a > 0
a< 0
Figure 1.1: GL free energy density function for a > 0 and a < 0.
we take a = ao(t — 1), where t = T /T c, a 0 > 0 and j3 independent of temperature.
One important quantity in the GL theory is the coherence length £. Let’s first
examine the GL differential equation which can be obtained by taking d f s/dip = 0
since a choice of ip should minimize the free energy. Without fields, the GL differential
equation becomes
aip + 0\ip\2tp - —
— V 2ip = 0.
2m *
(1.5)
Assuming ip — ipoo+Sip and keeping the leading terms in Sip, one can rewrite equation
1.5 as
fj2
aSip + 3/31^00\28ip - - — V 2Sip = 0.
2m *
( 1.6)
Substituting |-0oo|2 = — we obtain
(1.7)
2m '
4
CHAPTER 1. INTRODUCTION
The solution to this equation in one dimension is of the form 6xp ~
« T>- s i -
where
<18>
The expression of Sip shows that a perturbation of ip by a small Sip will decay in a
length of order £(T). Therefore f(T ) is called the GL coherence length, the charac­
teristic length for the variation of -0(r). £(T) diverges as T —>Tc.
1.1.1
Enhanced conductivity above Tc by am pli­
tude fluctuations
Thermodynamic fluctuations in 3D clean superconductors are generally small due
to the long characteristic coherence length and large superfluid density. These super­
conductors usually feature a very sharp transition between the normal and supercon­
ducting states. However, in dirty 3D superconductors, the coherence length is reduced
by the mean free path. As a result, thermal fluctuations round the sharp edges of the
superconducting transition and bring down the transition temperature. Moreover,
they make small but measurable contributions to superconducting properties above
Te.
We start by first estimating the size of thermal fluctuations of xp near the free
energy minimum. This can be obtained from
J^jjr) and one has (Sip)2 =
ksT /\a\. To better understand the effects of thermal fluctuations, we start with the
free energy in the momentum space. xp(r) can be expanded as xp(r) = ]T]k p \e ik r. At
5
CHAPTER 1. INTRODUCTION
T > Tc, the free energy density in equation 1.3 in the zero-field case can be rewritten
as
'
k x
We drop the f l^]4 term in the expression above because for the GL mean field theory
|^ (r)|2 oc ns is small in the fluctuation regime above Tc except in the narrow critical
region near Tc. The thermodynamic average of |^k|2 can be calculated as
12
I l^k|2exp( - f / k BT)d?ip______ k BT
f exp(—f / k BT)d?ip
< l ^ k| >
| a | ( l + **£*)
(L10)
£ again is the coherence length defined in equation 1.8. The spatial correlation func­
tion of -0(r) and ^(r') is
g( R = r - r') = <
< | ^ | 2 > exp(ik • R).
>=
(1.11)
k
Plugging in the expression for < 1^12 > and integrating over k, we get
g{R) = wtvrexpj im-n)
2irn
This shows that
(112)
R
r) is exponentially correlated over the length scale £(T) in the fluc­
tuation regime above Tc. One can interpret the thermal fluctuations as evanescent
short-lived droplets of superconductivity of size ~ £(T) [5]. The presence of these
superconducting droplets above Tc must affect the conductivity measured in experi­
ments. This excess conductivity, which is also called paraconductivity, is in general
small, but can be studied especially in low dimensional superconducting samples.
To calculate the enhanced conductivity above Tc due to the thermal fluctuations,
one has to include the lifetime of the fluctuations. This is because the contribution
CHAPTER 1. INTRODUCTION
of a fluctuation to the paraconductivity is proportional to the time during which
the fluctuation-induced superconducting pairs exist to be accelerated by the external
field. This can be modeled by the time dependent Ginzburg-Landau equation as
ah
dip
8kB(T - Tc) at
(
. l2
h2
2 ^ V 2) i/>.
(a + m 2-
(1.13)
This equation is equivalent to assuming the deviation of tp from its equilibrium value
to relax exponentially in time. The relaxation time for k = 0 mode is
T“ (r) - s M ^ j '
<U4>
The linearized TDGL equation can be written as
-T O IT J = (1 - f 2V V .
(1.15)
Thus, the high-energy modes with k > 0 decays more rapidly with a relaxation rate
i
Tk
= i± * £ .
(1.16)
TGL
In the absence of such fluctuations, the normal DC conductivity is on= ne2T/m
where r is the mean scattering time of the normal electrons.
Wemightexpect that
the fluctuation conductivity can be written as
_
_ 2e2
af luc ~ m * 2 s
k
<
f^ k l2 >
2
Tk
■
(
)
Inserting < \ip\n\2 > and Tk from expression 1.10 and 1.16 and integrating over the k
space, one finds the two-dimensional Aslamazov-Larkin fluctuation conductivity
AL _
20
e2
T
16h d T - T c'
7
/ i I q\
^
^
CHAPTER 1. INTRODUCTION
Another contribution (Maki term) to the fluctuation conductivity comes from the
enhanced normal electron conductivity induced by superconducting fluctuations. In
the presence of pair-breaking effects, the contribution from the Maki term is strongly
suppressed. This pair-breaking parameter should be large for 2D amorphous super­
conducting films [6,7] and one can safely drop this additional term for these materials.
1.2
K osterlitz-Thouless-Berezinsky phase
transition
The GL mean field phase transition described in section 1.1 takes place when the
magnitude of the order parameter first becomes non-zero. However, two dimensional
systems show transition temperatures that can be substantially below the GL tran­
sition temperature. Unlike 3D superconductors in the clean limit, superconducting
fluctuations are largely enhanced in 2D films due to reduced superfluid density, re­
duced dimensionality and reduced coherence length in the presence of disorder for
films such as InOx. In this case, the transition temperature is controlled mainly by
fluctuations in the phase of the order parameter induced by the presence of thermally
activated free vortices.
However, if one writes down the Hamiltonian for a two dimensional system with
continuous symmetries and sufficiently short range interactions, the Mermin-Wagner
theorem states that the system can not show spontaneous symmetry breaking. It
8
CHAPTER 1. INTRODUCTION
means that at any finite temperature it is impossible to have
G(r -» oo) = lim (e-M?) fM #)) ± 0.
r—
>oo
(1.19)
This expressions means that a true long-range order is impossible. To explain the
phase transition in two dimensions, Berezinsky (1971) [8] and Kosterlitz and Thouless (1973) [9] demonstrated that if one includes topological defects (vortices and
anti-vortices), there is a transition at T k tb separating power law correlated and ex­
ponentially correlated states. In this picture, there are thermally activated vortices
above Tk t b and G(r) decays exponentially in r; For T <
T Kt
b
,
vortices can only
exist in bound pairs with opposite vorticity and G(r) decays in power law in distance.
In this section, I will give a short overview of the 2-dimensional Kosterlitz-ThoulessBerezinsky (KTB) phase transition. Readers can refer to reference [9-15] for detailed
calculations and discussions as well as renormalization group (RG) equations for the
KTB transition and also the analogy of KTB physics to 2D Coulomb gas physics.
1.2.1
2D X Y m odel
The classical 2D XY model can be used to describe phase transitions in superfluid
4He, 3He films and superconducting films (note that the 2D XY model can only be
applied to superconducting films if the in plane magnetic penetration length is bigger
than the sample dimension [16]. Vortices and antivortices are thus held together
by logarithmical confining potentials). In this model, the system consists of spins
9
CHAPTER 1. INTRODUCTION
of unit length arranged on a 2D square lattice. These spins can be represented as
conventional vectors S* = S cos 0t\ + S sin <9j. Oi is the angle of the spin on site i with
respect to some arbitrary polar direction in the 2D vector space. Using |5 |2 = 1, the
Hamiltonian for the system can be written as:
H = - J ] T Si • ^ = - J ] T cos(0i - Oj).
<i,j>
<i,j>
(1.20)
Here, J is the coupling constant (spin stiffness) between the nearest neighbor spins.
J > 0 for ferromagnetic interactions. < i , j > sums over all the nearest neighbors on
the 2D square lattice.
The Hamiltonian remains the same for a rotation of all the spins by the same
angle, thus shows an 0(2) symmetry. However, its ground state with all the spins
pointing in the same direction must spontaneously break this 0(2) symmetry. If
we assume that the spins are nearly parallel from site to site at low temperatures
(T < J), we can have cos(0j —8j) ~ 1 —|(0, —8j)2 (spin-wave approximation). In
this approximation and in the continuum limit, equation 1.20 can be expressed as
H ~ i J <MV0)2,
(1.21)
Two dimension is a critical dimension for a lot of phenomena. To demonstrate the
particularity of two dimension, we consider a d-dimensional XY model and generalize
the Hamiltonian in equation 1.21 to a d-dimensional cubic lattice by extending the
10
CHAPTER 1. INTRODUCTION
integral over r to d-dimension. The average of Sx in d dimension is [14]
M
= < S x > = < cos0(r) > = < cos 0(0) >
J D9e~l" ’eie).
=
(1.22)
(1.23)
Here J D S = n J l , d 8 „ integrates over all possible configurations. After Fourier
transforming the phase fields and some algebra, < S* > reduces to [14]
— -s^ jC -^ )
(L24)
Sd is related to the function integral of all possible configurations of the phase field.
Integral I(L) =
dkkd~3 strongly depends on the dimensionality as following [14]:
m
L2~d
d < 2
ln(L/a)
d =2
5 W
- 2 ^>2
Here, the integration at short distances is cutoff by the vortex core size fo ~ o. while
at large distances is cutoff by the finite system size L. < Sx > must be zero in
the limit of large systems for d < 2 at any finite temperatures. This indicates that
no true ordered phase exists at low temperatures for d < 2, in agreement with the
Mermin-Wagner theorem.
One can also calculate the two point correlation function in d dimension [14]
exp(
G(r) = < S(r) • S(0) > ~ <
(l)^
r r)
d = x
d = 2
exp (—const. * T ) d > 2
11
CHAPTER 1. INTRODUCTION
where 77 = kBT/2irJ. As showed by G(r ->
0 0 ),
the system features an ordered phase
at low temperatures for d > 2. For d = 2, G(r) decays algebraically indicating that
the system is “critical” at any finite temperature.
The spin-wave approximation fails to account for the phase transitions in 2D.
This is because this approximation only describes small continuous deviation from
the ground state configurations. It rules out the possible existence of vortices and all
vortex like excitations. We notice that the Hamiltonian in equation 1.20 is invariant
under discrete local transformations 0i —» 0* ± 2ir. To the contrary, this additional
symmetry is not conserved in the spin-wave approximation. So the approximation
breaks down when we include vortices in the system. A vortex is a topological defect
in which the phase winds by ±2ir in going around the defect (Fig. 1.2) where
27m
if all paths encircle the center of the vortex
0
if all paths do not encircle the center of the vortex
V
Here, n does not depend on the choice of the path. The phase transition is topological
in nature by including vortex excitations.
To estimate the energy cost to introduce a vortex into the lattice, we assume
6(r) = 9(r).
Therefore, we get 27rr|W (r)| = 2irn from the previous close path
integral. Using equation 1.21, one obtains
E
(1.25)
(1.26)
Ec is the core energy. It is obvious that vortices with higher winding numbers are not
CHAPTER 1. INTRODUCTION
/
/
y
j
/
S
H
U
' '• - - X
^
\
W
\
U W W
W s t t t
\ \ \ w / /
\
\
/ /
/
Figure 1.2: The spin configuration of a vortex.
energetically favorable since the energy of a vortex is quadratic in the charge. For a
large enough system, the energy of one vertex diverges logarithmically with the size
of the system.
To obtain the free energy of a vortex, we estimate the entropy S = ka ln(L2/a 2)
from the number of independent places where a vortex of size a can be placed in a
system of size L. The free energy is given by
F = Ec + ( i r J - 2 k BT ) \n { - ) .
(1.27)
CL
F —Ec changes from positive to negative through T k t b = n J /2 k s from below. Above
Tk
tb
the system canlower the free energy by allowing vortex excitations.
In reality it requires less energy to create a bound vortex anti vortex pair (E =
2Ec + Ei ln(R/a) where E\ oc J and R is the distance between the vortex and an­
tivortex [14]). At T <
Tk
tb
vortices can exist only in bound pairs with opposite
13
CHAPTER 1. INTRODUCTION
vorticity. These pairs do not disrupt phase coherence because their net vorticity is
zero. The vortex and antivortex in the pair interact via a logarithmic confining po­
tential. Bound pairs of vortices at short distances screen the interaction potential
of pairs with longer separations, thus renormalize the bare spin stiffness J measured
on long length scales. KTB phase transition happens in a fashion that the vortex
pairs with the largest separation that are bound at
vortices begin to proliferate above
Tk
tb
T < T
ktb
unbind at
Tk
t b
■
Free
and the state with a quasi long range order
is destroyed.
1.2.2
Superfluid stiffness and th e universal jum p
In this thesis, I always use the superfluid stiffness (or the phase stiffness) Tg = J
to describe the effects of thermally activated vortex pairs in the superconductors. We
need to find the connection between the 2D XY model and 2D superconductors. In
superconductors, each vortex carries one quantized magnetic flux and has a phase
circulation 2n around the normal core of the vortex. Similar to spin stiffness, phase
stiffness describes the energy scale required to twist the phase of superconducting
order parameter 0(r) of superfluid or superconductors. The increase in the free energy
due to the twist is given by F(V0) - F(0) = \T e(V6)2. The gradient to the phase field
is related to the superfluid velocity us(r) = ^ V 0 . m* is the effective mass for the
bosons causing the superfluidity. For 4He films m* is the mass of the helium atom and
for a superconductor it is the mass of a Cooper pair. Together with F(V0) —F(0) =
14
CHAPTER 1. INTRODUCTION
| Nm*Vg, one concludes that phase stiffness is determined as Tg = N h2/m*.
At T > T k tb , the appearance of the first unbound vortex pair causes the phase
stiffness to drop discontinuously to zero. This so called “universal jump” is a par­
ticular feature associated with phase transitions of the KTB variety. Despite the
discontinuity in the stiffness the phase transition itself is still continuous. The uni­
versal jump occurs at
Tk
tb
= f Tg
{A T k
tb
= Tg for Tg defined by expression 3.2).
The two point correlation function G(r) starts to decay as G(r) ~ e~r/^ T) with free
vortices in the system. £(T) diverges in an unusual fashion as
£(T) ~ e-b\T-TKTB\~112
(L28)
when one approaches Tt k b from above. This divergence is much faster than any
power law. This exponential dependence comes from f 2 ~ l / n F where nF is the
vortex density. The vortices are thermally activated and hence nF oc exp{-(3E) with
/3 = l / k BT. This is just a simple picture for this fast divergence. A more rigorous
way to obtain the stretched exponential dependence on temperature is to use the RG
equations.
Another feature of KTB physics is that the temperature dependence of the re­
sistance curve above
TTk
b
can be well described by the Halperin-Nelson form [17].
A rough estimation gives R oc nF ~
£ -2
~ eib\T-TKTB\~1/2 Experimental confirma­
tion can be found in reference [12]. Although this agreement is often considered as
an indication that the phase transition is indeed induced by phase fluctuations, the
author in reference [12] pointed out that “this conclusion may be too rash”. Two
15
CHAPTER 1. INTRODUCTION
reasons: (1) the resistance form is for T very close to T k tb (the reduced temperature
should be way less than 1 ) when £ still has the unusual dependence on temperature.
However, most data are from outside this region. (2) The measured resistance curves
have little structure and the fitting form have too many free parameters. Therefore,
it is important for us to look at the AC response of 2D superconductors instead of
utilizing only DC probes to bear on this problem.
1.2.3
N onlinear IV characteristic
In additional to the universal jump in phase stiffness and also the universal func­
tional form of the resistance, KTB physics shows another jump in the nonlinear IV
characteristic. In the KTB model, in principal there are no free vortices present be­
low
T
k tb
, and hence no flux-flow resistance. However, each bound pair has a small
probability of being “ionized” provided a finite current j is imposed across the super­
conducting film. Therefore there is always some exceedingly weak dissipation in a 2D
superconducting film. The critical current is actually zero in 2D. The theoretic pre­
diction for the flux-flow resistance generated below
IV characteristic of the form V ~ J “ , where a =
1
Tk
+
tb
is equivalent to a nonlinear
2 7Arg .
V is linear in J for small
enough current when the system is in the normal state. At T — T K t b >a = 3. Thus
one expects a universal jump i n a = l - » a = 3 when T
Tk
tb
from above [18-21].
In 2D, the potential for the opposite charges depends logarithmically on distance.
As a result, the KTB physics can be mapped to a 2D Coulomb gas model. The
16
CHAPTER 1. INTRODUCTION
two models share the same grand canonical partition function and they fall into the
same universality class. The polarization of bound vortices by a current is equivalent
to the electric polarization in a medium. In the case of a Coulomb gas system, the
dielectric constant e is renormalized by the screening from dipoles of different intra­
pair separations. An increase in dielectric constant indicates the bound pairs become
more polarizable, corresponding to a decrease in the phase stiffness. At T k t b , free
charges present in the system and drive 1 /e (or Tg in the case of superconductors)
discontinuously to zero.
1.3
2D superconductor-insulator quantum
phase transition
One motivation of our project is to understand how superconductivity is destroyed
across the quantum phase transition boundary. For the 2D thermal superconducting
transition, as we detail in the previous section, the appearance of the first thermal
activated vortices kills the superconducting state. It has been shown experimentally
that certain 2D thin films undergo a quantum phase transition from a superconductor
to an insulator with infinite resistance at zero temperature. This quantum phase
transition is driven by a non-thermal parameter in the Hamiltonian, such as the
applied magnetic field, the film thickness, the disorder level, the pressure and so on.
Most of the measurements in this field are performed at sub-Kelvin temperature so
17
CHAPTER 1. INTRODUCTION
that the thermal energy K b T is sufficiently small. In most experiments, one measures
the resistance per square
R
q
through transport.
R
q
is obtained by setting the length
and width of the film equal to each other (to form a square). By doing so, /?□ should
not depend on the dimension of a film (except thickness).
One important theory model for a 2D superconductor-insulator transition (SIT) is
the “dirty boson” point of view which was proposed by Fisher [22]. A phase diagram
related to this model as a function of magnetic field, temperature and disorder is
displayed in 1.3. In this picture, the 2D quantum phase transition is a transition
directly from a true superconductor to a true insulator (defined at zero temperature
and a true insulator is only well defined at absolute 0 K with a diverging resistance).
A metallic state can only exist at the quantum critical point (QCP) with a universal
resistance of order h /4e2 ~ 6.4 Kfi. According to the scaling theory, the resistance
curve at the transition point should have little dependence on temperature at low
temperatures and also the magnetoresistance curves for a field tuned SIT should
have a well-defined crossing point, which is in general interpreted as the location of
QCP.
The dirty boson model describes a 2D SIT caused by the quantum fluctuations
purely in the order parameter’s phase. It suggests that superconductivity in 2D
is destroyed by phase fluctuations instead of by suppression of the magnitude of
the order parameter. If this is true, Cooper pairs can also exist in the insulating
phase, but they cannot move around freely. In 2D SIT, a duality transformation of
18
CHAPTER 1. INTRODUCTION
ELECTRON
GLASS-:
NORMAL
VORTEX
LATTICE
^SU PER OONOUCTOI
NORMAL
Figure 1.3: A proposed phase diagram for the dirty boson model. Reprinted from [22].
CHAPTER 1. INTRODUCTION
Cooper pairs and vortices can map superconductors and insulators onto each other
[23]: in the superconducting phase, we have Bose condensation of Cooper pairs, but
localized vortices; in the insulating phase, we have Bose condensation of vortices, but
localized Cooper pairs. The two phases are separated by a quantum critical point
at zero temperature. This duality view can also been reasoned from the Heisenberg
uncertainty principle between phase and particle number: AnAip > 1. Bosons can
stay in either an eigenstate of phase (A<p = 0 indicating a phase coherence) which is
a superconductor, or an eigenstate of particle number (An = 0 indicating localized
Cooper pairs) which is an insulator.
Theoretically one can argue that the 2D SIT is always bosonic. However, the
bosonic region can be arbitrary narrow and might not be feasible to probe directly in
some experiments. Therefore, this pure bosonic model still remains an open problem
in experiments. On the experimental side, while the transport, microwave cavity
and STM measurements on highly disordered InOx films [24-26] and the transport
on TiN films [27] seem to favor the scenario that pairing exists on both sides of
the transition, electron tunneling measurements on ultrathin quench condensed Bi
films near the SIT [28] suggest that the superconducting gap becomes very small and
approaches zero at the transition point. In this scenario, Cooper pairs are broken
at the transition and the insulating state is dominated by fermion physics [29]. The
appearance of a Fermion insulator across the QCP is not predicted by the dirty boson
model.
20
CHAPTER 1. INTRODUCTION
Another experimental observation that cannot be explained by the dirty boson
model is the possible existence of a range of metallic states at zero temperature. For
many samples,
at the crossing point is pretty close to the universal resistance. The
notable exceptions from
R
q
are MoGe and other low-disorder films. The transitions
in these samples are also sample dependent. Moreover, these samples show signs of
low temperature intervening metallic states between the superconducting and insu­
lating states. One possible explanation by Kapitulnik et al. [30] is that the effects of
dissipation come from the coupling of the system to a dissipative heat bath. However,
the source of the heat bath is still unknown.
Despite all the disagreements on the possible mechanisms how superconductivity
is destructed across the transition, one common feature for all the continuous QPT is
the diverging correlation lengths £ and the diverging correlation time £T in the vicinity
of the QCP [31,32]. Theoretically, a d-dimensional quantum system can be mapped
to a (d+l)-dimensional classical system, where the extra dimension is imaginary time
with a size set by h/3. For example, the time evolution of a ID Josephson junction
array (ID XY model) is equivalent to a configuration of a 1+1D classical XY model.
The coupling constant in the ID system corresponds to the temperature of the 2D
classical system. The ordered and disordered phases in the classical model represent
the superconducting and insulating phases in the quantum system. This analogy
enables us to generalize the critical behaviors near the critical point of a classical
transition. At T = 0 for the quantum system, £ and £r diverge as S = X — X c
21
0
CHAPTER 1. INTRODUCTION
in the following fashion
e ~ i«r
(1.29)
(1.30)
8 describes how far the system is away from the critical point X c (X = B if the
transition is tuned by applying a magnetic field). These diverging behaviors and the
associated scaling analysis represent universal behaviors of physical quantities and
should be insensitive to microscopic details of the quantum system.
Our experiments look at the dynamics of the 2D SIT. We explicitly measure the
frequency dependence of the complex conductance and impedance of the supercon­
ductor across the transition. As we will demonstrate in Chapter 3 that our broadband
microwave probe is sensitive to the temporal correlations of the superconducting fluc­
tuations. There, we show the critical slowing down (corresponding to the diverging
correlation length) as the system is approaching the thermal transition point, where
resistance goes to zero at the same temperature. This shows that the AC approach
is a suitable tool to study the quantum phase transition since one can determine the
true location of the QCP. This critical slowing down behavior for a continuous QPT is
the theoretical background for data presented in Chapter 4. There, we show data that
support a superconductor metal transition and the critical magnetic field might be
way below the phenomenological crossing point of the iso-thermal magnetoresistance
curves at low temperatures. The understanding of a 2D QPT in a superconduct­
ing film should help us to understand the dynamics of different phases of strongly
22
CHAPTER 1. INTRODUCTION
correlated electronic systems, including but not limited to quantum hall effects and
superconducting fluctuations in high temperature superconductors.
1.4
AC response of a superconductor
Since this thesis work is about properties of 2 D superconductors at finite frequen­
cies, we would like to understand the measurements performed in our experiments.
To this end, we need to know the optical conductivity of a superconductor, espe­
cially the low frequency response of a superconductor, when the probing frequency
is much smaller than the superconducting BCS gap of our sample. Technically a
Matthis-Bardeen approach should be used to treat the mean field AC conductivity
of a superconductor. However, I will mostly use a simpler two fluid approach, which
works quite well for states near Tc despite the simplicity of the model. This will be
demonstrated in the later chapters. We employ the Drude model to describe the gen­
eral electrodynamics properties and the optical response of metals. The conduction
electrons are modeled as a gas of particles with no Coulomb repulsion. The central
assumption of this model is the existence of a mean free time (also called the relax­
ation time), which is the average time that elapses between collisions of electrons.
One can write down the equation of motion of an electron of mass m driven by an
electric field E with mean free time r as
CHAPTER 1. INTRODUCTION
Here, v is the average velocity of the electrons. For a steady state, one has d v/d t = 0
due to the competition between the scattering and the acceleration by E. The current
density for n conduction electrons per unit volume is J = —new = ne2r /m E = <ToE,
and one produces the Ohms law for a metal.
We assume that the applied ac field is of the form E (t) — ’Eoe~lut. The solution
to equation 1.31 should be of the same form v = v(ui)rrlwt. Equation 1.31 can be
rewritten as
(——
r
dt
— —eE /m
(1.32)
Using J = —nev = cr(u;)E, one obtains a complex frequency dependent conductivity
o{u) = o0-— - — = oy(uj) + ia2(u) = (To 1 + l“ T .
1 - iujt
1 + ar r 1
(1.33)
This frequency dependence of the complex conductivity shows that ax is frequency
independent with a DC value ao well below the scattering rate
to decrease at this
7
7
= 1 /r. o\ starts
and falls off with u~2 at higher frequencies. a2 peaks at
7
;
<r2 oc u for low frequencies and a oc u ~ l for high frequencies. When ujt < 1, one has
cr0 « <Ti(u) 3> (t2 ( ( j ) . This is the low frequency limit of equation 1.33, the “so called”
Hagen-Rubens regime. For the InO* film, the scattering rate of its normal state is
way above our accessible frequency range and the Hagen-Rubens limit applies. We
can attribute the frequency dependence of the calibrated data for the normal state
InOx to the response of its substrate (see Chapter 2 for details).
24
CHAPTER 1. INTRODUCTION
In a zero-dissipation limit r —> oc, equation 1.33 becomes
TTT iP
O
(1-34)
TIC'^
a2(u) = —
THU
(1.35)
The / sum rule of the conductivity is given by
Io
ax(u)du = !<r0 = y •
(1-36)
w2
u p is the plasma frequency and - f is the spectral weight which is essential the area un­
der the conductivity spectrum. This sum rule demonstrates that the spectral weight
should be conserved and is a constant for all temperatures.
The AC response of a superconductor can be obtained by considering a two-fluid
model where n = ns(T)+ nn(T). ns{T) denotes the superconducting part and nn(T) is
the normal fraction with scattering rate rn. The BCS theory gives nn(T) ~ e_A/fcijT
at low temperatures. We take the dissipationless limit of the Drude form for the
superconducting electrons and assume urn -C
1
for normal electrons, and the complex
conductivity is given by
cr\ u3) =
irnse2
nne2rn
& \u ) 4
2m
m
~n—
--------------------------------
,nse2
mu
^ 1 ----------------- •
(
1 37
-
)
Under the condition hu <C 2A and T < T C, the AC response of a superconductor
can be given by simply the zero-dissipation limit of the Drude form. In this case, a x
is zero everywhere except at u = 0. According to equation 1.36, the coefficient of
the 8 function in ax should be the spectral weight. a2 a \ / u with a prefactor that is
25
CHAPTER 1. INTRODUCTION
proportional to the superfiuid density ns. We will refer back to these behaviors of o\
and
<72
in later chapters when we discuss our experimental data.
At higher frequencies and temperatures, the AC response of a superconductor
can be evaluated from the Mattis-Bardeen expression on the basis of the BCS theory.
Both thermal energy and the energy of the radiation can create pairs of quasiparticles
and contribute to dispassion. One can refer to [5] for a thorough discussion.
1.5
Thesis overview
In chapter 2, I give an introduction to the broadband Corbino microwave spec­
troscopy. In that chapter, I will discuss in length about the design of the spectrometer
in our group and also about the calibration procedures, both at room temperatures
and low temperatures. There also is a section dedicated to describe the characteriza­
tion of the Corbino spectrometer at finite magnetic fields.
In chapter 3 , 1 discuss the results of the AC measurements of thermal fluctuations
in an InOx film. There, I show data for the explicit frequency dependency of the
complex conductance of the InOx film and the phase stiffness over a range from 0.21
to 15 GHz at temperatures down to 350 mK. Superfluid stiffness acquires frequency
dependence at a transition temperature which is close to the universal jump value.
Our observations are consistent with Kosterlitz-Thouless-Berezinsky formalism. We
explicitly demonstrate the critical slowing down of the characteristic fluctuation rate
26
CHAPTER 1. INTRODUCTION
on the approach to the superconducting state and show in general the applicability
of a vortex plasma model above TKTg.
Chapter 4 focuses on the microwave measurement on a low disordered InO* film at
finite magnetic fields. Data are presented for a field tuned quantum phase transition
between the superconducting and the resistive states in the frequency range of 0.05
to 16 GHz. The relevant frequency scale of superconducting fluctuations approaches
zero at a field B sm fax below the field Bcross where different isotherms of resistance
as a function of magnetic field cross each other. The phase stiffness at the lowest fre­
quency vanishes from the superconducting side at B w B sm, while the high frequency
limit extrapolates to zero near B cross- Our data are consistent with a scenario where
B sm is the true quantum critical point for a transition from a superconductor to an
anomalous metal, while BcrOSS only signifies a crossover to a regime where supercon­
ducting correlations make a vanishing contribution to both AC and DC transport
measurements in the low-disorder limit.
In Chapter 5 , 1 summarize the work presented in this thesis.
Appendix A is for readers who are interested in knowing how to calculate sheet
impedance from the calibrated reflection coefficients. Appendix B details the exper­
imental procedures for both zero field and finite field measurements. It is specially
targeted at users needing to perform microwave measurements in the lab. Appendix
C contains more information about the repeatability of different scans at zero field.
Magnetic field distributions and their affects on temperature sensors are covered in
27
CHAPTER 1. INTRODUCTION
Appendix D. Appendix E is complementary for the data presented in chapter 4. It
contains almost all the data we have for the magnetic field measurements on the low
disordered InO* film.
I report a study of the complex AC impedance of CVD grown graphene in Ap­
pendix F. There, we measure the explicit frequency dependence of the complex
impedance and conductance of a single-layer graphene over the microwave and tera­
hertz range of frequencies using microwave Corbino and time domain terahertz spec­
trometers. We demonstrate how one may resolve a number of technical difficulties
in measuring the intrinsic impedance of the graphene layer that this frequency range
presents, such as distinguishing contributions to the impedance from the substrate.
From our microwave measurements, the AC impedance has little dependence on tem­
perature and frequency down to liquid helium temperatures. The small contribution
to the imaginary impedance comes from either a remaining residual contribution from
the substrate or a small deviation of the conductance from the Drude form.
28
Chapter 2
Broadband Corbino microwave
spectrometer
In the field of condensed m atter physics, spectroscopy is used to investigate the
response of various materials to the electromagnetic radiation as a function of fre­
quency. It employs reflection or transmission measurements to determine the physical
and electrical properties of those materials. In general, the characteristic energy scale
of materials under study extends over several orders of magnitude. For some mate­
rials, like normal metals, semiconductors and insulators, the characteristic energies
are typically of the order of eV. However, in some systems, especially for supercon­
ductors, the energy gaps go down to sub-millimeter and even microwave frequencies.
At sub-millimeter frequency ranges, the radiation can still be guided through free
space and conventional optical components like mirrors and lenses can be used; how-
29
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
ever, microwave radiations have to be guided by coaxial cables or waveguides and
their interaction with the sample has to be performed in a well-defined manner. This
is because the wavelength of a microwave (typically 0.001-0.3 m) is comparable to
the dimensions of the experimental components and samples. Broadband microwave
measurements are traditionally very difficult and have to be performed using different
techniques.
For the amorphous superconducting InOx thin film in our project, we can estimate
its superconducting gap from Tc, which is about 170 GHz for a T c « 2.4 K sample.
This falls into the microwave frequency range and thus requires us to probe the sam­
ple using microwave radiation. Most of the microwave measurements in the literature
have been carried out by using a microwave cavity. Although a microwave cavity is
able to measure both the real and imaginary conductivity with good sensitivity and
high accuracy, it only can examine a limited number of frequencies. The introduction
of the broadband Corbino spectroscopy to study strongly correlated system is a sig­
nificant advance in this field. This technique can give us true spectroscopic response
of the sample under study as well as provide phase information of the conductance
without referring to the Kramer-Kronig relation.
At Johns Hopkins University, we successfully incorporated the Corbino system into
a He3 cryostat and are able to obtain reliable data down to 300 mK as well as with a
perpendicular magnetic field up to
8
Tesla. The purpose of our study is to investigate
the thermal and quantum fluctuations in disordered superconductors and the intrinsic
30
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
electronic behaviors in the insulating side of the quantum phase transition. In this
chapter, we mainly focus on the experimental setup details. We demonstrate that we
overcome the challenges presented especially to this kind of broadband spectrometer
and are able to perform our experiment repeatedly and obtain reliable results.
2.1
Experimental setup overview
Lockin Amplifier
Model: SR830
-j
.
Network Analyzer J
/
Bias Tee
Model: Affleat N5230A
Copper cablo
Glass seal adapter y
Stainless
steel cable
Glass seal adaptor,
heat sunk 4.2 K
Glass seal adapter _
heat sunk 300 mK
cable
Charcoar
pump
Copper
cable
Heat sunk stage for
outer conductor of
-th e coaxial cables
He3 stage
Magnet
Sample Component
Figure 2.1: Schematic of the overall experimental setup showing all the coaxial cables
sections and connections in the Corbino microwave spectrometer.
The schematic drawing of the overall experimental setup in our group is shown in
31
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
Fig. 2.1. A microwave radiation is generated by a vector network analyzer (Agilent
model N5230A), and guided by coaxial cables. The network analyzer can generate
microwave signals from 10 MHz up to 40 GHz with 1 Hz resolution and is equipped
with two 2.4 mm male connectors. As showed in the scheme plot, 4 sections of
semi-rigid coaxial cables are used in our transmission line setup. The overall length
of the whole transmission line is roughly about 1.4 meters. We use a number of
different coaxial cables with different properties at different points in the system.
For the copper coaxial cables (Micro-Coax Company, part number UT-85C-TP-LL)
that are used outside the cryostat connecting the network analyzer and also used to
connect the superconducting cable and the Corbino probe, the outer conductor is tin
plated copper while the inner conductor is made of silver plated copper for better
conductivity. Inside the cryostat, the upper longest section in our transmission line is
the stainless steel coaxial cable (Micro-Coax Company, part number UT-085-SS). Its
outer conductor is 304 stainless steel and the inner conductor is silver plated copper
covered steel (SPCW). We added a NbTi superconducting coaxial cable (Keycom
Company, part number NbTiNbTi085A) into the system to better isolate the heat
load from the room temperature connections, especially from the inner conductor.
The transmission line is terminated by a customized Corbino probe, which was
made from a 2.4 mm Rosenberger in series adapter (part number: 09K121-K00S3).
As one can see from the left picture in Fig.
2 .2 ,
we took the thread away from one
end of the adapter and carefully machine away the extra materials so that the surface
32
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
Figure 2.2: Picture of a Corbino probe showing the flat surface of the probe and the
4 pin configuration of the inner conductor. However, This is not the Corbino probe
we used in our experiment as one can see that its surface is a little rough. Although
we do not think it will affect the results but we use another Corbino probe with much
smoother surface. Due to calibration issues, we cannot disconnect the one in use to
take a picture.
of the out conductor is flat within 0 .0 0 1 ” deviation.
Samples usually have a donut shaped pattern as demonstrated in Fig. 2.3. By
using the Corbino disk geometry, the currents in the film flow in the radial directions
and only produce magnetic fields that are parallel to the surface of the films. The
edge effects of the thin film are effectively eliminated compared to other experimental
setups with a square or rectangular geometry. The sample is tightly pressed against
the surface of the Corbino probe to make a direct electric contact between the outer
Au pad of the film and the outer conductor of the probe. For this particular type of
adapters, the inner connector is not flush with the outer conductor (roughly about
0.005” lower). To bridge the gap between the center Au pad of the sample and the
inner conductor of the Corbino probe, a small conical shaped center pin (see Fig. 2.4)
33
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
is plugged into the inner conductor.
Au c o n ta c t
4-
S u b s tra te
Figure 2.3: Au pattern of the sample prepared for microwave measurements and the
wire connections for 4 probe measurements. On the right we show a picture of one of
the samples we used in the experiment.
One reason to choose the Rosenberger adapter is because of the special design
of its inner conductor, which has 4 fingers as displayed in the right picture of Fig.
2.2. These fingers hold the center pin in place and also provide some springy force
necessary to keep the pin in the correct configurations when a sample is attached. The
interaction between the sample and the Corbino probe is also fixed by a spring behind
the sample. The spring force is carefully set to be the same for all the samples and
calibration standards by adjusting the length of the set screw. The same reference
plane is assured for each sample by using a caliper to set the set screw in place. This
improves the overall quality of our data dramatically.
A particular experimental challenge in performing these experiments in a He3
34
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
Corbino Probe
_ ^
C en ter Pm
Sam ple
'
C opper housing
\
S et screw
'
C opper su p p o rt
Figure 2.4: Scheme plot of the sample stage.
cryostat was to isolate the heat load from the top of the coaxial line which always stays
at room temperature. We thermal anchor the outer conductor of the transmission line
at two locations: the top flange of the inner vacuum chamber (IVC) and the He3 pot.
To ensure good thermal contact, we made a copper housing for the adapter between
stainless steel cable and NbTi cable and pressed this copper housing tight against the
top flange of the IVC. Another copper housing for the adapter between NbTi cable
and copper cable was screwed tightly on top of the He3 pot. This guarantees that
the section of the superconducting cable is thermal anchored at 4.2 K and at the
base temperature of the cryostat. This can be easily viewed in the picture in Fig.
2.5. For the connection between NbTi and copper cables, we also used copper wires
and the thermal grease around the cables, connectors and adapters to make sure that
35
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
the outer conductors of the cables are well thermally anchored along the connection.
The main reason for this is because we were concerned that the outer stainless steel
conductors of the microwave adapters may not provide enough heat transfer.
Figure 2.5: Pictures of the set up showing the manner how we heat sink the coaxial
cable. On the right we show the front image of the sample stage.
It is more difficult to heat sink the inner conductor since the dielectric in the
cable is usually made of teflon and in most adapters and connectors is just air. This
means that although we can manage to cool the outer conductor to 300 mK, the
inner conductor might stay at a much higher temperature due to the poor thermal
conductivity of air and teflon. Before we added the superconducting coaxial cable,
the base temperature of the set up was only 500 mK and the holding time was less
36
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
than half an hour. To thermally anchor the inner conductor properly, we incorporated
three hermetically sealed glass bead adapters (Kawashima Manufacturing Company,
part number: KPC185FFHA) as displayed in Fig. 2.1. Two of these special adapters
were heat sunk respectively at 4.2 K and the He3 stage that were separated by a
roughly 10 cm long superconducting NbTi coaxial cable as showed in Fig. 2.5. We
were able to reach 290 mK without microwave radiations and the holding time can
be at least a day, which is enough for one cycle of our experimental procedures.
The transition temperature of the superconducting cables is around
incident microwave power level was chosen to be -27 dBm (~
2
8
—9 K. The
//W and this is the
lowest power that we could set the network analyzer to). It turns out that this power
level is not heating up the sample too much at the base temperature yet high enough
compared with the noise level. All the cables, adapters and connectors (except the
section outside the dewar which stays at room temperature for all the measurements)
were thermal cycled a couple of times in liquid nitrogen before and after assembly to
reduce the effects of thermal contractions in later measurements.
After we assembled all the cables, we keep all the connections untouched since
undoing any of the connections would ruin the calibration and change the signal. We
use a lift and two rails to guide and move the dewar up and down for low temperature
experiments. The coaxial cables also get “aged” after many times of cooling. The
“aging” of the cables causes oscillations in frequency in the calibrated data. However,
the overall effects to the final results of the sample under study are minor as we will
37
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
discuss in more details in the later text.
Using a vector network analyzer, we measure the complex reflection coefficients
S'™ from the sample that terminates the otherwise open-ended transmission line. S™
is then calibrated by measurements of three standards: open, load, and short. I
will give more details about the calibration standards and procedures in the following
sections. The useful frequency range for our studies usually lies in between 50 MHz to
16 GHz. Above 16 GHz, microwave reflections are dominated by the resonance in the
sample holder stage or their interference with other components in the setup. At low
frequencies, usually lower than 45 MHz, microwave data appear to be contaminated
by the finite contact resistance (~
2
Ohms) of the Corbino press fit contact.
In addition to AC measurements, we can measure the two point DC resistance
simultaneously with a lock-in amplifier (Model SR830) by adding a bias tee (Agilent
11612B Bias Network, 45 MHz to 50 GHz) into the transmission line. The measure­
ment frequency of the lock-in amplifier for most measurements is 13 Hz. We found
that due to this pass frequency of the bias tee, setting the spectrometer to scan at
frequencies lower than 45 MHz would introduce a large amount of noise into the DC
measurements.
We also found out that an accurate determination of the resistance for the load
sample is very important. The uncertainty in measuring the DC value of the load
sample (either from the uncertainty in determining the contact resistance or from the
noise introduced by setting the spectrometer to scan below 45 MHz) will propagate
38
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
to the conductance of the thin films under study. The excitation currents for the
lock in amplifier that we used during different measurements were within the range
from 100 nA to 200 nA. Since it is a 2 probe DC measurement, to reduce the contact
resistant, 100 - 350 nm thick donut shaped gold contact were evaporated on to all
the samples except the open standard. An iron shadow mask was used during the
gold evaporation and was held on top of the sample with a permanent magnet. The
inner Ti and outer r 2 diameters of the donut shaped gold contact were 0.7 and 2.3
mm respectively. We also tried lithography at the beginning but photoresist needs
to dried at around 100 C and the heating process would anneal the InOz films under
study. So according to our experience, shadow masks do less harm to the samples
than lithography.
2.2
D ata analysis
The actual reflection coefficient from the sample surface differs from the measured
S™ due to the effects of extraneous reflections, damping, and phase shifts in the
transmission line. .S^ can be calculated as:
Sa =
*~*ii ~ ^ D
EH + E s i S R - E o Y
(2
(
l)
}
Here, the complex error coefficients E D, Es , and ER represent the effects of di­
rectivity (signal reaches the detector directly without interacting with the sample),
source match (signal coming from the sample is reflected again and adds to the signal
39
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
approaching the sample), and reflection tracking (damping and phase shift) in the
transmission lines. This implies that performing three reference measurements on
standard samples with known reflection coefficients is needed to determine the three
unknown error coefficients. Interested readers can refer to reference [33] for the full
inversion.
This calibration procedure however gives us another challenge that is the repeata­
bility of each measurement as the three error coefficients are very temperature depen­
dent in general and especially in our setup since we included a superconducting cable.
We need to make sure that the temperature profile is the same when we perform the
microwave measurements on three standards and the sample under study. To this
end, we established a particular cool-down procedure. Liquid nitrogen and then liquid
helium were introduced into the bath and temperatures were allowed to equilibrate
for over 12 hours before starting measurements. A very slow and repeatable scan was
performed for each sample from the base temperature up to 10 K in about 9 hours.
I will discuss the low temperature calibrations with more details in the next section.
The detailed measurement procedures can be found in Appendix B.
After we determine E d , E s , and E r for each temperature and frequency, S h
can be obtained via equation 2.1. To extract the sample sheet impedance Z eJ }, in
principle the standard equation
<2-2)
may be used. Here Z q = 50 ohms is the characteristic impedance of the cable and
40
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
g = 27r/ ln(r 2 /ri) is the geometric factor where T2 and r\ are the outer and inner radii
of the donut shaped sample. However, this Z R
J ? will be the impedance of the film
under study only when the substrate contribution is negligible. In what follows, Z§ub
is the effective substrate impedance from everything that lies behind the film. For a
thin film where the sample thickness is much smaller than the skin depth and under
the assumption that only TEM waves propagate in the transmission lines, the effective
impedance for a thin film of impedance Zs backed by a substrate with characteristic
impedance Z§ub [34] is
zf
S
For a sample that has Zs "C Z§ub,
= — 5*
1 + #us !
( 2 .3 )
'
'
~ 0 and equation 2.3 reduces to Z eJ ^ = Zs-
In our case both InO* in the normal state and graphene have a sheet resistance
comparable to the Si substrate so Z eJ i = Zs does not hold anymore. In order to
obtain the real response of the film, it is necessary to extract Z§ub. To isolate the
impedance of the sample under study, we assume that Hagen-Rubens limit holds for
InO* in the normal state since our probing frequency is in the microwave range and
it is way below the characteristic scattering frequency of the sample (in the range of
THz). This implies that Zs in the normal state should be pure real and independent
of frequency and could be deduced from the DC resistance exactly. The substrate
contribution Z§ub is then extracted from the calibrated InOx data at 5.6 K. With a
reasonable assumption that Z§ub is temperature independent at low temperatures,
the intrinsic response of the film Zs at any other temperatures can be calculated.
41
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
Complex sheet conductance G = ad is related to sheet impedance as G = \/Z * in
the thin film limit (See Appendix A).
2.3
Calibrations
Calibrations are very important if one wants to know the actual response from the
sample. We need three standards for the three unknown error coefficients and one
needs to carefully choose the three calibration standards. A blank high resistivity Si
substrate was used as an open standard (Sxx = 1). A 20 nm NiCr film evaporated on
Si substrate was used as a load standard. NiCr has a very high scattering rate, so one
can assume that the impedance of the NiCr standard is flat in our accessible frequency
range. Its S X1 can be evaluated from its simultaneously measured DC resistance R via
the relation S X1 =
A 20 nm superconducting Nb film (Tc ~
6
K, which can be
clearly seen from the DC resistance of Nb films) sputtered on Si substrate was used
as a short standard (Sxl = —1) for thermal fluctuation measurements. A Nb film
above Tc is not a good short standard since it is in the normal state and has a finite
resistance. Using a superconducting Nb film as a perfect short yields more reliable
results than using bulk copper for calibrating superconducting InO* films. This is
likely the case because copper is less reflective than the superconducting InOx sample.
The small calibration errors from the imperfection in the short standards result in
a small error in the phase of the calibrated conductivity thus for highly conductive
42
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
samples giving us some negative real component of conductivity. However, this fact
does not affect too much of the calibrated conductivity of InO* once the temperature
is above the transition temperature Tc and in the fluctuating regime since the sample
is very dissipative and copper turns to be a fairly good short in that case.
For measurements in magnetic fields, Nb films cannot be perfect shorts once we
apply a perpendicular magnetic field. Therefore we use bulk copper with thick Au film
on top as a short standard. This again does not affect our results since InO^ becomes
very dissipative with applied magnetic fields. Short only calibration as discussed in
reference [33] may not be possible in our setup since the three error terms have very
strong temperature dependence especially after the inclusion of the superconducting
cable. Different choices of open might affect the high limit of the cutoff of usable
frequency ranges. This was again discussed in length in reference [33]. A quick
test of different calibration standards shows that the choice of glass, ceramic or Si
substrate gives the same conclusion with regards to the experimental data so we did
not perform a systematic study of the effects on the cutoff frequency from different
open standards.
2.3.1
R oom tem perature calibrations
One room temperature calibration was carried out at the end of the first copper
coaxial cable using the commercial available calibration standards. Calibrations at
the sample stage were also performed. These room temperature calibrations were
43
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
used to characterize the system at the early stage of the building phase. The raw
signals are usually very noisy and lossy because we have a very long transmission
line system with many connections. The stainless steel cables and superconducting
cables are in general very lossy. Room temperature calibrations are very important
and we revealed two significant facts for later experiments: the importance of the
exact position of the set screw and the substrate correction.
The microwave configuration between the sample and the Corbino probe, as dis­
cussed in the previous section, is defined by the two springy forces provided by the
inner conductor of the probe and the spring behind the sample. To ensure we have
the same reference plane for all the samples, we should have the same springy force
for each configuration. Spring tension determines how hard the sample is pushed
against the probe. Different spring tensions may lead to different positions of the
center pin, thus change how the microwave radiation interacts with the sample. The
main purpose is to maintain the same length of the spring beneath the sample for all
the configurations. This approach would also make sure that the center pin will be
pushed into the inner conductor at the same depth each time. We pick up a standard
for the length of the set screw outside of the sample stage when the sample and the
Corbino probe have the best contact, which is usually defined from the minimum
point of the simultaneous DC measurement. This at the same time also sets the
force in the spring. The length of the set screw changes accordingly for samples with
different thickness. Although the reference plane might change once the IVC is in
44
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
high vacuum as discussed later and in low temperature environments, all the refer­
ence planes should still be the same for all the samples if we have the same starting
reference plane and same procedure to pump and cool down the system.
The choice of load and short are pretty standard. As pointed out by Ref. [33], the
choice of open, however, would affect the usable frequency range since it may change
the resonance frequency in the calibrated data. To check response from different open
standards, we performed a quick check on different samples with high impedance
using a very short copper cable. We attached an unmodified Rosenberger adapter
at the end of the copper cable and the whole connection was calibrated by the three
commercially available standards. We then replaced the Rosenberger adapter with
our Corbino probe. This exchange process changed the signal level some, but the
overall change should not be too significant and this room temperature test still gave
us some ideas of responses of different possible open standards as shown in Fig. 2.6.
Comparing the magnitude of S n of all the open samples, air (no sample is used to
terminate the otherwise open transmission line) is a much better open with resonances
at much higher frequencies. Highly doped Si, on the other hand, has an impedance far
away from a perfect open thus cannot be used as an open standard. Although air is a
perfect open, it cannot be used for low temperature calibrations when the commercial
standards are not applicable. This is because the center pin would protrude somewhat
compared with the situation when there is a sample terminating the transmission line,
which changes the reference plane. We can clearly see this from Fig. 2.7 where we
45
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
1.0
0.8
0.6
co*
0.4
Highly doped Si
Ceramic
Plastics
G lass
Air
Silicon
0.2
0.0
0
20
10
30
40
Frequency (GHz)
Figure 2.6: Magnitude of S n for different open substrates. Magnitude of Sn for
highly doped Si is just an illustration for the possible response of a lossy sample.
show the calibrated real (Z{) and imaginary (Z 2 ) impedance for 40 nm NiCr film on
high resistive Si substrate as a function of frequency at room temperature. Substrate
correction is not needed for this sample since its sheet resistance is negligible compared
with the one of the Si substrate. Different colors in this plot indicate different sets
of calibration standards. Data calibrated with air as open clearly deviates more from
the expected value which is derived from the co-measured DC value, especially at
high frequencies. The fast increases at around 16 GHz in Z x and Z 2 are caused by
the resonance in the sample stage. The huge deviation of data calibrated by air
as open from their expected results can be accounted by the strong interaction of
the microwave radiation with the setup component and the different position of the
46
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
center pin from all other calibration standards and samples when there is no sample
to terminate the line. We also used ceramic and glass to calibrate the sample as
displayed by the green and purple curves in Fig. 2.7. From Fig. 2.6, we know
that both glass and ceramic develop a resonance with the sample holder at a higher
frequency compared with Si. This is also confirmed from the calibrated data of 40
nm NiCr film. The resonance using Si as open formed at about 17 GHz while the
first resonance did not show up until 21 GHz for data calibrated by ceramic and glass
(not shown in the plot).
250
40
40
40
40
200
nm
nm
nm
nm
NiCr using air a s open
NiCr using Si a s open
NiCr using ceramic a s open
NiCr using glass a s open
-50
0
2
4
6
8
10
12
14
16
18
Frequency (GHz)
Figure 2.7: Real and imaginary impedance of 40 nm NiCr on Si as a function of
frequency. 20 nm NiCr film as load and bulk copper as short were used for all the
calibrations. Red dashed lines mark the expected impedance for this sample from the
co-measured DC value.
As discussed above applying the same force in the spring for all the samples is
important. A large difference in the reference plane, of course, would give us different
47
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
results of the sample under study. A small difference on the other hand would give
some wiggles in frequency in the calibrated data as shown in Fig. 2.8. The difference
in experimental setup for the two sets of data is that the set screw was just about
0 .1
mm away from its assumed position for the red curves. As we can see the absolute
magnitude and shape of the two sets of data are overall very similar. However, the
data in red have small oscillations in frequency in both Z\ and Z<i and this is caused
by the small deviation of the reference plane when we set the set screw differently for
the 40 nm NiCr film for that test run.
150
.............
_
'3?
E
sz
O
8
£<0
100
•40 nm NiCr with fixed force in the spring
' 40 nm NiCr with set screw about 0.1 mm away
from its supposed position
50
0
r ,- r ,
-50
6
_L
_L
8
10
12
14
16
Frequency (GHz)
Figure 2.8: Real and imaginary impedance 40 nm NiCr on Si as a function of fre­
quency. Black curves are the frequency dependence of the calibrated Z\ and Z 2 with
the set screw in the supposed position maintaining the same spring force for all the
calibration standards and the sample under study. For the red lines, the set screw is
in the right position for all the three calibration standards, but it is about 0 .1 mm
off from the presumed position for 40 nm NiCr.
48
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
We performed a more systematic study of the position of the set screw as shown in
Fig. 2.9. 10 nm A1 film on a Si substrate was calibrated by 20 nm NiCr, bulk copper,
and ceramic or glass. Red dashed lines are guide to the eye of the expected value of
the impedance. The position of the set screw was chosen by closely watching the DC
resistance for 20 nm NiCr film when we tightened the set screw. The 20 nm NiCr has
350 nm thick Au pattern which ensures good electric contact between the sample and
the Corbino probe. We took the location when the resistance was the minimum point
of reading as the standard position for the set screw. The force of the spring was
adjusted to be the same by this standard position for the three calibration standards.
For the 10 nm A1 film, we tried different spring configurations as described by the
color legend in Fig. 2.9. For different forces in the spring, the calibrated impedance
slightly deviates from the expected value, especially at higher frequency. Here we
did not distinguish which open standard we used since both ceramic and glass yield
the same calibrated data as demonstrated in the plot. A loose spring may result
in unreliable data as shown by the blue curve. In that case, the sample and the
probe may not even have good contact. The drop of the real impedance at very low
frequencies is the contamination from the contact resistance.
Fig. 2.10 shows the sheet impedance for one of the InOz films calibrated at room
temperature. In the plot, the real impedance is demonstrated by solid curves and the
corresponding imaginary part is shown by a dashed line with the same color scheme.
The set screw stayed at the same position for the four sets of measurements. Three
49
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
10 nm Al using glass as open
10 nm Al using ceramic as open
10nm Al with spring 0.25 mm tighter
10nm Al with spring 5 mm tighter
10nm Al with spring 0.25 mm looser
10nm Al with spring 0.5 mm looser
guild to the eye of the expected impedance
8
10
Frequency (GHz)
Figure 2.9: Impedance of a 10 nm Al film at room temperature with different spring
configurations.
1500 -
1000
—
InO, in air
lnO*when we started pumping the IVC
lnOxwhen we pumped the IVC for 10 mins
lnO„ in vacuum
-
500 -
-500 -
10
Frequency (GHz)
Figure 2.10: Impedance of InOz at room temperature in the range of oj/2-k — 0.05 20 GHz. Solid lines are the real parts of the sheet impedance at four different trials
and the dashed lines are the imaginary parts.
50
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
calibration standards were all measured when the IVC was open (glass, 20 nm NiCr
film and bulk copper). The difference in each data set was the environment of the
InOz film. The black line was taken when InOx stayed in the atmosphere. After we
close the IVC and start pumping, we took another set of data and calibrated it by
the same error coefficients. This set of data is displayed by the green curves. Data
in blue show the calibrated InOx data after we pumped the IVC for about 10 mins
and both the real and imaginary impedance for this run yield the same results as
the green lines. After pumping the IVC overnight (~ 12 hours) by a turbo pump,
InOz film was in a high vacuum environment. We took another room temperature
data of InOx and calibrated them with the same error terms. The calibrated data
are displayed by the red curves. The set of calibrated InOx data definitely has more
wiggles after the IVC was pumped over night as shown in Fig. 2.8. After we pumped
the air out, the cryostat was compressed a little bit and also the air dielectric in some
adapters changed to vacuum. These caused a shift in the sample reference plane, thus
introduced a small calibration error in the final data.
We also notice that the impedance for InOx has very strong frequency dependence.
However, InOx film in the normal state at room temperature is a very disordered
metal and its scattering rate (usually in the range of THz) is much higher than the
upper limit of the frequency range we can access in this setup. According to the
Drude model for a dirty metal with a very high scattering rate, this implies that
the impedance of the InOx sample should be flat in our frequency range. Therefore,
51
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
the frequency dependence of the calibrated data can only come from the substrate
contribution. Although Si substrates have no effects on 40 nm NiCr films and Al
films, the assumption of
£s
~ 0 is no longer valid for this InOx film. We obtained
the substrate contribution as discussed in the previous section. An example of the
substrate contribution is displayed in Fig. 2.11 where we demonstrate the magnitude
of Z§ub obtained using equation 2.3.
3
I
£
%
ie
I
0.1
Figure 2.11: Magnitude of Z§ub from one of the low temperature measurements.
2.3.2
Low tem perature calibrations
Low temperature calibrations are more complicated than room temperature cal­
ibration procedures, especially after we added the superconducting cable. One ex­
52
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
perimental challenge when we were characterizing the system was the repeatability
of the measurement of each sample since all the error coefficients have strong tem­
perature dependence. To correctly remove the cable contributions, we have to make
sure the temperature profile along the transmission line be the same for the three
calibration standards and the sample. For that reason, the repeatability of the cool­
ing down procedures for the three calibration standards and each measurement of the
sample is essential. The detailed information of the experimental procedures can be
found in Appendix B. In this section, we mainly discuss the characterization of the
spectrometer with the superconducting cable.
2.3.2.1 Effects of the superconducting cable
We added the superconducting NbTi cable to better thermally isolate the sample
from the heat flow down the inner conductor of the transmission line. There is some
tradeoff between a lower base temperature and better repeatability in the design of
the whole transmission line. We originally had an approximate 20 cm NbTi cable
in the system and had a faster scan in temperature. As a result, we could reach a
very stable base temperature at about 290 mK. However, it took a very long time for
this 20 cm superconducting cable to reach a thermal equilibrium. To characterize the
cables’ response, we looked at SJ? = |S„|e<* (tp is in radian for the following graphs)
as a function of temperature for a standard which should have little low temperature
dependence. Therefore, the changes in S'™ as we scan the temperature are mainly
53
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
the contributions from the cables to the reflected signals.
In Fig. 2.12, we display the temperature dependence of l^nl and ^ at 7 GHz for
the bulk copper for two different warming up and two cooling down procedures with
the same temperature controlling parameters. These measurements are faster scans
in temperature than the standard procedures as described in Appendix B. For the
two warming up scans, the difference in |5 n | is about 2 %. For the two cooling down
procedures, this difference can be as big as 4 %, especially at high temperatures.
However, we concerned most about the discontinuity in the signal at about 2 K in
which the l^n j signal can be different up to
8
%. This is caused by the long relaxation
time the superconducting cable needs to reach its thermal equilibrium as well as the
different level of liquid helium in the dewar for each scan (the order for each scan can
be found in the plot of iSnl in Fig. 2.12). This required us to figure out a strategy
so that we can continuously scan from base temperature to
6
K or higher. In that
case, we just need to maintain the same cryogenic environment in the dewar and the
same temperature setting for all the measurements.
Knowing that the superconducting cable demands a long time to reach its thermal
equilibrium, we shortened the cable by half and carefully thermally anchored all the
connections as detailed in the experiment overview section. The base temperature
now is 300 mK, 10 mK higher than before. We strictly follow the experiment pro­
cedures in Appendix B for each measurement cycle where we show how we control
the temperature repeatedly and reliably from 300 mK up to 10 K. Microwave data
54
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
0.45
0.44
0.43
of
-
0-42
0.41
0.40
0.39
- 0.20
-
0.22
-
0.24
-
0.26
-
0.28
-
0.30
-
0
5
10
Tem perature (K)
15
20
Figure 2.12: Four temperature scans of the bulk copper at 7 GHz with a 20 cm
superconducting cable in the transmission line. Red and black lines are data for two
different warming up procedures and green and blue ones are for two cooling down
scans. The order in which these scans were taken is showed by the number next to
the curves in the plot of |S n|.
55
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
0.42
first warming up
cooling down
second warming up
0.41
0.40
0.39
0.38
0.37
0.36
2.50
2.48
2.46
2.44
2.42
2.40
2.38
0
5
10
Temperature (K)
15
20
Figure 2.13: Magnitude and phase of .S1^ of Si as a function of temperature at 7
GHz with a 10 cm superconducting cable in the transmission line. Different runs are
indicated by the color legend.
56
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
at zero field were taken 3 times per cycle: first warming up (from base temperature
to about 4 K), cooling down (from 20 K to 2 K) and second warming up (from base
temperature to around 10 K). We always start the first warming up at helium level
55 cm (~ 20 liters liquid helium in the dewar), the cooling down scan at helium level
50 cm and the second warming up at a helium level roughly around 26 cm. To re­
produce the same temperature profile, we wait the same amount of time before each
measurement for every sample.
We also found out that the bulk copper might be a cause for some negative
responses in the real conductance of InOz. After those initial characterizations of
the spectrometer, we switched to a superconducting Nb film below
6
K as a short
standard for low temperature measurements. Since the Nb film might have some
temperature dependence at low temperatures, especially around 6 K when it becomes
superconducting, we focused on S™ of Si to characterize the response of the cables for
different runs. In Fig. 2.13, we show the temperature dependence of iS'nl and ip of
51 standard for the three measurement runs in one cycle. The two warming up scans
have a difference in |S'n| that can be close to
2
%, especially at low temperatures. This
is caused by the difference in the helium volume in the dewar: the helium level for
the second warming up is almost half of the one for the first warming up. Therefore,
the cryogenic environments for the two scans are different. The overall temperature
of the coaxial cables for the second warming up is usually slightly higher than the
first warming up. However, this also indicates that our data should not have very
57
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
strong dependence on the helium level. The maximum difference in jS'n | is about 2
% for a huge change in the helium level for the two warming up scans. If the helium
level differs by
1
or
2
cm for the second warming up for different standards, the error
introduced in this case is small and should be negligible. This conclusion may also
be supported by the fact that all the coaxial cables (except the one connecting the
network analyzer) sit in high vacuum without a direct contact with the liquid helium
bath.
The difference in |S n| between the cooling down and second warming up reduces
to
1
% or less over the whole temperature range. The overall temperature dependence
of S’™ for the two runs feature the same shape except some offset in the y axis. Both
sets of data have little dependence in temperature above 9 K indicating the cables have
little thermal response in this temperature range. Below 9 K, the NbTi cable becomes
superconducting, hence introduces a change in S™. Fig. 2.13 demonstrates how the
NbTi cable reaches its equilibrium for one scan. Its response can be reproduced
despite some offset in the absolute value.
For data analysis, we use standards from one scheme to calibrate the sample
measured in the same fashion. It is in general easier during the second warming up
for us to control how long the system stays at the base temperature to eliminate some
external disturbance (we need to transfer liquid helium before the first warming up).
Our data in Chapter 3 and Appendix F were taken at the second warming up and were
calibrated by standards measured also at the second warming up scans. Although
58
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
0.4
lnOxfilm first warm ing up
lnOxfilm cooling down
lnOxfilm seco n d warm ing up
0.3
tn
0.2
0.1
3
-2
-
0
5
10
15
20
Temperature (K)
Figure 2.14: Magnitude and phase of S™ of InOx as a function of temperature at 7
GHz before calibration. Different runs are indicated by the color legend.
59
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
there is some difference in Sft in first warming up, cooling down and second warming
up scans, the error due to the calibration procedures should be way less than
1
%
for the same type of scan in temperature with almost the same helium level in the
dewar.
The InOx film that we present data in this thesis has a very sharp feature in the
raw 5"! data for its finite temperature superconducting transition (see Fig. 2.14).
The change in the reflection signal in the InOx film is much more dramatic than the
changes in the three standards. We still can see the changes in the raw S™ due to
the existence of the superconducting cable around
8
- 9 K, but this change is very
small compared to the overall signal change in the InOx film. So error introduced by
the possible difference in the helium level in the dewar should be totally negligible in
the final analyzed data and do not affect the physical interpretation.
2.3.2.2
Error coefficients
After the measurements of the three standards, we obtain three equations between
S fx and S'™ for the three calibration standards using the formula 2.1. Three unknown
error coefficients
E
r
,
Es , and
E D
can be determined by the three equations and are
plotted as a function of temperature for the exemplary frequency of 9 GHz in Fig.
2.15. The data are from a low-temperature calibration using a bulk copper as short, a
blank Si substrate as open, and a 20 nm NiCr film as load. The data suggest that
Er, and Ed change very little with temperature below
6
E
r
,
K. Although the transition
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
0.4
49
c
0 0.3
1O
w
§ 0.2
111
o
T«J
1
C 0.1
o>
«
X
-
0.0 Ik
0.5 F
f49
c
.2
00
io «
w
§
Ul
o
S
£<0
a.
-1.0
- 1.5
•2.0 Ik
0
1
2
4
3
Temperature (K)
5
6
Figure 2.15: Temperature dependence of the magnitude and phase of the three error
terms at 9 GHz.
61
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
temperature of the superconducting cable is around
an equilibrium below
6
8
— 9 K, it more or less reaches
K. Therefore, a short only calibration might be possible for
low temperature measurements (see reference [33] for more details) that one only
uses a short standard to calibrate the sample’s reflection coefficients. This short
only procedure would greatly reduce the errors due to different measurement runs of
different standards. If the sample under study is a very good superconductor (with
high superfluid density), one in principle can use the sample in the superconducting
state as a short standard. In this case, no extra measurements are needed. However,
we cannot use the InOx sample in the superconducting state as a short standard
since the film is very disordered with a low superfluid density compared with other
clean superconductors such as A1 films. The weak temperature dependence of the
three error terms demonstrate the potential possibility of performing a short only
calibration in our spectrometer and could be investigated in the future.
2.3.3
M icrowave m easurem ents in a perpendicular
m agnetic field
The detailed dimensions of the He3 cryostat coupled with the magnet can be
found in Appendix D. The superconducting magnet consists of superconducting NbTi
wires. A power supply supplies a current to energize the magnet. The magnet can
reach
8
Tesla with a current of 47.62 Amperes. Contour plots of the magnetic field
62
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
distributions in the axial and radial directions as well as the magnitude of the field
are also displayed in Appendix D. The magnet can stay in a “persistent” mode. Once
we reach a desired magnetic field, we turn off the persistence switch heater which is
effectively a superconducting short across the power leads within the cryostat. Once
the persistence mode has been entered, we ramp down the current in the power supply
quickly. The magnet still stays at the desired value even after the current in the power
supply becomes zero. Our microwave scans in temperature are performed at constant
magnetic fields that the magnet is set to stay in the persistent mode. This way, the
magnet will be stable throughout the whole measurements. Also, the current in the
power supply of the magnet is zero, thus does not affect the readings of all the other
electric components that are close by such as temperature controllers and the lock
in amplifier. We found some small effects from the current leads on the temperature
and lockin readings when we were charging the magnet.
A magnet tends to trap magnetic fluxes during magnetic field sweeps and thus
shows hysteresis. The nominal “zero field point” is not true zero due to the trapped
flux. To perform measurements at true zero magnetic field, one has to oscillate the
magnetic field to 0 to remove any possible trapped flux. In experiments, we need
to estimate the magnitude of the hysteresis to make sure that it has little effect on
the experimental data. To do so, we swept the magnetic field at a fixed temperature
and searched for the minimum point of the resistance reading of a film. In Fig. 2.16,
we show the sheet resistance of an InOx sample as a function of field at 2 K. This
63
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
700
A field scan from -1 T to 1 T
A field scan from 1 T to -1 T
too
800
§
I
S
aP
300
200
100
•
1.0
-0.5
0.0
1.0
urn
Figure 2.16: The hysteresis of the magnet in our system. The curve in red is a scan
of the InOx film in field from -1 Tesla to 1 Tesla at 2 K. The curve in blue is a scan
of the InO* film in field from 1 Tesla to -1 Tesla at 2 K
64
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
shows that the hysteresis of the magnet in our system is about 0.036 T, which is small
enough and can be safely ignored for our measurements.
0.8 GHz
0.251
4 GHz
0.453
- 0.035
0.280
0.482
0.259
0.030
2
0.259 <
0.257
0.481
- 0.025
0.480
0.255
0.470
0.255
7 GHz
0.020
55x10'
12.5 GHz
0.434
2.02
0.310
0.433
2.01
-£ 0.432
2.00
«
2.00
0.431
0.300 -
2.00
0.430
2.07
0.420
2.00
0
2
0
4
■«D
2
4
0
> r o
Figure 2.17: Magnitude and phase of S'™ of a Si standard as a function of field at
300 mK for 0.8, 4, 7, and 12.5 GHz. Different frequency values are indicated in the
legend for each plot. In all the plots, data in red are the magnitude of S™ and data
in blue are the phase of S'™ (in radians) for that particular frequency.
For calibration purpose, we need to find out the field dependence of the coaxial
cables, especially because the superconducting cable in the system might be affected
by the applying magnetic field. One can in principal, examine this dependence by
looking at S'Jj of a Si standard for a magnetic field scan at fixed temperatures. Fig.
65
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
1.20
o, (T * 0.3321 K) calibrated by standards at 2 T/o, (T « 0.3323 K) calibrated by standards at 3.5 T
o, (T » 2.6658K) calibrated by standards at 2 T/o, (T = 2.5652 K) calibrated by standards at 3.5 T
1.16
1.10
I
1.06
1.00
0.06 L _ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ L
0
2
4
0
0
10
12
14
10
Frequency (OHz)
Figure 2.18: Ratio of real conductance of InOx measured at 3.5 Tesla but calibrated
by standards measured at 2 Tesla and 3.5 Tesla. The blue dashed line marks the
expected ratio when the calibrated data from both calibration sets are the same. One
ratio was calculated from data at about 330 mK and another was obtained from data
at about 2.67 K.
2.17 shows the magnitude and phase of S™ of a Si standard as a function of field at
300 mK for 4 frequencies. Since the Si standard should have little temperature and
magnetic field dependence, the changes in S'™ showed in Fig. 2.17 for frequencies at
0.8, 4, 7, and 12.5 GHz are mainly the contributions from the cables to the reflected
signals as we sweep field. Both the magnitude and phase of S'™ are not linear in field
and they show minimums at different fields for different frequencies. However, the
overall changes in the response of the cables are still small compared with the change
in InOx signal showed in Fig. 2.14. Fig. 2.18 plots the ratios of the real conductance
of InOx measured at 3.5 Tesla but calibrated by measurements of standards at 2
Tesla and 3.5 Tesla. As one can see an interpolation in field can still yield reasonable
calibrated data as long as the sample under study has a very large change in signal
66
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
at different magnetic fields. This validates the calibrated InOx data at 4 Tesla using
an effective calibration interpolated from 3.5 Tesla and 5 Tesla calibration standards
in Chapter 4 due to a missing set of calibration curves.
2.4
Review of the Corbino spectrom eters
from other groups
The broadband Corbino spectroscopy was introduced by Anlage’s group in Uni­
versity of Maryland in the 90s [34,35]. They used this technique to study thin films of
high temperature superconductors YBCO [36-39] as well as colossal magnetoresistive
manganites [40]. Later, a group in University of Virginia used the same technique
to study the microwave ac conductivity spectrum of a doped semiconductor in its
Coulomb glass state [41]. This technique has also been used to study the dielec­
tric response of liquids and soft condensed matter [42] in a group in Leiden. Those
experiments went down to 4 K or just liquid nitrogen temperature.
The limit in temperature was pushed down to 1.7 K by Marc Scheffler at University
of Stuttgart in Germany [33,43]. There, they reported the frequency dependence
of microwave conductivity of the heavy fermion metal [44] and superconducting A1
films [45]. Kitano et al. also constructed a Corbino spectrometer [46] to investigate
the critical behavior of LSCO [47,48] as well as NbN films [49]. Recently, Pratap at
Tata institute also has constructed a Corbino spectrometer that goes down to 2.3 K
67
CHAPTER 2. BROADBAND CORBINO MICROWAVE SPECTROMETER
to study NbN films [50].
The Corbino spectrometer in our group is the first of this kind that can go down
to 300 mK with an applied magnetic field up to
8
Tesla. As I have shown in this
chapter that we can repeat the measurements in a reliable fashion and we are the
first group reporting reproducible data at 300 mK and
8
Tesla. We have applied our
setup to investigate 2 D superconducting InO* films, graphene and 2D quantum phase
transition [51-53].
68
Chapter 3
AC studies of the zero field phase
transition
In this chapter, I will discuss the dynamic effects of the KTB phase transition
probed by the home-built microwave spectrometer. I will start this chapter by dis­
cussing the expectations of such a transition.
3.1
Superconducting fluctuations
In Fig. 3.1, sheet resistance in the unit of Ohms of one InOx film is displayed
as a function of temperature. This plot is a guide to understand superconducting
fluctuations in different regimes showing how the sample evolves from the normal
state towards the superconducting state at low temperatures. The superconducting
69
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
transition is much broader than the 3D case due to the existence of strong fluctuations.
Following Ginzburg and Landau, a pseudowavefunction ip(r) — A ellf is the order
parameter which describes how deep the system is into the superconducting phase.
A and <p are the magnitude and the phase of the complex order parameter. Within
conventional wisdom, one expects that the sample evolves to a superconducting state
from the normal state by first entering an amplitude fluctuation region, which can be
well described by the GL mean field theory. At some temperature scale Tco, we expect
the amplitude of the order parameter is well defined to support fluctuations in the
phase (p. In this regime, the resistance is still finite due to phase fluctuations. At T
= T k t b >the transverse phase fluctuations are frozen out and KTB phase transition
takes place. The system enters a superconducting state with bound vortex-antivortex
pairs.
To establish the temperature Tc0 where the superconducting transition might oc­
cur without phase fluctuations, we fit the DC resistance curves to the AslamazovLarkin form for the fluctuation conductivity for two dimensional systems. The total
conductivity in the amplitude fluctuation regime can be written as atotai = <
tn + &a l ­
gal
is given by equation 1.18. Converting the expression to the measured sheet
resistance Rmeas>one obtains
=
=
(31)
Here R n = l / a Nd is the sheet resistance for normal electrons. In Fig. 3.2, we fit the
R m from Rmeas of an InO* film using
as a fitting parameter. Curves in black and
70
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
1000
800
□ 600
£ 400
200
1
2
3
T(K)
4
6
6
Figure 3.1: Ra as a function of temperature for one of the InOx films.
71
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
1*00
1400
1200
1000
800
—
600
Manured DC atiaat Resistance
Effective Normal sto a t Rasistanca
with Tj, ■ 2.76 K
Effective Normal sto a t Rasistanca
with T . « 2.58 K
400
200
1
3
2
4
S
6
T (K)
Figure 3.2: Measured sheet resistance as a function of temperature along with two
fitted effective normal sheet resistances.
blue are the fittings for Tc0 = 2.55 K and T& = 2.75 K. Under the assumption that R m
should not have a minimum in the temperature range we measured, Tc0 is determined
as the fitting parameter when the minimum in fitted R n first disappears. In this
fitting analysis, one actually only obtains a minimum value for T ^. It is technically
possible that T& could be larger. However, this Tco is just a parameter to show the
possible crossover behaviors. Equation 1.18 is only valid when the ensemble average
of |i/>|2 is small. That is why the extracted
R n {T )
in the data has a notable upturn
which is obviously not physical.
For this particular sample, Tco is 2.75 K and is marked by the dashed vertical green
line in the plot. Note this temperature Tco is way above the temperature
T k tb
~
2.4 K, when Rmeas is indistinguishable from the contact resistance. Below Tco, we
72
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
consider the superconducting fluctuations are mostly fluctuations in the phase of the
superconducting order parameter.
3.2
Dynamics of KTB phase transition
The remarkable properties of superconductors and superfluids arise from the
macroscopic quantum-mechanical coherence of their complex order parameter (OP),
\J) — A el<t>. In conventional superconductors, fluctuations of the OP amplitude and
phase occur in temperature regions only infinitesimally close to Tc. In contrast, in
disordered materials with reduced dimensionality, the situation may be considerably
different. Their low superfluid density gives a small phase stiffness and phase fluc­
tuations that may be particularly soft [54]. In such systems, phase plays the role of
a dynamic variable and may result in a situation where the transition results from a
phase disordering of the order parameter while its amplitude remains finite. Effects
such as zero resistivity are lost when the phase is no longer ordered on all lengths.
However, phase correlations may remain over finite length and time scales resulting
in significant precursor effects above Tc.
As discussed in Chapter 1, in strictly two dimensions (2D), such a transition has
been proposed [17,55] to be of the Kosterlitz-Thouless-Berezinsky (KTB) variety, as
in 4He films [8,9,12,56]. In such a transition, thermally excited free vortices are
not possible below the transition temperature
73
T ktb as
the vortex-antivortex binding
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
energy increases logarithmically with separation. However, above Tktb it becomes
entropically favorable for vortices to unbind. Vortex pairs with the largest separation
unbind first and the phase stiffness measured in the long-length and low-frequency
limit suffers a discontinuous drop. Vortex unbinding reduces the global phase stiffness
and renders the system increasingly susceptible to further vortex proliferation. At
temperatures just above Tktb such systems can be described as a two-component
vortex plasma and may be realizations of the 2D XY model.
Because free vortices are the topological defects of the phase field, their spacing
plays the role of a Ginzburg-Landau correlation length £, which diverges as T —>T k t b The role of free vortices as topological defects and their finite energy cost give an
exponentially activated vortex density (nF oc l/£ 2). Asymptotically close to the
transition, this results in an unusually stretched exponential dependence of f on
temperature, £ ~ es/ T'^ T~TKTB\ which is in stark contrast to the power laws typically
expected near continuous phase transitions [9]. Similar dependence is expected in the
“critical slowing down” of the phase correlation time 1/D, which in a vortex plasma,
is proportional to the time £2/D required to diffuse the intervortex spacing (where D
is the vortex diffusion constant) [17].
Although the conventional wisdom is that a KTB-like transition occurs in ultrathin superconducting films [55], the issue is still in fact controversial. For instance,
it has been proposed that, unlike 4He films, superconducting films are in a regime
of low core energy (the high “fugacity” limit), that causes the transition to acquire
74
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
a nonuniversal character or even be first order [12,57,58]. There has been a great
deal of work looking for KTB physics in the linear and nonlinear dc transport char­
acteristics of thin films superconductors [21,59,60]. But it is not clear to what extent
these experiments are influenced by inhomogeneous broadening [61] and if even KTB
physics would be detectable in such experiments [12]. In contrast, finite-frequency
measurements can directly probe temporal correlations and can be explicitly sensi­
tive to phase fluctuations right above Tc. Important information has been gained
from measurements at discrete frequencies [21,25,47,62], but only true spectroscopic
measurements can give important information concerning critical slowing down.
In this paper, we present a comprehensive study of the complex ac conductance
of effectively 2D amorphous superconducting InO* films. We make use of our recent
development of a broadband Corbino microwave spectrometer, which can measure the
explicit frequency dependence of the complex conductance of thin films over a range
from 0.21 - 15 GHz at temperatures down to 350 mK. These unique measurements
allow true spectroscopy in the microwave range at low temperatures. We explicitly
measure the temporal correlations of the fluctuation superconductivity and demon­
strate the manner in which their time scales diverge on the approach to the transition.
The temperature dependence of the critical slowing down is consistent with a contin­
uous transition induced by the freezing out of vortex-like phase fluctuations.
75
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
3.3
Sample Details
For these measurements, high-purity (99.999 %)
1^03
was e-gun evaporated un­
der high vacuum onto clean high-resistivity silicon substrates held at liquid nitrogen
temperature to a thickness of approximately 30 nm. InOa, samples were prepared
at Sambandamurthy’s lab at University of Buffalo. Their synthesis derive from the
work of Ref. [63], where it was shown that amorphous InO* can be reproducibly
made by a combination of e-beam evaporation of 1 ^ 0 3 with optional annealing. Es­
sentially similar films have been used in a large number of recent studies of the 2D
superconductor-insulator quantum phase transition [24,62,64-67]. We believe that
the films are morphologically homogeneous with no crystalline inclusions or largescale morphological disorder because of the following: TEM-diffraction patterns are
diffuse rings with no diffraction spots, AFM images are completely featureless down to
a scale of a few nanometers (the resolution of the AFM), and R versus T curves when
investigating the 2D superconductor-insulator transition [24,62] are smooth with no
reentrant behavior that is the hallmark of gross inhomogeneity. In Fig. 3.3, we show
the AFM image, TEM diffraction pattern and TEM image for a co-deposited insulat­
ing InO* film showing that InOx films evaporated in this fashion are amorphous and
homogeneous. The temperature dependence of the sheet resistance of a granular InOz
film can be found in Fig. 3.4, which shows the very noticeable reentrant behavior of
a granular film.
The in-plane penetration depth — the so called Pearl length (2A|D/d) [16] — can
76
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
O
Figure 3.3: The AFM image, TEM diffraction pattern and TEM image of a co­
deposited insulating InOx film. The TEM diffraction pattern shows diffuse rings
indicating that there are no crystalline inclusions. Both AFM and TEM images
show that films prepared by the quench condensed e-gun evaporation do not include
mesostructure and are not granular as well.
6000
A granular inaulating lnOxfilm
6000
5
m
i 4000
JZ
O
7
3000
2000
1.0
1.6
2.0
2.6
3.0
3.6
4.0
T(K)
Figure 3.4: Sheet resistance of a granular InOx film showing the noticeable reentrant
behavior.
77
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
be calculated from the data below to be approximately
6
mm near Tc, which is well
in excess of any sample dimension. Vortices and antivortices are thus held together
by logarithmical confining potentials. In this case, superconducting films are similar
to the case of 4He films.
3.4
Results
Experiments were performed in the home built broadband Corbino microwave
spectrometer described by the previous chapter. We concentrate on a particular
In 0 2 film with a Tc = 2.36 K, but the data is broadly representative of samples
with this normal-state resistance. In what follows, Tc is defined as the temperature
at which the simultaneously measured dc resistivity becomes indistinguishable from
zero (shown in Fig. 3.6). A small ± 5 mK uncertainty in this determination does
not affect our conclusions. In Figs. 3.5 (a) and 3.5 (b) we plot the real (G\) and
imaginary (G2) conductance as a function of frequency at different temperatures.
Well above the transition, Gi is flat and featureless and G 2 is small, as one expects
for a highly disordered metal at low frequencies. When the sample is cooled toward Tc,
the real conductance initially becomes enhanced and its spectral weight shifts to lower
frequencies. At lower temperatures, the imaginary conductance grows dramatically
and its frequency dependence becomes close to 1/u. This is the low-temperature
behavior expected for a superconductor. As seen clearly in plots of the same data
78
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
0.5
1.0
1.5
2 .0
2 .5
3.0
T (K)
0 .5
1.0
1.5
2.0
2 .5
3.0
T (K)
Figure 3.5: (a) and (b): frequency dependence of the real and imaginary conductance
in the ranges uj/2 t: = 0.21 —27 GHz and T = 0.35 —4 K. A color scale representing
different temperatures is displayed in (a). The black curves are the conductance at
Tc. The features in (a) at about 22 GHz are residual features imperfectly removed
during calibration, (c) and (d): temperature dependence of the real and imaginary
conductance in the frequency range uj/2n = 0.660—15 GHz. A color scale representing
different frequencies is displayed in (d). The dashed black lines mark Tc = 2.36K.
79
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
as a function of temperature (Fig. 3.5 (c) and (d)), the region immediately above
Tc is dominated by superconducting fluctuations. As shown by comparison to the
dashed line, the real and imaginary conductance begin to show an enhancement in the
temperature region above Tc. Our measurements are explicitly sensitive to temporal
correlations. This is seen for instance in the fact that the near-Tc “dissipation peak”
in Fig. 3.5 (c) is exhibited at lower frequencies for lower temperatures; the maximum
in dissipation is expected when the characteristic fluctuation rate Q /27r is of the order
of the probing frequency uj/2n.
Above the transition temperature, due to the thermal energy, one expects short
range fluctuations with length £ which persists for time r. On approaching the tran­
sition point, both £ and r become larger and larger. The characteristic fluctuation
frequency
0
, which is defined as 1 / r , becomes smaller and smaller and vanishes in
the vicinity of the transition. This phenomenon, so called critical slowing down, is
generic to any continuous phase transition. One interesting feature in our very raw
data is that we can see the signature of the critical slowing down from the movement
of the peak in real conductance in temperature as shown in Fig. 3.5 (c).
A particularly important quantity for quantifying fluctuations is the phase stiff­
ness, which is the energy scale to twist the phase of the OP. The phase stiffness, Tg, is
proportional to the superfluid density and can be defined (in units of degrees Kelvin)
through the imaginary conductance G2 as
,
,
G2
kBTg(u) - —
N (u)e2Hd
- ——— — ,
80
(3.2)
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
1000
800
73
600 %
ZJ
400
3
"
□
200
0
1.0
2.0
3.0
4.0
T (K)
Figure 3.6: Temperature dependence of the phase stiffness at o;/27r = 0.21 — 15
GHz plotted against the vertical axis on the left. A color scale representing different
frequencies is displayed as well. The black curve shows resistance per square of the
same sample plotted with the vertical axis on the right. The dashed pink line is the
KTB prediction, 4Tktb = Tg, for the universal jump in stiffness. The dark purple A
markers are Tg s obtained via the scaling analysis described in the text. Tc is marked
by the black dashed line.
where G q
—
^ is the quantum conductance for Cooper pairs and N (u) is a frequency-
dependent effective density. In Fig. 3.6, we plot the stiffness versus temperature
measured at frequencies between 0.21 to 15 GHz. Tg(u) defined through Eq. (3.2)
measures the stiffness on a length scale set by the probing frequency, which is typically
proportional to the vortex diffusion length during a single radiation cycle,
A
is a constant on order of unity and D is the diffusion constant. At temperatures well
81
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
below Tc, there is essentially no frequency dependence to the phase stiffness, consis­
tent with the scenario that the phase stiffness is rigid on all lengths. At temperatures
slightly above Tc, the phase stiffness is largest at high frequencies. In the fluctuation
regime, the system retains a phase stiffness on short length scales. Plotted alongside
the stiffness data is the co-measured resistance per square, R q. Within experimental
uncertainty, the phase acquires a frequency dependence at the temperature where the
resistance appears to go to zero. In keeping with our discussion in the introduction,
this is reasonable as a superconductor can only exhibit zero resistance when its phase
is ordered on all lengths.
KTB theory predicts that at the transition temperature,
TKt b ,
the stiffness in the
zero-frequency limit will have a discontinuous jump to zero with a magnitude Tg =
4Tktb- However, because finite frequencies set a length scale, the ac stiffness should
go to zero continuously. We generally expect that a signature of the discontinuity
will manifest in a strong frequency dependence in the stiffness that onsets at
Tk t b -
In Fig. 3.6, the dashed diagonal line gives the prediction [9,17] for the universal
relationship between Tktb and the stiffness. It crosses the stiffness curves very close
to where they start to spread. This, along with the fact that the resistivity goes
to zero at this temperature, leads us to assign the transition to a vortex unbinding
transition of KTB-like character. Note that a careful inspection of Fig. 3.6 on linear
scale reveals that the stiffness is in fact approximately 30% (see Fig. 3.7) greater at
Tc than the universal prediction. We cannot be sure at this time whether this is a
82
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
20
4
8
g )/2 jc
15
12
(GH z )
T6 critical
“^ p r e d ic te d
5-
ok
2.0
Figure 3.7: A plot of Fig. 3.6 on a linear scale near Tc reveals that the stiffness is
approximately 30% greater at Tc than the universal prediction.
systematic deviation (due perhaps to the dissipative motion of bound vortex pairs or
evidence of a non-universal jump [1 2 ]) or a small calibration error.
Above
Tk t b ,
the conductance due to fluctuating superconductivity is predicted
[17,48,68,69] to scale with the form
g <3
_ (* s 3 )s (u /n ).
'
m
(3.3)
In this scaling function, all temperature dependencies enter through f!(T), the char­
acteristic relaxation fluctuation rate, and Tg, an overall amplitude factor related to
the total spectral weight in the fluctuating part of the conductivity. Note that in Eq.
83
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
co/n(T)
Figure 3.8: (a) Phase of G (u /fi) as a function of frequency, (b) Magnitude of G(ui/Q)
as a function of reduced frequency. A color scale representing different temperatures
for both plots is displayed in (b).
(3.3) the prefactors T f and Q are real quantities, so that the conductance phase angle,
V?, must be equal to the phase angle of S(u/Q ). In Fig. 3.8 (a), we plot phases of the
measured conductance at different temperatures above the transition temperature for
this particular sample from 460 MHz to 10 GHz in a temperature range from 2.398 to
2.993 K. Those phases are extracted directly from the measured conductance. Phases
below 2.398 K are not shown because one needs to accurately determine the three
error coefficients to higher precision near and also below the critical temperature.
Thus phases obtained after calibration at those temperatures are a little off from the
actual phases of the sample. To collapse all the phases into one single universal curve,
we divide the probing a; by a different D(T) at each temperature, thus extract the
characteristic fluctuation frequency fi for each individual temperature. In Fig. 3.8
(b), we show the magnitude of conductance for all the temperatures as a function of
84
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
reduced frequency u/Q . The amplitude of the universal function l^l can be directly
obtained by multiplying the amplitude of the conductance by M I /^ b T^G q ).
1.0
0.8
^
£S
0.6
* 0.4
0.2
0.0
0.01
2 4 6
0.1
2 4 6
1
2 4 6
0.01
co/fXT)
2 4 6
0.1
2 4 6
1
2 4 6
co/n(T)
Figure 3.9: (a) Phase of S(w/Q) normalized by 7r/ 2 as a function of reduced frequency
w/O. (b) Magnitude of S(u/Q ) as a function of reduced frequency. A color scale
representing different temperatures for both plots is displayed in (a). Each plot is
comprised of data measured at temperatures from 2.398 to 3 K and frequencies from
0.46 to 10 GHz.
In Fig. 3.9 (a), we show this phase angle, <p, collapsed into a function of reduced
frequency ui/Q for each temperature. The data collapse reasonably well down to 38
mK above Tc. At lower temperatures, the fluctuation frequency begins to enter the
low-frequency end of the spectrometer (about 500 MHz for this set of data). In the
low scaled-frequency limit, which corresponds to high temperatures and normal-state
response, the phase approaches zero as expected. In the high scaled-frequency limit,
which corresponds to low temperatures and the superconducting state, the phase
approaches 7r/ 2 also as expected. This analysis allows us to extract fi(T). Here,
85
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
we have isolated the fluctuation contribution to the conductance by subtracting off
the dc value from well above Tc (at 5.6 K). Having determined U(T), we adjust Tg
and normalize the magnitude of conductance by
l
jO
to get the magnitude (|Sj) of
S(u/Q.) so that they fall onto one curve as demonstrated in Fig. 3.9 (b). In 2D, Tg is
equivalent to the high-frequency limit of the stiffness. We plot it alongside the finitefrequency stiffness in Fig. 3.6. One can in principal compare the scaling forms for the
1.0
1
0.8
^
2.4 2. 5 2.6 2.7 2. 8 2.93.0
0.6
I
*
CO
0.4
2DAL
0.2
0
.
0.0
0.1
1
10
100
oVQ(T)
0.1
1
10
100
w/Q(T)
Figure 3.10: Comparison of the scaling forms with the 2D AL formalism.
InOz film with the 2D AL formalism (see Fig. 3.10). The explicit expression of the
frequency dependent complex conductance was calculated by Schmidt [70]. Unlike
the NbN films [49], one can see that we cannot fit the phase and magnitude of the
scaling function simultaneously by the 2D AL expression. This also suggests that the
transition in our InOx cannot be explained by the 2D AL formalism only.
86
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
The monotonic decrease of D(T) [Fig. 3.11 (a)] as Tc is approached from above, is
an indication for the critical slowing down expected near a continuous transition. In
Figs. 3.11 (b) and 3.11 (c), we fit Q(T) to the stretched exponential form expected
near a KTB transition D0 exp(—y /4 T '/(T —Tc)) as well as to a generic power-law
form D0(l —
The fits were performed over different temperature ranges from
Tc on up (147 and 112 mK, respectively) such that the same reduced x 2 is achieved
on both fits.
The stretched-exponential fit gives coefficients of Qq/2 tt — 181 GHz and T ' = 0.23
K, while the power-law fit gives Q,q/2 tt = 90 GHz and zu — 1.58, which are all rea­
sonable parameters. Unfortunately it is hard to distinguish definitely between these
scenarios in our data. However, we do favor the KTB scenario since the stretchedexponential fit covers a wider temperature range and fits better near Tc. For instance,
within the ansatz of Ref [17]. it is predicted that V — 7 (1 ]*} —TKtb)i where
a constant of the order of unity. Ref. [61] predicts more specifically that
7
7
is
= 4a2,
where a is the ratio of the vortex core energy, //, to the vortex core energy in the 2D
XY model, fixy. Viewed in this regard, our value of V is consistent with a reasonably
small value of the core energy.
For both functional forms, one expects that the prefactor D0 will be of order of
the inverse time needed to diffuse a vortex with core size fo- Using the BardeenStephen [17] approximation for D, one can derive the expression hfi0 = 2ttXh D / ^ =
27tA^A:bTc, where A is a constant of the order of unity and G n is the normal-state
87
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
dc conductance. For the present sample, this gives Q0/27t
rj
48A GHz, which is
consistent with both fits. Due to its larger fitting range and its consistency with the
universal jump, we favor the stretched exponential form, but in practice, it is difficult
to definitively exclude power-law dependencies.
However, we can note a number of additional aspects consistent with a vortex
plasma regime. Over a more extended temperature range above Tc, one expects
that Q(T) will obey the relation n 0 exp(—^-). Here, T* is half the energy needed
to thermally excite a free vortex-antivortex pair. In Fig. 3.11 (d), we plot T* =
Tln(Do/U). As expected this quantity appears to diverge as T —» Tc. It reaches a
high-temperature limiting value of about 0.27 K. One expects [59] that kr>T* — n +
\ksTg ln(£/f0) as the logarithmic interaction has a cutoff at f. Here, (jl is the vortex
core energy and the second term is the vortex interaction energy. At temperatures
well above Tc where £ is of the order of £0> the logarithmic term is negligible and
the excitation energy should be proportional to the core energy alone. Within the
BCS model, the core energy can be shown [59] to be approximately fcflT^T)/ 8 .
A comparison with T# from Fig. 3.6 gives an estimate of ii/k B ~ 0.3 K in this
temperature range. The agreement with experiment is essentially exact, but one
should not take the exactness too seriously as there are a number of neglected factors
of the order of unity.
88
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
3.5
Conclusion
We have presented a comprehensive study of the complex microwave conductance
of amorphous superconducting InOx thin films. Our data explicitly demonstrate
critical slowing down close to the phase transition and, in general, the applicability of
a vortex-plasma model above Tc. This technique opens up the possibility of studying
dynamic scaling of phase transitions at low temperatures and frequencies of a number
of material systems.
89
CHAPTER 3. AC STUDIES OF THE ZERO FIELD PHASE TRANSITION
<t*N20
JN
2.4
2.6
T (K)
2.8
2
5
8F
*
010
£
a£
4
5
6 7 8 9
3
4
5
6
0
0.1
0.2
0.3
0.4
0.5
0.6
oVO(T)
Figure 3.11: (a) Fluctuation frequency as a function of temperature from 2.398 to
2.993 K. In (b) and (c), we plot U (T)j2 ti versus l /y /T — Tc and T —Tc, respectively,
along with the fitting (black curves), (d) Excitation energy in units of degrees Kelvin
as a function of T —Tc.
90
Chapter 4
2D field tuned
superconductor-m etal quantum
phase transition
A quantum phase transition (QPT) is a zero temperature change of state as a
function of a non-thermal parameter. The two dimensional (2D) superconductorinsulator transition (SIT) is an emblematic example of a QPT and has been the
subject of many theoretical and experimental studies [22,71]. As conventionally en­
visioned, superconductivity can be suppressed by applying magnetic fields beyond
some critical value whereupon the system transitions to an insulating state with a
diverging resistance at T = 0. One possible scenario is that this transition occurs
via a “fermionic” mechanism by destroying the amplitude of the superconducting
91
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
order parameter. At the transition point, superconducting correlations are strongly
suppressed. Another possibility is a “bosonic” mechanism in which the Cooper pairs
become localized through quantum disordering of the order parameter’s phase. In
this case, one expects pairing to exist on both sides of the transition.
Within the latter bosonic description, a universal resistance of order the quantum
resistance for Cooper pairs
Rq
=
h / 4e2
~ 6450
il
is expected at the quantum critical
point (QCP) [22]. Although some indications for superconducting correlations in the
insulating state have been reported [24-26], there has been little definitive evidence
in favor of this pure bosonic model. Some systems do exhibit a resistance of order
Rq
at the transition; however, others show a much smaller critical resistance [72]. More­
over, instead of a direct transition to an insulator many materials appear to exhibit
an intervening metallic region that features a small yet finite saturated resistance at
the lowest measured temperatures [73-78]. This effect is usually more pronounced in
low-disorder films. In these cases the transition appears to be one from a supercon­
ductor to a strange metal with superconducting correlations. A true metallic phase
as such may be surprising because one might naively expect that delocalized bosons
would ultimately condense at the lowest temperatures. The physics is still unclear
despite various theoretical efforts to demonstrate the possibility of a zero-temperature
dissipative state with superconducting correlations [30,79-82]. On the experimental
side, the possibility exists that the apparent zero-temperature DC dissipation could
be a consequence of insufficient cooling of the carriers despite careful experimental
92
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
checks. For these reasons, it is important to utilize experimental probes other than
DC transport to investigate this problem.
Microwave spectroscopy gives an advantage in studying the 2D SIT in that one can
be explicitly sensitive to temporal correlations. AC measurements provide detailed
information about the critical slowing down of the characteristic frequency scales
approaching a transition, which may reveal the true location of the QCP. Furthermore,
we can study the dynamics of the possible intervening metallic state. Through the
imaginary conductance, AC measurements of superconductors also allow access to
the phase stiffness Tg(u), which is directly related to the phase coherence on a length
scale set by the probing frequency.
4.1
Dynamics of 2D quantum phase tran­
sition
In this chapter, we present novel measurements of the frequency, temperature and
field dependence of the complex microwave conductance on a particularly low-disorder
superconducting InOx film through its QPT. From the simultaneously measured DC
resistance, a well-defined field BcrOSS is identified as the crossing point of different
isotherms R(B). In most interpretations of similar data, Bcr0as is taken to be the
location of the QCP. However, quite contrary to expectations for the slowing down of
fluctuations near the presumed continuous transition at Bcr0SS, the relevant frequency
93
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
scales extrapolate to zero at a much smaller field B sm. The phase stiffness Tg at the
lowest frequency vanishes from the superconducting side at B « Bsm, while the high
frequency limit approaches zero at B « Bcr0SS. Our data support a scenario in which
B sm is the true QCP for a transition from a superconductor to an anomalous metal,
while Bcross only signifies a crossover to a regime where superconducting correlations
are strongly suppressed.
Samples are morphologically homogeneous InOx films prepared by e-gun evapora­
tion of In 2 0 3 to a thickness of approximately 30 nm onto high-resistivity Si substrates
as described in Chapter 3 . The nominal 2D SIT in InOx can be tuned by applying
perpendicular magnetic fields [24,25,66,67]. Broadband microwave experiments were
performed in a home-built Corbino microwave spectrometer coupled into a He-3 cryostat. We measured the complex reflectivity of the sample in the microwave regime,
from which complex impedance and conductance can be obtained. Three calibration
samples with known reflection coefficients (20nm NiCr on Si, a blank high-resistivity
Si substrate and a bulk copper sample) were measured to remove the contributions
from the coaxial cables to the reflected signals [38,41,44,48,51]. Two terminal DC
resistance can be simultaneously measured via a bias tee. The DC resistance without
the microwave illuminations was used to carefully check and correct for any microwave
induced heating. The sample at 5 Tesla had a impedance closely matched to the cable
thus had a maximum absorption of microwave radiations. The lowest temperature
for that field was 426 mK under microwave radiation. The heating effects at other
94
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
fields were negligible. Calibrations were performed at each displayed magnetic field
unless otherwise specified. With substrate corrections 2.3, the true response of the
InOx film can be isolated at all fields and temperatures.
0
1
2
3
4
0
T(K)
2
4
6
8
B (T)
Figure 4.1: (a) Temperature dependence of sheet resistance Rq at different fields as
indicated by the color legend, (b) R q as a function of field at 6 fixed temperatures
as shown by the color legend. The crossing point of the two lowest temperature
isotherms is approximately 7.5 Tesla.
In Fig. 4.1 (a) we plot the two-terminal sheet resistance Rq as a function of
temperature at fixed magnetic fields. The particular InOx film in this paper shows a
transition to a zero resistance state at Tc = 2.36 K at zero field. Previous microwave
studies have demonstrated that its zero-field thermal fluctuations are consistent with
a 2D Kosterlitz-Thouless-Berezinsky transition [51]. The normal state resistance per
square
R
n
is about 1200 fh This number is far below R q and is comparable to
R
n
for
thin films like a-MoGe [72]. It implies that this film has a much lower disorder level
(kpl is in the range 3 ~
6
[83]) compared to the InOx films used in many previous
95
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
QPT studies and falls into the same class of lower-disorder thin films superconductors
such as a-MoGe [30]. At low temperatures, the slopes of the resistance curves change
sign at about 7.5 Tesla (see Fig. 4.1 (a)). For the two lowest temperatures, the
data also exhibit an isoresistance crossing point Bcr0sa at 7.5 Tesla. This field is
conventionally interpreted as the location of QCP. Like a-MoGe, this sample exhibits
an exceedingly weak “insulating state”, with barely a
10
% rise in the resistance
from 4 K to the lowest measured temperatures at B > Bcr0gs as shown in Fig. 4.1
(a). This is again very different from strongly disordered InOx films that show an
enhancement of the resistance upwards of
109
Q at similar magnetic fields at low
temperatures [24,66]. Like other weak disordered superconducting films, the DC
data show an apparent trend towards saturation at low temperatures for fields above
3 Tesla. This saturation was confirmed in separate two-terminal measurements of
this sample down to dilution fridge temperatures.
A separate measurement of the same sample down to 60 mK was carried out
roughly half a year after the microwave measurements. Unfortunately, a direct com­
parison between the two DC data sets cannot be made as the InOx sample anneals
even at room temperature over the course of intervening months. As shown in the
supplementary information, although the sample changed with a systematic move­
ment of curves with the same resistance to higher fields, the data sets are overall very
similar with continuing trend towards saturation at low temperatures. The sample
has been kept in the dry box the whole time and has lost some oxygen over the course
96
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
of intervening months. In Fig. 4.2 (a), we display Ra as a function of temperature
at different fields for the sample after a few months aging. The level of disorder has
changed with a new Tc = 2.68 K at zero field. We show R q as a function of field at
75 mK and 150 mK in Fig. 4.2 (b) and one can clearly see that BcrOSS changes to 7.86
Tesla.
In Figs. 4.3 (a) and (b), we plot the real (Gi) and imaginary (G2) conductance
respectively as a function of frequency at the base temperatures for each field («
426 mK for 5 Tesla and « 300 mK for all other fields). Due to a missing set of
calibration curves, the data at 4 Tesla were calibrated using an effective calibration
interpolated from 3.5 Tesla and 5 Tesla calibration standards. We estimate that the
error introduced by this interpolation is less than
with a slope of
-1
1
%. As shown by the straight line
on the log-log plot in 2 (b), at zero field and 300 mK, G 2 shows
the 1/u frequency dependence expected for a superconductor at frequencies below
the gap. This dependence is consistent with Gi = 6(uj) via the Kramers-Kronig
relation. Indeed, Gi at zero field is small with a value that is at the limit of our
experimental sensitivity, thus is not plotted. At B <tc BcrOSs, G% remains linear with
the same slop in the log-log plot but its magnitude drops dramatically as the field is
increased. This implies that the 5-function in G\ is preserved, although its spectral
weight (proportional to the superfluid density) is greatly decreasing.
At intermediate field strengths (B ~ 3 Tesla), a maximum in G 2 appears. Accord­
ing to the Kramers-Kronig relation, this implies that a significant spectral component
97
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
1400 r
1200
1000
□
0T
800
2T
3. 5 T
4T
5T
6T
6. 5 T
600
400
7. 5 T
8T
11 T
14 T,
200
4
3
2
1
0
T (K)
1400
1200
75 mK
150 mK
1000
cross
800
£
600
400
200
0
4
6
8
BCD
10
12
14
Figure 4.2: (a) Temperature dependence of the R q measured in the range T = 0.06
- 4 K at fields represented by the color legend, (b) Ro as a function of field at two
fixed low temperatures.
98
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
Figure 4.3: Frequency dependence of (a) real (Gi) and (b) imaginary (G2) conduc­
tance respectively in the ranges u /2 n = 0.08 - 16 GHz. Gi and G2 have the same
color legend at finite fields except that G\ at zero field is not plotted. The dashed
grey line in (b) is a guide to the eye of G2 oc l /u .
in G\ has a finite width. As shown previously [51] , the frequency of the maximum
in G2 corresponds to the characteristic fluctuation rate D in a fluctuating supercon­
ductor. The decrease in the frequency of the peak in G2 as the field is reduced is
an unambiguous signature of critical slowing down of the fluctuation frequency while
approaching a transition. However, the peak in G2 is developed at a field that is
well below Bcross and the fluctuations are clearly speeding up as we approach Rc-oss
from below. This behavior is inconsistent with the conventional wisdom for QCP
phenomenologies if Reross is a QCP because one generally expects a slowing down of
the internal fluctuation frequency scales as one approaches a continuous transition.
When B ~ Rcross> we cannot distinguish the superconducting signal from the normal
state background as Gi is flat and featureless and G2 is small. The data is reminiscent
of what one expects for a disordered metal with a high scattering rate [51].
99
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
50M H z 1
125M Hz
500M H z
1G H z
5G H z
10G H z
14G H z
&/2n (GHz)
B (T)
Figure 4.4: (a) Frequency dependence of the phase stiffness in the ranges u/2 ir =
0.08 - 16 GHz at the base temperature for each field. The color legend for fields is
the same as in Fig. 4.3 (a), (b) Phase stiffness as a function of field at different
frequencies at the base temperature for each field.
An essential quantity for analyzing superconducting fluctuations is the phase stiff­
ness
Tg ,
which is the energy scale required to twist the phase of the superconducting
order parameter. Within a parabolic band approximation,
Tg
<x N, the superfluid
density. More precisely (and in a model independent fashion), it is proportional to
the spectral weight in the zero frequency delta function and can be measured through
Gi
Tg
as
Tg(ui)
=
, where
Gq
=
1/ R
q
.
This relation expresses the energy scale
in degrees Kelvin, and gives the phase stiffness on a length scale set by the probing
frequency. Fig. 4.4 (a) shows
T g (u )
above for each field. At low field,
at the respective base temperatures described
Tg
shows essentially no frequency dependence,
which suggests that the phase is ordered on all lengths. We see a dramatic drop in
Tg
at B -C Bcross. For intermediate fields,
100
Tg
starts to acquire a strong frequency
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
dependence at low u/, which reflects that Cooper pairs have short-range correlations
that can be resolved at high probing frequency while the long-range correlations are
suppressed. At high u the frequency dependence becomes less pronounced showing
that one approaches a well-defined high frequency limit.
The rapid decrease in the overall scale of
(b) where we display field dependence of
Tg
Tg
can be clearly observed in Fig. 4.4
at several frequencies cuts from Fig.
4.4 (a). Above 2 Tesla, the curves start to spread, indicating the superconducting
correlations gain a length dependence. At the lowest frequency (50 MHz, which probes
the longest length scale), Tg drops dramatically around 3 Tesla indicating that long
range ordered phase coherence is suppressed by increasing fields. Note the strong
suppression in Tg in this field range; at some frequencies the suppression in
followed over 7 orders of magnitude. Unlike the low frequency behavior,
Tg
Tg
can be
at high
frequency extrapolates towards zero near Bcr0as. This latter finding greatly differs
from previous microwave cavity measurements on a much more disordered InOx film
(Bcross = 3.68 Tesla) [25]. In that work the finite-frequency
Tg
was non-zero well
past the phenomenologically defined Bcross into the strongly insulating phase. This
was interpreted as a state that while strongly insulating on long length scales, has
superconducting correlations on short ones. In contrast,
Tg
for our low-disorder film
vanishes on approaching Bcross instead of staying finite well beyond it. This indicates
that the superconducting correlations do not survive appreciably through 5cro«s and
the superfluid density is indistinguishable from zero into the weakly insulating state
101
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
at the lowest temperatures we can access.
To form a more quantitative understanding of the fluctuations, we fit G\ and G2
to a model where the fluctuation contribution is given by a zero-frequency Lorentzian
lineshape [84]. This is simpler, but essentially equivalent to the scaling analysis we
performed in Chapter 3 to obtain the characteristic fluctuation rate approaching
a finite temperature transition at zero field [51]. Lorentzian fits are equivalent to
assuming that time correlations are exponentially diverging while approaching the
transition. The fits agree well with the data, thus justifying this assumption. The fit
of G\ and G2 to a Lorentzian lineshape goes as
, Nie2d n
1
N2e2dr2
1
G = od=
---- :------ 1----------------- :---m
1 —iuti
m
1 —tur2
Essentially, we use two Drude models to describe the combined contributions from the
normal electrons and superconducting fluctuations to the complex conductance. The
scattering rates of the normal electrons for this film are much bigger than Q and exceed
our frequency range. One can estimate the scattering rate r for the InOx in the normal
state from the normal resistance by using the equation R N = l / G N = l/C2^ ^ ) . Here,
d = 30 nm is the film thickness. Using n ~ 1020 cm-3 [63,85,86] and R n « 1000
Ohms, we estimate the scattering rate for the normal electrons is on order of 100 THz.
Therefore, the Drude term of the normal electrons just makes a constant contribution
to G\ and a negligible contribution to G2. In Fig. 4.5, we show the fitting to G\ and
G2 at B = 3.5 Tesla, T = 850 mK as an example and the fitting agrees well with
our experimental data. For this particular data set, the complex conductance can be
102
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
12x10
B = 3.5 Tesla
T = 850 m K
T
M
E
£
0
Fitted G,
Fitted G ,
9
£
§3
1
o
0
2
4
6
8
10
a>/2n (GHz)
12
14
16
Figure 4.5: Experimentally measured real and imaginary conductance and fitted data
to a Lorentzian lineshape model.
fitted as,
_
+
0.0094
l-i(£ /1 .2 3 9 )G H z
0.00172
l- i( £ / 2 2 .7 8 7 ) G H z
In this model the fitted width of G\ is the characteristic fluctuation rate D(T), while
its integrated area is equivalent to the high frequency limit of
Tg
(in appropriate
units).
In Fig. 4.6 (a), we plot D(T) for fields up to 6 Tesla. Data above 6 Tesla exhibit
fluctuation rates that axe far above our accessible frequency range. At zero field, Q
goes to zero when T approaches T c from above showing the critical slowing down that
confirms our previous results [51]. Q drops in a much slower fashion at finite fields and
103
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
even begins to saturate to a finite value at the lowest temperatures for B is greater
than or equal to 2 Tesla. Fig. 4.6 (b) is a contour plot of
in field and temperature.
It is safe to conclude that the D = 0 contour falls below the lowermost curve that has
n = 0.2 GHz. In general, small fi contours at low temperatures extrapolate to zero
at a field less than 3 Tesla, which is again far below Bcross- To form a global view, we
bring a number of these quantities together in a single phase diagram in Fig. 4.6 (c).
For all quantities, energy scales and frequencies have been converted to energy units
(in degrees Kelvin). Upward and downward triangles show the low and high frequency
limits of Tg in our setup’s accessible frequency range. Squares are f2(T) taken at base
temperatures. Circles demonstrate the temperature when
Rq
= 0.3%
R
n
,
where
R
n
is the sheet resistance at 4 K. It denotes a region in phase space where the resistance
is “small” . It is clear from this plot that
Tg
in the zero-frequency limit (shown by
the bold line in Fig. 4.6 (c)) and Q converge towards zero at B « 3 Tesla. This “V”
shaped phase diagram is exactly what one expects near a QCP where energy scales
extrapolate to zero from either side. Again, BcrOSS, which is conventionally considered
to be a QCP, appears to be completely unrelated to the actual critical behavior. One
can see that
B ^oss
is the field scale where the high frequency Tg is suppressed. Due to
the lack of evidence for a diverging
R
q
at
B sm
< B<
Bcross
in the zero-temperature
limit, one reasonable interpretation of the phase diagram is that this low-disorder
InOx film has a true QCP located at
B sm
« 3 Tesla between a superconducting and
an anomalous metallic state. In this picture BCTOSS only marks a crossover in behaviors
104
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
0.2 5
10
15
20
25
cross
.
1 .0
1 .5
2 .0
2 .5
0
T(K)
T dt R qt s 0.3 A> Rn
Te at 50MHz
Te at 14GHz
Q in Kelvin
cross
Te at 0 Hz
Figure 4.6: (a) Temperature dependence of Cl at different magnetic fields, (b) Contour
plot of f! in temperature and field. Solid vertical black lines are the actual B and T
values where data were taken. Color indicates the magnitude of interpolated values
of Cl from the fitted data, (c) A phase diagram of all the quantities converted to
unit of Kelvin. The dashed vertical black lines in (b) and (c) mark Bcross- (d) An
alternative scenario for contour plot of Cl assuming Bcross is the QCP.
105
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
between a dissipative state with strong superconducting correlations on short length
scales and one with vanishing such correlations.
The view that B sm instead of Bcross is the true QCP in weakly disordered films
runs counter to much prevailing dogma in the field. If one is to insist that Bcross
controls the critical behavior, it requires at least two additional conditions to explain
our data, both of which we consider unlikely. First, one has to posit that the contours
of constant Q must have particular shapes like those shown in Fig. 4.6 (d) and what we
are seeing is a finite temperature effect. If, as discussed above, a true superconducting
state only exists for T ^ 0, B = 0 and T = 0, B ^ 0, the true Q = 0 contours are
the two red lines lying on the B = 0 and T = 0 axes. All the finite Q contours
should intercept the two axes at T > Tc, B = 0 or B > B^oss, T = 0. In our case,
this means that these contours would have to have exceedingly long tails extending
over Bcross• Although we cannot exclude this possibility, it is a challenge within
the current theoretical framework to explain why the contours would pick up this
particular shape. Secondly, if Bcross is the actual QCP, the AC dynamics must be
completely insensitive to the existence of the QCP as we have seen no evidence of
quantum critical signatures in our AC data extrapolating towards Bcross• This may
be possible, but only in a scenario where the QPT is a transition associated with pure
classical percolation [30,87].
106
CHAPTER 4. 2D FIELD TUNED SUPERCONDUCTOR-METAL QUANTUM
PHASE TRANSITION
4.2
Conclusion
To conclude, we find evidence for a possible scenario where a 2D QPT occurs at
a field
B am
between a superconductor and an anomalous metal with superconducting
correlations. The lowest temperature
Q
and
Tg
at the lowest frequency extrapolate
to zero from both sides of B sm. The lack of evidence for finite-frequency
Tg
surviving
Bcross shows that BcrOSS is a crossover above which superconducting fluctuations make
a vanishing contribution to both DC transport and AC measurements. A careful and
complete investigation of a more disordered film is needed to compare the effects of
different disorder levels, which is a future direction of our project.
107
Chapter 5
Summary
We have studied the dynamics of the superconducting transition in two dimen­
sions, of both thermal and quantum types. To carry out the experiments, a broadband
Corbino microwave spectrometer was developed. To the best of my knowledge, this
is the first time this type of spectrometer was applied to study 2D quantum phase
transition. We were able to push the temperature limit of the spectrometer of this
kind to 300 mK, which enable us to access a regime where Hu < k g T (1 GHz « 50
mK).
The InOx sample we reported data in this thesis features a broad transition in
sheet resistance for B = 0 transition. We fit the sheet resistance to the AslamazovLarkin form in 2D to determine Tco, the temperature where Cooper pairs start to
form. Below Tco, phase fluctuations dominate. By doing a true spectroscopy, we
obtain the dynamical information about the superconducting fluctuations and find
108
CHAPTER 5. SUMMARY
evidence for critical slowing down along the T axis. We also demonstrate that the
thermal phase transition is a transition of KTB type and can be described by a vortex
plasma physics.
We can extract T tk b as the temperature where the superfluid stiffness acquires
a frequency dependence. However, the superfluid stiffness at
Tktb
is about 30%
greater than the prediction of the universal jump of the KTB physics. It needs a
more systematic study on 2D superconductors with different transition temperatures
to determine whether this deviation is caused by small calibration errors or effects of
non-universal jump nature of the KTB transition.
The study of thermal fluctuations helps with our understanding of the fluctuations
across the quantum critical point. Our microwave data of the low-disorder InOx
sample look very much like a thermal transition, for example, the evolution of the
real and imaginary conductance at 300 mK in magnetic fields looks similar to the
transition in temperature at B = 0. One can compare Fig. 3.5 and Fig. 4.3 and easily
come to the same conclusion. We reveal that while the DC transport is consistent
with previous measurements of SIT with a crossing field BCTOSS which is the nominal
quantum critical point, we find no evidence of the phase stiffness past this field.
More anomalously we observe the characteristic fluctuation rate is “speeding up”
as we approach BcrOSS from below. Our data show that the true quantum critical
point might be at a field B am where both the low frequency phase stiffness and the
fluctuation rate extrapolate to zero. This field J3sm is far below jBc-oss and it seems
109
CHAPTER 5. SUMMARY
that the sample displays some intermediate metallic behavior above B am.
Compared with the microwave cavity data from R. Crane et al [25], where they
showed that the superfluid stiffness is finite above Bcross in a high-disorder limit, we
observed that the low-frequency superfluid stiffness vanishes well below B^-oss in the
low-disorder limit. Interesting questions to ask are that there has be to one disorder
level where superfluid stiffness vanishes at 5 CTOSS, what is the significance of this
disorder value? Is it possible this disorder value is the critical parameter? We might
be able to find answers to those questions once we take spectroscopic measurements
on InOx films with higher disorder values. This is planned for future measurements.
110
A ppendix A
Transmission line model
This chapter is dedicated for readers who want to know how to obtain impedance
and conductance from the reflection coefficients. A transmission line exhibits proper­
ties of capacitance, inductance, resistance, and conductance and can be schematically
represented by Fig. A.I. R is the distributed resistance of the conductors. The mag­
netic field produced by the currents contains magnetic energy as an inductor L. The
capacitance C describes the electric field between the conductors. G represents the
conductance of the dielectric material separating the inner and outer conductors.
•
nnrr\_A^_
Figure A.l: A schematic plot of the transmission line model.
Ill
APPENDIX A. TRANSMISSION LINE MODEL
In the transmission line model, it is more convenient to use voltages (V) and cur­
rents (I) instead of electric and magnetic fields to describe the propagating waves
since the characteristic impedance of a line is always the ratio of V and I, but some­
times not directlythe ratio of E and B fields. This modelbest suits transmission lines
carrying TEM waves.
V and I fora transmission line depend on the position and
time as V(z,t) and I(z,t). The telegrapher equations for the voltage and current [88]
are:
=
W
^
H KC + L G ) ^ + RGI{z,t)
The time dependence of the voltage and current are
V (z,t)
= V(z)eiut
I(z,t) =
/(z)ei<Jt
The telegrapher equations thus simplify to
= 0
(A.l)
= 0
(A.2)
where
7 = y /(R + iuL){G -I- iuC) = a + i/3.
112
APPENDIX A. TRANSMISSION LINE MODEL
The solutions to the telegrapher equations are of the form
V{z) =
V’+e-7* + V~e+,yz
I ( z ) = ~ e ~ lz - ^ - e +7Z = 7+e~7* + 7~e+7*
Zq
(A.3)
(A.4)
Zq
where ZQ = y/(R + iijjL)j{G + iuC) is the complex characteristic impedance of the
transmission line. V + and 7+ are the amplitudes for a wave propagating along the
positive z direction and V~ and 7~ are those for a wave propagating along the negative
z direction.
A .0.1
Im pedance m atching
In a homogeneous line, the characteristic impedance is independent of z and thus
is a constant for a TEM wave. If there is a discontinuity in the line, the impedance
changes and a reflection occurs. This discontinuity may come from a change in the
line dimensions and materials which has a different impedance from the original line.
Suppose a load terminates the line at z = 0. The characteristic impedance of the
line is Z0 and the impedance of the load is Z i. According to equations A.4, a voltage
wave propagating toward the load is expressed as F +e~"72, and the corresponding
current wave is I +erlz (Z0 = V +f I +). In general, only part of the propagating wave
power is reflected and the other part is absorbed. The reflected voltage and current
waves are V~e+,yz and I~e+lz with Z q = V~ j l ~ . V +, I +, V~ and I~ are all complex
amplitudes. At z = 0, the voltage and the current of the line and those of the load
113
APPENDIX A. TRANSMISSION LINE MODEL
must be equal:
= vL
+
i+- r
=
iL
Note that 7+ and I~ have opposite directions. Using Z L — Vl / I l , the equation
for current can be written as
Y1_Y=. = Yl
Zq
Zq
Zi
The reflection coefficient of the load is defined as S = V ~ / V +. Eliminating V ~ ,
we obtain the equation for refection from a load as
_ ZL -
Zq
Z l + Zq
For the geometry of the samples we use in our experiments, we have Z L =
[ln{b/a)/2Tx]Zs, where surface impedance Zs is defined as Zs = Er/H$. The surface
impedance of a bulk material is related to its conductivity as Z \f =
For a film
with thickness d, the effective impedance is Zs = Z f coth(/cd) where k = y/i^Qua.
In the thin film limit where the thickness is much smaller than the skin depth, one
obtains Z$
1/{ad).
In general, the confusion comes when we try to calculate the conductance from the
impedance. We need to know whether we should take the conjugate of the final results.
If we define E =
the conductivity is defined as <r = <Ti —zcr2 from the Drude
model. What if we define V (z , t ) = V(z)e~VJt for the transmission line model? The
114
APPENDIX A. TRANSMISSION LINE MODEL
voltage and current now can be written as V (z,t) = Vi(z)e~tu>t, I(z, t) = Ii(z)e~lu>t,
which satisfy the following equations:
«
- «
. >
- »
where
71
= \ / ( R —iojL)(G —iuiC) = a —i/3 =
7
*
Complex conjugate of equations A.2 are written as:
-
0
-
0
Then the solutions to the above equations are the conjugate of the solutions to equa­
tions A.2: Vi(z) = V*(z), h (z ) — I*(z). Therefore, the reflection coefficients under
this convention is S = r t =
■ The conductivity is defined as
a —
o\ + i a 2 from
the Drude model in this convention. In the thin film limit, Zs is still proportional to
1/a.
Therefore, if we take the time dependence to be elu)t, a = o\ — i a 2. If we take
the time dependence to be e~lu)t, what we calculate from the reflection is Z*L and
a
=
0
\+
i a 2.
And we obtain a correct sign of a 2 for both cases.
115
Appendix B
Experimental procedures
B .l
Measurements at zero field
As discussed in the main content, experiments for the three standards and samples
under study have to be carried out in the same manner repeatedly. Our cryostat is
a one shot He3 system. When the sorption pump (charcoal pump) is heated over
about 25 K, it releases enough He3 gas so that we can condense liquid He3 into the
He3 pot by keeping the temperature of 1 K pot below 2 K. After the temperature of
the He3 pot reaches below 2 K for over 20 minutes, we can start cooling down the
charcoal pump. For our system, once the temperature of the charcoal pump becomes
less than 10 K, it functions as a very good pump and pumps on the condensed liquid
He3 in the He3 pot. After charcoal pump reaches its base temperature (usually at
about 4.3K) for about half an hour, He3 pot would approach its base temperature ~
116
APPENDIX B. EXPERIMENTAL PROCEDURES
300 mK.
First warming up
Cooling down
Second warming up
£
£
I
&
£
.2
olE
0
100
200
300
Time (arbitrary unit)
400
Figure B.l: Examples of temperature profiles of the sample stage for the standardized
three zero field measurements. Each data point was recorded every minute for this
set of experiment. Therefore, the X axis is proportional to time. The total time span
of the X axis is about 450 mins.
So in general, for T > 2 K, the system is cooled by He4. From 2 K to the base
temperature, the system is cooled by He3. It is difficult to control the temperature
well from 1.5 K to 2 K and this was a challenge for us since we need to be able to
reproduce the temperature profile for each sample. We could not simply control the
temperature by controlling the heater on the He3 pot. Once the liquid He3 starts
evaporating, the whole condensed He3 liquid would be gone in just a second and we
would get a jump in the sample temperature. Nor could we control the temperature
through purely controlling the 1 K pot temperature cause we could not get down to
117
APPENDIX B. EXPERIMENTAL PROCEDURES
300 mK in that way.
The most applicable way to control the temperature of our He3 pot is to actually
control the temperature of the charcoal pump. If we slowly warm up the charcoal
from about 4.3 K (lowest it can reach) to 30 K, the He3 exchange gas is slowly warmed
up and it would gradually warm up the sample stage. We use a very slow warming
up procedure and one warming up scan usually takes a couple of hours to reach 5 to 6
K. But as one can see from Fig. B.l that the temperature profile is very reproducible
for two different warming up procedures with different liquid He4 levels in the dewar.
B .1.1
O perating sequence
It is very important for us to reproduce the cryogenic conditions in every temper­
ature sweep due to the sensitivity of the calibration procedure on cryogenic environ­
ment. The full procedure we followed is below:
Day One:
1. Pump the vacuum jacket of the dewar and also the vacuum jacket of the liquid
He4 transfer tube to high vacuum (HV) if it’s necessary. The turbo pump we use
is Pfeiffer vacuum turbo pump TPH 062 or similar. It’s highly recommended
to pump the vacuum jackets every a couple of months. We usually pump them
overnight (more than 12 hours) or longer for the vacuum jacket of the dewar
(at least 24 hours) since it has a huge surface area.
118
APPENDIX B. EXPERIMENTAL PROCEDURES
Day two:
1. Attach the samples into the sample stage. Use a caliper to fix the length of
the set screw. For a sample of thickness 380 microns, the length of the set
screw outside the sample stage is 4.5 mm. Adjust it accordingly if the sample
thickness is different so that the spring force is the same for every single sample
so that the references plane is the same. (Note the length might change after we
changed the Corbino probe in the summer of 2012.) Check the DC reading to
make sure that the Corbino probe and the sample are in good electric contact.
Also check the microwave spectra at room temperature.
2. Attach the IVC cylinder to the insert and seal the IVC with an indium wire.
3. Pump the IVC to HV using a turbo pump. We usually pump the cylinder
overnight (more than 12 hours).
Day three:
1. Leak check at room temperature.
2. Fill the IVC with N2 gas. We do not have a pressure gauge for the IVC. In
general the regulator for the N2 gas tank reads about 1 or 2 Psi. We usually
fill the IVC with two pump tubes of gas N2 (the pump tube is connected to the
IVC from the turbo pump and it’s about 2.5 meters long).
3. Back flow the 1 K pot and the charcoal pump with He4 gas since during this
step, the dewar is still cold in most cases.
119
APPENDIX B. EXPERIMENTAL PROCEDURES
4. Lift up the dewar. This is usually a two persons’ job. One person cranks up
the dewar and another person needs to make sure that the baffles of the insert
are not hitting the baffles in the dewar. The tolerance between them are very
small. Attach the dewar to the insert which is fixed onto a frame. We leave
the platform of the lift up to provide extra support to the dewar. Check the
pressure reading of the He3 tank. It usually reads 220 Psi at room temperature.
The back flow to the 1 K pot and the charcoal pump is on all the time during
this process.
5. Fill the dewar with liquid nitrogen. Once liquid nitrogen starts to accumulate
at the bottom of the dewar, stop the back flow to the 1 K pot and the charcoal
pump and start pumping them. Check the flow for both absorption lines for
the 1 K pot and the charcoal pump and make sure they are not clogged. Close
the flow meter of the charcoal pump.
6. Stop transferring once the level of liquid nitrogen reaches about 12” — 15”.
7. When 1 K pot reaches about 100 K, fill the 1 K pot with He4 gas.Turn off the
pumps for the 1 K pot and the charcoal pump.
8. Leave the system overnight to reach thermal equilibrium at 77 K
Day four:
1. Pumping out the gas nitrogen in the IVC.
120
APPENDIX B. EXPERIMENTAL PROCEDURES
2. Attach the IVC to a leak check. Push out the liquid nitrogen in the dewar by
overpressurizing the dewar with He4 gas while leak checking. The regulator of
the He4 tank reads about 1 or 2 Psi. But we never use pressure higher than 3
Psi. The whole IVC is immersed in He4 gas environment during this procedure.
3. When all the liquid nitrogen in the dewar is pushed out, flush the 1 K pot and
the charcoal pump with He4 gas for about 10 seconds. Overpressurize the dewar
again to get rid of any possible residue of liquid nitrogen.
4. Transfer liquid helium. Once liquid helium starts condensing at the bottom of
the dewar (when the He level meter yields positive reading. It usually takes
about 5 — 10 mins when the pressure in the liquid He tank reads about 1.5 Psi.
But the time is longer after we installed the magnet.) start the pumps for the
1 K pot and the charcoal pump. Open the flow meter for the charcoal pump
and the needle valve for the 1 K pot. Check the readings for both to make sure
they are not clogged.
5. Stop transferring when the helium level monitor reads about 55 — 60 cm for
the initial cool down.
6. Start cooling the sample stage to base temperature. Set the set point to 32
K and turn on the heater in medium range for the charcoal pump. Open the
needle valve for the 1 K pot between 1/2 and 2/3 turn open. Since we do not
have exchange gas in the IVC, it takes about 2 — 3 hours for the sample stage
121
APPENDIX B. EXPERIMENTAL PROCEDURES
to reach 2 K from 77 K. We maintain the temperature of the 1 K pot below 3
K during the process by either opening or closing some amount of the needle
valve.
7. Once the temperature of the sample stage reaches 2 K, we set the set point
for the charcoal to 25 K and keep the 1 K pot below 2 K. After condensing
He3 for about half an hour, turn off the heater and open the flow meter for the
charcoal pump to cool down the charcoal pump below 5 K. Close the needle
valve of the 1 K pot. After about half an hour, the sample stage reaches its
base temperature, usually at about 300 mK.
8. Close the flow meter of the charcoal pump and turn off the pumps. Leave the
system overnight to reach thermal equilibrium at low temperatures.
Day Five:
1. Come in the early morning. Sample stage usually stays below 1 K.
2. Transfer up to 56 cm liquid helium in the dewar.
3. Set the set point of the charcoal pump to 25 K and control the 1 K pot below
2 K.
4. After condensing He3 for about 25 mins, start cooling down the He3 pot to the
base temperature by turning off the heater and opening the flow meter of the
charcoal pump. Close the needle valve of the 1 K pot.
122
APPENDIX B. EXPERIMENTAL PROCEDURES
5. We always start the first warming up when the helium level in the dewar hits
55 cm. Start the labview program to record the temperature, resistance and
microwave data. Close the flow meter for the charcoal pump. Set the set point
of the charcoal pump to ramp from 3.5 K to 35 K at a ramping rate 0.1 K/min.
PID values for the heater are P = 30, I = 20, D = 0. Turn on the heater for
the charcoal pump and the heater setting is medium. When the set point gets
6 K, we turn off both rough pumps to the 1 K pot and the charcoal pump and
open the needle valve for the 1 K pot 1 turn open. To make sure the helium in
the dewar has almost the same boil off rate, the helium level monitor measures
the liquid helium level every 15 mins.
6. Stop the program for the first warming up measurement when the helium level
becomes 50.5 cm. Usually the sample stage is at 4 K. First warming up in
general takes a little more than 4 hours.
7. Prepare for the cooling down procedure by setting the set point of the charcoal
pump to be 25 K. Turn on the pumps for the 1 K pot and the charcoal pump.
When helium level gets to 50 cm, warm up the sample stage to 20 K. PID values
for the heater are P = 30,1 = 100, D = 0. 10 mins after the sample stage reaches
20 K (to reach temperature stability), set the set point of the sample stage to
ramp from 20 K to 2 K at a ramping rate 0.05 K/min. Heater range is set to
be 2.5 Watt. Start the program to record data for the cooling down procedure.
123
APPENDIX B. EXPERIMENTAL PROCEDURES
We keep charcoal pump at 25 K and 1 K pot between 2.35 K — 2.55 K (needle
valve about 1/2 to 2/3 turn open) during the whole cooling down scan.
8. Stop the program for the cooling down measurement when the sample stage
reaches below 2 K. Turn off-the heater for the sample stage. This measurement
takes about 6 hours. Helium level usually reads at about 27 —27.8 cm. Prepare
for the second warming up measurement.
9. The charcoal pump still stays at 25 K and the 1 K pot stays below 2 K. Condense
He3 for about 30 mins (count from the time when the sample stage gets 2 K
from the previous cooling down procedure). Turn off the heater of the charcoal
pump and close the needle valve for 1 K pot. Cool down to base temperature
by opening the flow meter to cool the charcoal pump. Start the second warming
up measurement 52 mins after we start cooling down to the base temperature.
Start the program to record data for the second warming up procedure. Close
the flow meter and set the set point of the charcoal pump to ramp from 3.5 K
to 35 K at a ramping rate 0.1 K/min. PID values for the heater are P = 30, I
= 20, D = 0. Turn on the heater for the charcoal pump and the heater setting
is medium. When the set point gets 6 K, we turn off the pumps and open the
needle valve for the 1 K pot 1 turn open. The helium level monitor measures
the liquid helium level every 15 mins. The helium level usually is between 24
— 25 cm. Leave the system to slowly warm up overnight.
124
APPENDIX B. EXPERIMENTAL PROCEDURES
Day Six:
1. Stop the program when the sample stage reaches the desirable temperature
(usually above 10 K).
2. remove the cryostat from the insert. Fill the IVC with gas nitrogen. Wait
until temperature of the insert is close to room temperature. Vent the IVC
and remove the cylinder. Remove the samples. Cover the dewar to prevent ice
forming inside.
One complete measurement cycle on one sample can be performed within a week.
As one can see, we have more controls in the second warming up. Also, we do not need
to transfer liquid helium for that measurements. As it turns out, the temperature of
the He3 pot changes a little bit when we step on the frame where the insert is attached
to. That is why we normally analyze data from the second warming up. However, as
demonstrated in the main text, the difference from the three different temperature
scans is minor. Since our insert is sitting in HV and has no direct contact with the
liquid helium bath, the small deviation in helium level for the second warming up for
different samples has insignificant effects.
B.2
M easurements at finite fields
We purchased the magnet through the Janis company and successfully installed
it in the He3 cryostat. The dimension for the whole cryostat and the magnet can
125
APPENDIX B. EXPERIMENTAL PROCEDURES
be found in Fig. D.l. The whole magnet has to be immersed in liquid helium to
avoid possible magnetic quench which happened once when we were charging the
magnet for a measurement on the low-disorder InO* film. We transfer enough liquid
helium in the dewar to make sure the magnet is still covered by liquid helium after
the temperature scan. All the helium transfers have to be done when the magnet is
off so that there would not be any local heating effects to cause a possible quench.
The IVC cylinder was made from copper before we installed the magnet. We
changed it to a stainless steel one for measurements in magnetic fields since the eddy
currents generated by a changing magnetic field in the copper would be too large.
The sample stage is also made of copper. Therefore, when we scan field at fixed
temperatures, to maintain temperature stability, the sweep rate cannot be too fast.
The sweep rate in current for magneto-resistance measurements is 0.0079 A/S from
0 to 5 Amperes, 0.0158 A/s from 5 to 45 Amperes and then 0.0079 A/S again from
45 to 47.62 Amperes. The scan time is about an hour from 0 to 8 Tesla and the
temperature is pretty stable during the whole scan.
B .2.1
O perating sequence
For measurements at finite magnetic fields, since the whole magnet has to be
covered by liquid helium, we cannot perform the three measurement scans described in
the previous section at one field. However, we know that as long as we can reproduce
the temperature and control the time the system staying at the low temperature
126
APPENDIX B. EXPERIMENTAL PROCEDURES
environment, errors due to different runs are negligible. A measurement cycle at one
field takes about 12 hours. We generally take data at two magnetic fields per day.
For the data we demonstrate in Chapter 4, it takes about a week and half to finish
the measurements for one sample.
Day 1 ~ Day 4: the same as the procedures for zero field measurements.
Day Five:
1. Refill the liquid helium level to about 50 cm.
Let the system stay at low
temperature for extra a number of hours. After the initial filling the cryostat
has to sit in cryogenic environment for at least 24 hours.
2. Prepare for measurements in magnetic fields after the system stays in the liquid
helium environment for at least 24 hours.
3. Transfer liquid helium up to 55.2 ~ 56 cm. The amount of transferred liquid
helium depends on the strength of the desired magnet field. The persistent
heater stays on longer when we are charging the magnet to higher magnetic
fields thus boils off more liquid helium. We normally transfer to 55.2 cm liquid
helium for measurements at B = 0.1 Tesla and 56.2 cm for B = 8 Tesla. For
B = 0 Tesla, we transfer up to 56.3 cm liquid helium since we need to oscillate
the field to zero to get rid of any possible trapped flux. By doing so, we start
our data acquisition typically at a helium level between 53 — 53.6 cm.
4. 20 mins after we finish transferring liquid helium, turn on the pumps and keep
127
APPENDIX B. EXPERIMENTAL PROCEDURES
the charcoal pump at 25 K and the 1 K pot below 2 K (needle valve about half
turn open). Condense He3 for 30 mins (count from the time when the sample
stage gets 2 K). Open the needle valve a little more to warm up the 1 K pot to
2.2 K (make sure we have enough liquid He4 in the 1 K pot when we cool the
system to the base temperature) and then close the needle valve. Turn off the
heater and open the flow meter for the charcoal pump slightly over range. Cool
the system to the base temperature of the cryostat. When the temperature
sensor of the sample stage reads 400 mK, turn on the persistent heater and
ramp up the field to the desirable value. Typical ramping rate is 0.02 A/S from
0 to 40 Amperes, 0.0158 A/S from 40 to 45 Amperes and then 0.0079 A/S from
45 to 47.62 Amperes. Turn off the persistent heater and put the magnet in
persistent mode after the magnet reaches the set point.
5. 2.5 hours after we finish transferring liquid helium, start the program to record
all the data. Close the flow meter and set the set point of the charcoal pump
to ramp from 3.5 K to 35 k at a ramping rate 0.1 K/min. PID values for the
heater are P = 30, I = 20, D = 0. Turn on the heater for the charcoal pump
and the heater setting is medium. When the set point gets 6 K, we turn off the
pumps and open the needle valve for 1 K pot 1 turn open.
6. Sample reaches about 6 K roughly after 6 hours. Stop the program. The helium
level normally is at 40 cm. Ramp down the magnetic field. Proceed to the next
128
APPENDIX B. EXPERIMENTAL PROCEDURES
measurement at another magnetic field or prepare to remove the cryostat from
the insert.
One thing I need to point out is that the magnetic field has very strong effects
on the silicon diode temperature sensors. The silicon diode temperature sensor at
the charcoal pump reads 3.5 K at 8 Tesla, while it reads 4.35 K at 0 Tesla. So the
warming up procedure is in general shorter for measurements at high fields. However,
this is the same for every sample since we calibrate the system at every single field. To
avoid a sudden change in the output of the heater of the charcoal pump, we typically
set the set point of the charcoal pump to be 3.5 K, turn on the heater and then start
ramping the set point. This order, however, does not m atter for measurements at
low fields since 3.5 K is normally much lower than the reading of the charcoal pump’s
base temperature.
129
A ppendix C
Reflection coefficients for zero-field
measurements
As we discussed in length in Chapter 2, the difference for the first warming up,
cooling down and second warming up scans for zero field measurements are not that
significant. In this appendix, I would like to present the raw S™ for a superconducting
Nb film (Fig. C.l), a 20 nm NiCr film on Si (Fig. C.2), and a glass substrate
(Fig. C.3). We can clearly see the Nb film becomes a superconductor at about 6
K in Fig. C.l. For the three graphs, one can again see the effects from the NbTi
superconducting cable at around 8 — 9 K. The |5 n | for open and short standards
have almost the same magnitude. We found that the differences are not significant
at all for the three scans.
130
APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD
MEASUREMENTS
Nb film first warming up
Nb film cooling down
Nb film second warming up
0.42
0.40
0.38
0.36
-0.15
- 0.20
-0.25
0
5
10
Temperature (K)
15
20
Figure C.l: Magnitude and phase of
of one Nb film as a function of temperature
at 7 GHz. Different runs are indicated by the color legend.
131
APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD
MEASUREMENTS
0.106
NiCr film first warming up
NiCr film cooling down
NiCr film second warming up
0.104
0.102
CO
0.100
0.098
0.40
0.35
0.30
0
5
10
Temperature (K)
15
20
Figure C.2: Magnitude and phase of S'™ of one NiCr film as a function of temperature
at 7 GHz. Different runs are indicated by the color legend.
132
APPENDIX C. REFLECTION COEFFICIENTS FOR ZERO-FIELD
MEASUREMENTS
0.42
G la ss first w arm ing up
G la ss cooling dow n
G la ss s e c o n d w arm ing up
0.40
CO
0.38
2.60
2.55
2.50
0
5
10
Temperature (K)
15
20
Figure C.3: Magnitude and phase of S'™ of a glass substrate as a function of temper­
ature at 7 GHz. Different runs are indicated by the color legend.
133
A ppendix D
M agnetic field distributions
The center of the magnetic field is carefully adjusted to be the exact location
of the sample. Knowing the field distribution in the system is essential to estimate
the effects of the magnetic field on the superconducting cable, all the microwave
connections and also on the temperature sensors. We also notice that readings for
both RuO* temperature sensors (one located on top of the He3 pot and another one
located at the back of the sample stage close to the sample) also were suppressed
by the increasing magnetic field. Knowing the field strength at the location of those
temperature sensors enables us to correct the temperature reading under applied
magnetic fields.
Data of magnetic field dependence on the axial axis (Z direction) and the radius
axis (R direction) from the center of the magnet were provided by Janis, the manu­
facturer company of our cryostat. We plot them in a fashion of contour plots. Fig.
134
APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS
D.2 is the contour plot of the B z in axial and radial directions. Fig. D.3 displays the
contour plot of the B r . B z is much stronger than B R at locations that are close to
the field center so that the magnitude of the total field is about the same as B z in
that region as showed in Fig. D.4.
We roughly estimated the distance of the two RuOx sensors from the magnet
center. When B = 8 Tesla at the center, the field at the sensor closer to the sample
is about 8 Tesla, and the field at the sensor on top of the He3 pot is about 3.6 Tesla.
The temperature therefore can be corrected by a universal correction of temperature
in a magnetic field [89] for RU600 series temperature sensors as:
T(actual) = (1 + 0.009B) * T (B ).
135
(D.l)
APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS
mi
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Figure D.l: The design of our cryostat with the magnet from Janis company. Cour­
tesy of Janis company.
136
APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS
30
0.2
25
0.4
0.6
Axial (cm)
20
0.6
0.8
1.6
15
2.4
3.4
1 = 3 .6
10
4.4
3.8
5.4
E 6 .4
6.6
2.8
3.2
52
4.4
6.8
6.6
7.4
7.2
5
7.6
2.2
3.6
7.6
7.8
8.2
0
3
4
Radius (cm)
Figure D.2: Contour plot of B z .
137
6.4
APPENDIX D. MAGNETIC FIELD DISTRIBUTIONS
Axial (cm)
30
Radius (cm)
Figure D.3: Contour plot of B R.
138
APPENDIX d . m a g n e t i c f i e l d d i s t r i b u t i o n s
Axial (cm)
M a g n itu d e
= ®±=66 = 6.8
n
Radius (cm)
Figure D A Contour plot of the magnitude of the magnetic field.
139
A ppendix E
Microwave conductance of InOx at
finite fields
In this appendix, I present almost all the data we took for the measurements at
finite fields of the low-disorder InOx film. I show the complex conductance and the
phase stiffness at B = 0, 1, 2, 3.5, 4, 5 and 6 Tesla. One can notice the suppression
of the complex conductance with an increasing magnetic field at the lowest measured
temperatures. The sharp universal jump feature in stiffness shown in Fig. E .l (d)
also gets smeared for higher and higher magnetic fields.
140
APPENDIX E. MICROWAVE CONDUCTANCE OF INOx AT FINITE FIELDS
o
O
1 0 '1
E
.c
O 10’2
(D
10-3
10-4
8
12
<ii>/2jt (G H z)
co/271
4
w/271
(G H z)
8 12
(G H z )
Figure E.l: Complex conductance and phase stiffness at B = 0 Tesla. Frequency
dependence of (a) real and (b) imaginary conductance are shown in the ranges u;/27r
= 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a).
Temperature dependence of (c) real conductance and (d) phase stiffness respectively
in the ranges T = 0.3 - 4 K. The color legend for frequency for (c) and (d) is shown
in (d).
141
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
o)/2rc(GHz)
to/2n(G H z)
Figure E.2: Complex conductance and phase stiffness at B = 1 Tesla. Frequency
dependence of (a) real and (b) imaginary conductance are shown in the ranges o;/27t
= 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in (a).
Temperature dependence of (c) real conductance and (d) phase stiffness respectively
in the ranges T = 0.3 - 4 K. The color legend for frequency for (c) and (d) is shown
in (d).
142
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
1.0
2 .0
3.0
' C2
_c
T (K)
S
■
10-4 PP
w
10 X
1 0 ‘3
O
1
*
2
0.5
4
1.0
6 8 10 12 14 16
<u/2ir (GHz)
1.5
2.0
T (K)
2.5
1 1 11 ll
6 8
co/2jt (GHz)
3.0
Figure E.3: Complex conductance and phase stiffness at B = 2 Tesla. Frequency
dependence of (a) real and (b) imaginary conductance are shown in the ranges a;/27r
= 0.05 - 16 GHz. The color legend for temperature for (a) and (b) is shown in
(a). Real conductance and phase stiffness as a function of temperature are plotted
respectively in (c) and (d). The color legend for frequency for (c) and (d) is shown
in (d).
143
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
0.51.0 1.5 2.0 2.5 3.0
T( K)
2
4
6 8 10 12 14 16
g>/2ji (GHz)
8
0.1
2
4 6 8
2
4 6 8
4
8
1
C0/27C(GHz)
0
10
12
co/2rt (GHz)
^10
0.5
1.0
1.5
2.0
2.5
3.0
T (K)
Figure E.4: Complex conductance and phase stiffness at B = 3.5 Tesla. Frequency
dependence of (a) real and (b) imaginary conductance are shown in the ranges w/27r =
0.08 -1 6 GHz. The color legend for temperature for (a) and (b) is shown in (a). Real
conductance and phase stiffness as a function of temperature are plotted respectively
in (c) and (d). The color legend for frequency for (c) and (d) is shown in (d).
144
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
_i
0.5
i
i
l
t
1.0
1.5
2.0
2.5
-1
3.0
T (K)
L_i
i ii ii. in 1111 11 mi iin ii 1
1.0
2.0
3.0
4.0
T (K)
Figure E.5: Complex conductance and phase stiffness at B = 4 Tesla. Frequency
dependence of (a) real (G i) and (b) imaginary (G2) conductance respectively in the
ranges u)/2n = 0.08 - 16 GHz. The color legend for temperature for (a) and (b) is
shown in (a). Real conductance and phase stiffness as a function of temperature are
plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is
shown in (c).
145
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
1.0
2.0
3.0
T(K)
2
4
6
8
10 12 14 16
4
6 8
2
4
6
1
(o/2n (GHz)
<o/2 k (G H z)
4
2
8
12
co/2n (GHz)
0.5
1.0
1.5
2.0
2.5
3.0
T (K)
Figure E.6: Complex conductance and phase stiffness at B = 5 Tesla. Frequency
dependence of (a) real (Gi) and (b) imaginary (G 2 ) conductance respectively in the
ranges uj/ 2 tx = 0.12 - 16 GHz. The color legend for temperature for (a) and (b) is
shown in (a). Real conductance and phase stiffness as a function of temperature are
plotted respectively in (c) and (d). The color legend for frequency for (c) and (d) is
shown in (c).
146
APPENDIX E. MICROWAVE CONDUCTANCE OF INO* AT FINITE FIELDS
10'3
-
0.5
1.0
1.5
2.0
2.5
3.0
T (K)
0.5
1.0
1.5
2.0
2.5
3.0
T(K)
Figure E.7: Complex conductance and phase stiffness at B = 6 Tesla. Frequency
dependence of (a) real (Gi) and (b) imaginary (G2) conductance respectively in the
ranges w/27r = 0.1 - 16 GHz. The color legend for temperature for (a) and (b)
is shown in (a). Temperature dependence of (c) real conductance and (d) phase
stiffness respectively in the ranges T = 0.3 - 3 K. The color legend for frequency for
(c) and (d) is shown in (c).
147
Appendix F
AC conductance of CVD grown
graphene
Graphene is a material consisting of a single atomic carbon layer arranged in a
2D honeycomb lattice, discovered in 1969 [90] and studied extensively since then [91]
by the surface science community. It has attracted wide-spread interest for both its
novel electronic properties and Dirac band dispersion as well as its broad application
potential [91-98]. Due to its high mobility, it also has been proposed to show great
promise for high speed switching in microwave and terahertz devices [99-101] and
terahertz plasmon amplification [102].
148
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
F .l
Electrodynamics of single layer
graphene
Recently it has been demonstrated that large-area monolayer graphene films can
be grown by chemical vapor deposition (CVD) on copper foils [103], following the
precipitation-based growth of somewhat non-uniform few-layer graphene films on Ni
foils [104,105]. This method [106-109] allows the growth of large scale graphene
films that can be transferred to various substrates, which will be essential for any
practical device applications. The availability of large area uniform graphene also
allows access to these materials from a greater variety of experimental techniques
including studies of their long wavelength electromagnetic response. Their complex
microwave and terahertz frequency dependent response are of particular interest and
their understanding is crucial in order to use graphene for fast electronic devices.
However, it has traditionally been difficult to get significant broadband spectral in­
formation in these frequency ranges, particularly in the microwave regime. Microwave
experimental techniques are typically very narrow band and may at best allow the
characterization of materials at only a few discrete frequencies.
In this study, we make use of our newly-developed microwave “Corbino” spec­
trometer to measure the broadband microwave response in the frequency range from
100 MHz to 16 GHz of CVD grown graphene at temperatures down to 330 mK.
This technique allows one to gain broadband spectral information in the microwave
149
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
regime. Microwave techniques are typically very narrow band. We present data for
both the sheet impedance and complex conductance. The measurement of the in­
trinsic impedance of a single atomic layer film on an insulating substrate presents a
number of experimental challenges in the microwave regime. As a non-resonant tech­
nique, the Corbino spectrometer requires an intricate calibration procedure. More
importantly, any attem pt to measure the intrinsic impedance of graphene will be af­
fected by capacitive coupling to its dielectric substrate. At microwave frequencies, the
impedance associated with the substrate can be a substantial fraction of the graphene
impedance. We detail the manner in which substrate effects may be calibrated for,
as well as a number of other difficulties peculiar to this frequency range that must be
overcome. We also compare our data to that we measure at higher frequencies using
time-domain terahertz spectroscopy. Both measurements techniques are capable of
measuring the complex optical response functions as a function of temperature and
frequency, without resorting to Kramers-Kronig transforms.
CVD grown Graphene was prepared by Yufeng at UT Austin. The samples were
prepared by methane and hydrogen at pressures of 1.5 mbar over a 25 /xm thick Cu foil.
The graphene films are coated with PMMA and then the Cu foils are dissolved in an
aqueous solution of FeCl3 . The graphene is rinsed several times with de-ionized water
and can then be scooped out of solution onto a 380 jum thick clean high resistivity
Si substrate. High purity Si was used as it has a purely dielectric contribution to
the impedance of the graphene-silicon multilayer. The sample is allowed to dry and
150
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
300
^
250
I
200
2D
150
‘<5
jj
c
-
100
50
0
1500
2000
2500
3000
Raman shift (cm'1)
2400
— R d before an n ealing
— Rc a fter an n ealing for 24 ho u rs
^
(/>
E
.c
2200
+ R n after an n ealing for 24 h o u rs a t 260M Hz
O 2000
□
CU
1800
0
50
100
150
200
250
300
T em perature (K)
Figure F.l: (a) Raman spectra of graphene on Si. The Raman signal from a bare
Si substrate has been subtracted. The Raman spectra were taken by Yufeng at UT
Austin, (b) Resistance per square Ro for one of the CVD grown graphene samples
as a function of temperature. The black crosses are the microwave data at 260 MHz.
After the same sample was annealed at about 60 C in vacuum for about 24 hours,
the temperature dependence of its resistance shifted as shown by the red curve.
151
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
adhere to the Si and then the PMMA is removed by acetone. The resulting films are
verified to be of high-quality, predominantly single layer graphene from the intensities
and positions of the G- and 2D-band peaks in their 532 nm Raman spectra [110] (Fig.
F .l (a)).
To isolate the impedance of the graphene layer, we use Eq. 2.3 with Z§ub from our
previous independent study of thin superconducting films on identical Si substrates
[51]. In that study, we used amorphous metal films with scattering rates (« 100
THz) so high that the intrinsic AC impedance of the film itself in the normal state
was purely real and could be deduced from the DC resistance exactly. Thus, Z§ub
can be calculated by comparing the measured AC impedance of the metal film with
its known value. Knowing Z fu6 one can isolate the intrinsic broadband impedance of
the graphene layer.
In addition to measurements at microwave frequencies, the Corbino spectrometer
system can measure the two contact DC resistance simultaneously using a lock-in
amplifier and a bias tee. Multiplying the measured resistance between the inner and
outer conductors of the coaxial cable by the geometric factor g, we obtain resistance
per square. As shown in Fig. F .l (b), the resistance per square as a function of tem­
perature is approximately temperature independent ( « 3% over the range) with only
an upturn below 30K as the principal distinguishing feature. The sample properties
are changed only slightly by annealing in vacuum for about 24 hours at 60 C degree.
As shown in Fig. F.l(b) the resistance per square increased by 5 % with almost the
152
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
|
4000
8
2000
JZ
T = 296 K
Without substrate correction
With substrate correction
8
c
(0
■8
g
-2000
</)
E
-C
O
3000
2000
©
O
c
(0
1000
S
a.
E
20
15
a
O
3
5K
19 K
330 mK
15 K
296 K
10
.
0
2
4
6
8
10
12
14
16
Frequency (GHz)
Figure F.2: (a) Calibrated impedance with (black) or without (red) correction for
the substrate contribution at room temperature. Corrected (b) impedance and (c)
conductance (normalized by G q = ire2/2h) as a function of frequency in the range of
100 MHz to 16 GHz at different temperatures. Different colors in both panels indicate
different temperatures. In all the panels, solid lines are the real parts while dashed
lines are for the imaginary parts.
153
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
same temperature dependence. We ascribe the change in the overall scale due to a
change in carrier density by driving off absorbed gases.
In Fig. F.2, we present the results of our broadband microwave measurements on
one particular graphene sample from 100 MHz to 16 GHz at temperatures down to 330
mK. The small oscillations are the residual effects of standing wave resonances in the
transmission line that have been imperfectly removed by the calibration procedure. In
Fig. F.2(a), we compare the effective impedance at the sample surface calculated from
Eq. 2.2 with the impedance of the sample after the substrate correction described
above. One can see that the correction becomes significant at higher frequency where
the effective capacitance of the dielectric substrate plays a larger role. After correction
the impedance becomes primarily real and frequency independent as expected for a
conductor with a scattering rate larger than the measurement range. One can see
that it is essential to perform such a correction to quantify the impedance correctly.
In Fig. F.2 (b), we plot both real (Zi) and imaginary (Z2) sample impedance
corrected for the substrate contribution as a function of frequency at 5 different
temperatures. We can see that the frequency dependence of impedance at the base
temperature of 330 mK and room temperature are almost the same. The vertical dif­
ference in those temperatures matches with the difference of measured DC resistivity.
As clearly seen from Fig. F.2 (b), the real and imaginary parts of impedance have lit­
tle dependence on frequency down to low temperatures. This can also be seen in Fig.
F .l (b), where the resistance at the low frequency of 260 MHz at 4 different tempera-
154
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
tures is also plotted. These data follow the DC values closely indicating a consistency
of AC and DC measurements. Also, the real and imaginary parts of conductance have
little dependence on temperature which means that the Drude response of electrons
in graphene does not change a lot over this wide range of temperatures (from room
temperature to 330 mK). It also indicates that the scattering rate r - the average
time between scattering events - bears little dependence in temperature.
0.5
0.4
T=15 K
0.3
^
0
0.2
-
0.1
8x10
Frequency (Hz)
Figure F.3: Ratio (red) of real and imaginary parts of conductance as a function of
frequency in the range from 700 MHz to 9 GHz at 15 K before annealing. The black
curve is a linear fit with zero y axis intercept.
In Fig. F.2 (c), we plot the complex conductance obtained by using the inverse
of the data Fig. F.2 (b). Since in the thin film limit, the complex conductance is
the reciprocal of complex impedance, it also has almost no dependence on frequency.
Here we have ratioed this data to the quantum of conductance
155
Gq
= ire2/ 2 h
—
1/
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
16433 Ohms-1 expected for a graphene sample with its chemical potential tuned to
the Dirac point. Its large dimensionless scale shows that the carrier density for this
sample is high with the chemical potential far from the Dirac point.
Although the imaginary part of the conductivity is very small in Fig. 2, it is
not zero, which gives a measurable r from the data. Within the Drude model, the
complex conductivity from a charge responding to a time varying external oscillating
electromagnetic field with frequency u> is o(u) —
where n is the electron
density. Inspection of this equation shows that the ratio of the imaginary to real parts
of 0 2 / ( 7 1 gives w r a s a function of c j such that
r
can be determined from its slope. In
Fig. F.3, we plot this ratio for the conductances (G = od) as a function of frequency
at 15 K. G 2 /G 1 at other temperatures give similar results as both G2 and G\ have
little dependence on temperature. The slope of G2 /G 1 gives us an estimate of the
relaxation time, which is about 25.7 ps. A rough estimation of scattering rate is then
r = l / r = 38.9 GHz. This is quite close to the value for the Drude scattering rate
of 36.4 GHz obtained independently through fitting our data using a Drude-Lorentz
model [84]. Using this extracted scattering rate, we can estimate the mean free path
I = Vp * t = 28.3 fan with Fermi velocity vp = 1.1 * 106m /s [111]. However, we
should note that this small value of the scattering rate is interesting as it is at odds
with that inferred from previous studies using higher frequency time-domain terahertz
spectroscopy [112] or far-infrared reflectivity measurements [113].
The small value of the scattering rate is also different from that inferred from our
156
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
6K
7K
8K
9K
10 K
15 K
20K
25K
90K
100K
150K
200K
O
o
0
-10
0.2
0.6
0.4
0.8
1.0
Frequency (THz)
Figure F.4: Real (solid) and imaginary (dashed) parts of normalized conductance as a
function of frequency in the range from 150 GHz to 1.0 THz at different temperatures
for a similar graphene sample.
own time-domain terahertz spectroscopy (TDTS) measurements on another similarly
prepared sample. In TDTS an ultrafast laser pulse excites a semiconductor switch,
which generates an almost single cycle pulse with frequencies in the terahertz range.
The transmitted terahertz pulse’s electric field is mapped out as a function of time.
The ratio of the Fourier transform of the transmission through the sample to that
of a reference (usually the substrate on which the sample is deposited) gives the
complex transmission function of the film under study. This can be inverted using
standard formulas in the ‘thin film approximation’ T(u) = Y+n+z^ilJjd6^'' t° Se^ the
complex conductivity. Here
is the phase accumulated from the small difference in
thickness between the sample and reference substrates and n is the substrate index
157
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
of refraction. We have measured in terahertz frequency range from 150 GHz to 1.0
THz at temperatures down to 6K with flowing He4 gas. It is possible that the He4
gas environment for graphene sample in TDTS might make a slight difference for its
scattering rate compared with the Corbino system where the sample is sealed in high
vacuum.
In Fig. F.4, we show terahertz complex conductance data for a sample from
a different batch. This sample has a higher carrier concentration as evidenced by
its larger conductance. The sample also shows little dependence on frequency of
the real part of the conductivity and a small imaginary conductivity in this range.
As the conductance is expected to drop off dramatically around the frequency of the
scattering rate within the Drude model, this sample has a scattering rate greater than
1.0 THz. This is consistent with previous work [112,113] on CVD grown graphene
and on few-layer epitaxial graphene [114].
F.2
Conclusion
The two different scattering rates found in our microwave and TDTS measure­
ments show either the limitations of the Drude model for describing the fine details
of electron transport in graphene at low frequencies, or alternatively the inherent
difficulties of extracting out the precise complex impedance of graphene. In the first
case it may be that the details of scattering Dirac electrons give a conductivity line-
158
APPENDIX F. AC CONDUCTANCE OF CVD GROWN GRAPHENE
shape that is not precisely Lorentzian. Then the slope of G 2 /G 1 cannot be taken as
a measure of r. In the second case the scattering rate may be underestimated by
imprecisely removing the effects of the substrate contribution. It may be that this
overestimates the size of the imaginary contribution to the impedance and thereby
giving a larger contribution to G 2 . Future work will attem pt to measure changes
in the impedance by back gating the sample. It is likely that it will be possible to
measure changes in the impedance very precisely as the bias is swept from positive
to negative. Moreover, by going to the high bias regime/large conductivity regime
we should be insensitive to influence of the substrate impedance and will be able to
isolate the graphene conductance more precisely.
159
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Vita
Wei Liu was born in Le’an, a small town in Jiangxi
Province in China. After high school, she moved to
Beijing for her undergraduate and master degrees. She
attended Beijing Normal University, receiving a B.S
degree in Physics.
She then enrolled in the master
program in physics at Peking university, working on
Monte Carlo simulation of lattice QCD and tried to
understand the properties of elementary particles. She
moved across the world to Baltimore to enroll in the Ph.D. program in physics at
Johns Hopkins University. Her research has mostly focused on applying a homebuilt microwave spectrometer to study the thermal phase fluctuations and the twodimensional superconductor-insulator quantum phase transition in disordered InO*
films.
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