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Microwave surface impedance measurements on heavy fermion superconductor uranium beryllium-13 and low temperature scanning tunneling microscopy

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NORTHWESTERN UNIVERSITY
MICROWAVE SURFACE IMPEDANCE MEASUREMENTS ON
HEAVY FERMION SUPERCONDUCTOR U Bel3
AND
LOW TEMPERATURE SCANNING TUNNELING MICROSCOPY
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHELLOSOPHY
field o f Physics
BY
CHRONG-CHU TSAI
EVANSTON, ILLINOIS
DECEMBER 2000
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UMI Number: 9994754
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Abstract
This thesis is composed o f two parts. The first part concerns microwave surface
impedance measurements in the heavy fermion superconductor UBei3.
The
manifestations of order parameter collective modes in this compound were explored
using the microwave cavity perturbation technique. We present results o f the observation
of an absorption peak whose frequency- and temperature-dependence scales with the
BCS gap function, A(T). This was interpreted as a resonant absorption into a collective
oscillation. This observation provides strong evidence of an unconventional Cooperpairing mechanism in this superconductor.
The second part of this work gives a detailed description of the development of
a low temperature scanning tunneling microscope. Although many groups using various
approaches have worked on this subject, the design of a high performance low
temperature scanning tunneling microscope (LTSTM) is still very challenging.
We
present here the design o f our LTSTM, which has a special configuration and can be
mounted on one o f our dilution refrigerator cryostats. The system also employs a novel
and effective vibration-isolating mechanism. We also present images obtained with this
instrument.
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ACK NO W LEDG EM ENTS
First o f all, I would like to express my gratitude for the support from my family.
My parents have been a constant source o f an encouragement and unconditional support
even though we were separated by the Pacific Ocean. It is their prayers that keep me
sane during my graduate career. Thanks also go to my brothers, Tsung-Wei and TsungYih for their understanding and encouragement.
I would like to thank my thesis advisor, Prof. John B. Ketterson, for providing
me the opportunity to work with him . I have benefited from his great expertise and broad
knowledge. In addition, I would like to thank professors Arthur Freeman and Anupam
Garg for serving on my thesis committee.
I would like to thank Dr. Arthur Schmidt for his understanding and arranging
for substitute to take over my TA duties during a very difficult time o f mine. I want to
thank Mr. Mian-Zhong Lin for providing much advice in electronics and I would also
like to express my appreciation for his friendship. Dr. Jeffery R. Feller was a great help
with the microwave portion o f my work.
Finally, I would like to dedicate this thesis to my family, especially my wife,
Sue-Fen, and my daughter, Jocelyn. I gratefully thank them for their constant support
through all o f the successes and disasters.
h i
.
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Table of Contents
A cknow ledgem ents
ii
A bstract
iii
T able o f Contents
iv
L ist o f Tables
vii
L ist o f Figures
viii
C hapter 1
Introduction ................................................................................1
C hapter 2
A R eview o f the Experim ental Properties o f ................ 4
H eavy Ferm ion System s
2.1
2.2
Heavy Fermion. Systems ................................................................................. 5
2.1.1
Definition ............................................................................................... 5
2.1.2
Superconductivity in the Heavy Fermion Systems .......................... 6
A Review of Experimental Work on UBen ................................................... 9
2.2.1
Crystal Structure o f UBei3 .....................................................................9
2.2.2
Electrical Resistivity and Specific Heat of U B en............................... 11
iv
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2.2.3
2.3
Upper Critical Field o f UBei3 ............................................................14
Collective M odes............................................................................................. 15
C hapter 3
Experim ental Techniques and Instrum entation............21
3.1
Cryogenic System.............................................................................................22
3.2
Experimental Setups.........................................................................................23
3.2.1
C avity.................................................................................................. 24
3.2.2
Procedures For Lead Electroplating on C opper...............................29
3.2.3
The Antenna and Transmission Lines................................................31
3.3
Tuning Procedures..........................................................................................32
3.4
Cavity Perturbation Technique ................................................................... 34
3.4.1
Principles of Microwave Cavity Perturbation Technique
3.4.2
The FM Detection Method................................................................ 37
C hapter 4
35
R esults and D iscussions.......................................................42
4.1
Sam ples.......................................................................................................... 43
4.2
Data A nalysis.................................................................................................43
4.3
Results and Discussions................................................................................. 47
C hapter 5
In tro d u ctio n ..................................................................................58
C hapter 6
Basic Theory and O perating Principles o f S T M
6.1
61
Basic T heory..................................................................................................61
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6.2
6.1.1
Elastic Tunneling Through a One-Dimensional ............................61
Rectangular Potential barrier
6.1.2
Tunneling in More General Cases.................................................... 63
6.1.3
Scanning Tunneling Spectroscopy in the.........................................66
Superconducting State
Principles of STMOperation ........................................................................ 67
6.2.1
STM Imaging ................................................................................... 67
6.2.2
Operational Principles o f STS......................................................... 68
C hapter 7
LTSTM System ................................................................... 72
7.1
Cryogenic System............................................................................................ 75
7.2
STM Scanners............................................................................................... 78
7.2.1
Piezoceramic C ro ss.......................................................................... 78
7.2.2
Piezoceramic Tube .......................................................................... 82
7.2.3
Repolarization of the Piezos..............................................................85
7.3
Configuration of the LTSTM ...................................................................... 85
7.4
Vibration-Isolating System .......................................................................... 95
7.5
Preparation of STM T ip s ............................................................................... 97
7.6
Electronics ................................................................................................... 100
Reference ............................................................................................................... 109
vi
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List of Tables
Table 1. Bath compositions for lead-plating........................................................29
Table 2. Compositions o f solutions for cleaning. .............................................. 29
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List of Figures
I-l
Crystal structure o f UBe 13 ...........................................................................10
1-2 Temperature dependence of the electrical resistivity o f UBei 3 ............... 12
1-3 Electronic specific heat of U B eo............................................................... 13
1-4 Upper critical field o f UBei3 ...................................................................... 16
1-5 Power absorption by the collective mode for UPt3 for an Eig
Representation............................................................................................ 20
1-6 Schematic diagram o f the resonant cavity ..............................................25
1-7 (a) The Schematic drawing of one o f the two loop antennas; (b) The
assembly used for sample top loading.......................................................26
1-8 The resonant frequency and the Q o f the cavity for the TEoi i mode
at different sample positions.................................................................... 28
1-9 Block diagram o f the tuning arrangement using the FM
sweeping mode ......................................................................................... 33
I-10 The Lorenzian resonance line shape and modulation sig n a l................ 39
I-l 1 The electronics for the FM spectrometer ...............................................40
1 -1 2
A sketch o f the TE 012 cylindrical resonant cavity mode.........................43
1-13 A typical fit to the derivative o f a Lorenzian curve................................. 46
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1-14 Normalized surface resistance as a function of temperature................. 48
1-15 Surface resistance curve fitted with the Mattis-Bardeen theory
for the TEon mode.................................................................................... 50
1-16 The normalized surface resistance curves of TEoip modes vs. the
reduced temperature after subtraction o f the backgrounds.....................51
1-17 The normalized surface resistance and reactance measured at
27.09 G H z ................................................................................................. 53
1-18 Proposed temperature dependence of the collective mode
frequency ................................................................................................ 55
I-19 A plot of the line width vs.the position of the resonance peak
57
fi-1 The Abrikosov flux lattice o f NbSe 2 ...................................................... 71
II-2 Block diagram o f our LTSTM ............................................................... 74
II-3 A sketch of the thermal anchoring post...................................................77
II-4 Diagram of the piezoceramic cross.......................................................... 79
fi-5 Hysteresis o f the piezoceramic cross in the z-direction..........................81
H-6 The temperature dependence o f dsi for a PZT-5H tube.......................... 83
H-7 The piezo tube used in our LTSTM ..................................................... 84
H-8 Configuration o f two piezo crosses..................................................... 86
0-9 An image o f a “raised-square” sample................................................. 88
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n -1 0 A schematic drawing o f the new configuration o f the
piezo actuators.......................................................................................... 90
11-11 LTSTM housing......................................................................................91
11-12 Quadrant designations on the piezo cross and the piezo tube............. 93
11-13 A schematic drawing o f the vibration-isolating system ......................96
11-14 A drawing o f the apparatus used for etching W tips .......................... 98
11-15 A scan showing two different images................................................... 99
II-16 Circuit diagram of tunneling current preamplifier..............................101
11-17 HV operational amplifier circu its........................................................ 103
11-18 STM scan o f the raised-square sam ple................................................102
11-19 Electronic circuit of the modified HV op amps................................... 106
II-20a Image of raised squares (2 -D )............................................................ 107
II-20b Image of raised squares (3 -D )........................................................... 108
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C hapter 1
Introduction
The
phenomena
of
superconductivity
continues
to
fascinate
both
experimentalists and theorists more than 80 years after its discovery, in mercury by
Kamerlingh Onnes in 1911 [1]. The theory o f superconductivity formulated by Bardeen,
Cooper and Schrieffer [2] in 1957 successfully explained this phenomena with the
mechanism o f Cooper pairs, which bind in a so-called "s-wave state”, which will be
briefly discussed in chapter 2. Since the superfluid phases of 3He were identified as
being paired in a "p-wave state”, much effort has gone into examining whether pairingstates with orbital angular momentum / > 1 could be found in superconductors. The
possibility o f this nontrivial paring in the heavy fermion superconductors has attracted
much attention since the discovery
o f superconductivity in CeCuiSij-
The
superconducting states in heavy fermion compounds such as UPt 3 and UBen exhibit
behaviors that differ substantially from the predictions of the BCS theory. Quantities
such as the specific heat and ultrasonic attenuation display a power-law dependence on
temperature in the superconducting state. This suggests the existence o f nodes in the
energy gap. Also the existence o f multiple superconducting phases in UPt 3 [3] suggests a
possibly multi-component order parameter for this superconductor. The observation o f
1
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2
collective modes in the frequency range hco < 2A in heavy fermion superconductors
would furnish further proof of an order parameter transforming according to a multi­
dimensional representation.
The Cooper pairs in the heavy fermion superconductors carry a charge and may
couple with electromagnetic waves. One expects the characteristic energies o f the order
parameter collective modes to be on the order of the energy gap A ~ keTc, which
translates to frequencies in the microwave range for heavy fermion superconductors. To
observe the electromagnetic excitation o f collective modes , the obvious experiment is to
measure the microwave surface impedance employing a resonant cavity. In our work, the
surface impedance o f UBen has been studied in the frequency range 6 — 40 GHz
employing a lead plated copper cylindrical resonant cavity. The use o f a superconducting
cavity resulted in a quality factor, Q, o f order 10s at temperatures below the Tc o f lead.
This thesis is organized as follows. In chapter 2, we survey the experimental
properties o f the heavy fermion compounds in their normal and superconducting states,
preceded by a definition of the so-called "‘unconventional” superconductors. In chapter 3,
we describe, in detail, our microwave impedance measurement setup, which involves a
cryostat, a microwave resonant cavity, and various electronic and microwave
components. The so-called “microwave perturbation technique” as well as the frequency
modulation technique employed in this work are explained in detail as well as the tuning
procedures. At the end of this chapter, we describe the “FM detection method” used to
measure both of the frequency shift and the Q. The procedures for converting the
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frequency shift and the Q to the real and imaginary components o f the surface impedance
are also described. This was done mainly in an effort to explore the manifestation o f the
order parameter collective modes. In chapter 4, we present the results o f our microwave
surface impedance studies on UBei3 in the superconducting states. We also discuss the
observation of a frequency- and temperature-dependent absorption peak. We describe a
procedure which provides a simple relation between the energy gap function and the
electromagnetic power absorption. In conclusion we interpret the observed absorption
peaks as a power absorption into an order parameter collective mode.
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C hapter 2
A Review of the Experimental Properties of
Heavy Fermion Systems
Superconductivity in a heavy fermion system was first discovered in CeCu 2 Si2
by Steglich et al. [4]; later it was found in UBen by Ott et al. [5] and in UPt3 by Stewart
et al. [6]. The superconducting properties of these compounds may be said to exhibit the
most significant unconventional features among various exotic superconductors resulting
in much experimental and theoretical activity. In using the term unconventional here, we
follow the definition given by Rainer [7] and Fulde et al. [8], who require that the
symmetry o f the order parameter o f an unconventional superconductor be lower than the
symmetry of the Fermi surface (or underlying crystal). The symmetry group o f such
materials in the superconducting state is different from that in ordinary superconductors.
In this chapter we will give a brief introduction to the definition and general properties of
heavy fermion systems.
We will also review some o f the anomalous experimental
properties o f UBei3 .
4
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2.1 Heavy Ferm ion Systems
2.1.1 Definition
Heavy fermion systems are characterized by the fact that
the Sommerfeld y
factor, associated with the linear part of the low-temperature electronic specific heat, is
anomalously large (e.g., y = 1100 mJ/K2mole for UBen [5], and y = 430 mJ/K2mole for
UPt3 [9]).
The y factor is related to the density of states by the relation, y =
(Tr2/3)kBiN(sF), where N (sF) is the density o f states at the Fermi level. It is common to
define a specific heat effective mass through the relation N(sF) = m*kF/h37t2. Therefore a
large y implies a large effective mass, m*, in these systems (usually in the order o f 102103 times the bare electron mass, me). For an ordinary metal, m* is o f the order o f me
(e.g., m* = 1.3 me for Cu [10]).
Heavy fermion materials are intermetaliic compounds containing lanthanide
(notably cerium) or actinide (uranium and neptunium) ions with partially filled 4f- or 5felectron shells.
It is generally believed that it is the interaction among these highly
correlated f-electrons and between the f-electrons and the conduction electrons that is
responsible for the large effective mass.
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2.1.2 Superconductivity in the H eavy Ferm ion Systems
Many experimental results on heavy fermion superconductors point toward
possible unconventional electron pairing in these materials.
Before discussing this
subject, we will give a brief sketch o f the conventional BCS theory.
It was a breakthrough in the understanding o f superconductivity when in 1956
Cooper [11] presented a pairing model for the electronic ground state of a
superconductor. What Cooper showed was that when there was an attractive interaction
between two electrons, having opposite spin and momenta, at the Fermi energy in the
presence of the remaining electrons in the Fermi sea (the interaction of which was
ignored), a two-electron bound state would form, no matter how weak the interaction.
In a simple model, which assumes that the scattering matrix element is a constant
-V (with V positive) for s F -tia)c < e < e F +ficoc , and zero otherwise, where coc is a
cutoff frequency for the interaction, Cooper found a binding energy given by
s^-2hcoce ~ ^ 0) < 0 .
(I-l)
The associated wave function is spherically symmetric corresponding to s-pairing. Here
N(0) now corresponds to the density o f states at the Fermi level.
With the concept o f electron pairing in mind, Bardeen, Cooper, and Schrieffer [2]
constructed a microscopic theory of superconductivity in 1957 (the BCS theory). They
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7
assumed the attraction mechanism is due to a polarization o f the background ionic lattice
by the electrons, the electron-phonon-electron interaction, which provides a strong
“overscreening” effect resulting in a net attraction between the electron pairs. Through a
clever choice o f a variational wave function they minimized the expectation value o f a
Hamiltonian that retained only an attraction between pairs. An important characteristic
of the BCS theory is the so-called “gap parameter”. At finite temperatures, the gap
parameter is defined as
Ak = - £ ( l - 2 / k.K .v k.Vk„ ,
k'
(1-2)
where Vk’k is the scattering matrix element, / i s the Fermi distribution function, v2 is the
probability o f pair occupancy, and ic is the probability o f pair vacancy. The BCS selfconsistent equation takes the form
Ak = - Z ( l - 2 / k) ^ - V t t
(1-3)
k’
where E k, = -y/ak. + Ak.
The predictions of the BCS theory for many superconducting properties are in
good agreement with experimental results for ordinary metals.
As noted above the
attraction mechanism in the BCS theory was assumed arise from the electron-phononelectron interaction. However, according to their theory, any mechanism leading to an
attractive interaction near the Fermi surface can lead to the superconductivity.
One concludes from the large measured value o f the jump in the specific heat at
Tc, AC / yTc, in the heavy fermion compounds [5, 6], that superconductivity o f these
materials arises from the heavy f-electrons. It hasbeen argued that, in the picture o f the
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8
BCS theory, it would be difficult for these electrons to form the ordinary s-wave Cooper
pair, due to the strong Coulomb repulsion o f f-electron. To avoid a large overlap o f the
making up pair wave function, the pairing would occur in a higher angular momentum
“channel”, such as a triplet p-wave pairing, in 3He, or a singlet d-wave pairing, where the
centrifugal barrier tends to keep the electrons further apart.
The unconventional
properties o f the well-studied superfluid 3He may give some insight into the
unconventional heavy fermion superconductors.
However, there are significant
differences between this superfluid system and the superconductor system. To name a
few, neutral atoms are paired in 3He, while charged electrons are paired in heavy fermion
superconductors.
The strong correlation effects and the spin-orbit interaction in the
heavy-fermion system may also change the picture from superfluid JHe. Furthermore,
the presence o f the crystal field in heavy fermion systems induced a discrete rotational
symmetry, while it is continuous for 3He. We emphasize that it is still unclear what the
pairing mechanism is in heavy fermion superconductors and many experimental results
are obscure and do not allow an unambiguous determination o f the order parameter.
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2.2 A Review o f Experim ental W ork on UBei 3
2.2.1 Crystal Structure o f U Bei3
The crystal structure of UBe^ is the cubic NaZni3 configuration as shown in
figure I-1(a). The crystal symmetry of UBen belongs to the cubic
Of ,
point group. There
are eight formula units contained in one primary cell with a lattice parameter a = 10.252
A, which gives a U-U distance o f 5.13 A. Each uranium atom is surrounded by 24 Benatoms (the open small circles in figure I-l (a)), forming an icosahedral cage, and 8 Be[atoms (the solid small circles in figure I-l(b)), forming a cube structure around the Uatom. One o f the 13 Be-atoms of a formula unit sits on the Bei site and is surrounded by
a polyhedral arrangement of 12 Ben-atoms (figure I-l(b)). Also shown in the figure are
two different U-atoms (denoted by the large white spheres and the large black spheres) in
one primary cell. Each set of the U-atoms are connected by a conventional fee unit cell
structure.
The difference between the two sets o f U-atoms is that
the Ben atoms
surrounding them are mirror images of each other. A pair o f black and white U-atoms,
each with a corresponding Bei and 12 polyhedral Ben, forms a basis o f the lattice.
It appears that the superconductivity in this material is very sensitive to any kind
o f chemical impurities, especially to substitution on the Be sites [12].
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(a)
¥-44
(b)
Figure I-l. Crystal structure o f UBe13. (a) Unit cell of UBe13.
U-atoms are shown as large spheres. Be-atoms are shown as
small spheres, (b) Section o f the unit cell. Open spheres are BeEatoms surrounded by Bel-atoms (closed spheres). These figures
are taken from Hiess et al. [64] and Knetsch [65].
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2.2.2 Electrical Resistivity and Specific Heat o f U B ei3
Figure 1-2 shows the temperature dependence o f the electrical resistivity o f UBei3
(after Ott et al. [5]). Upon cooling below room temperature, the electrical resistivity o f
UBei3 rises steadily with deceasing temperature and passes through a broad maximum at
about 30 K. Below 10 K, p again rises and reaches a maximum value o f 234 pQcm at
2.35 K. At still lower temperatures, p decreases with decreasing temperature and then
vanishes at the superconducting transition at about 0.86 K.
The specific heat measurements by Stewart et al. [13], Ott et al. [5] and Mayer et
al. [14] are shown in figure 1-3. Ott et al. reported a y factor o f 1100 mJ/K2mole. From
their experimental data, the jump in the specific heat at Tc, relative to the normal state
specific heat o f 2.4 [14], exceeds the weak coupling BCS value o f 1.43 (this jump is
higher than other heavy fermion superconductors). On this basis it was suggested that
“strong coupling” effects need to be considered in UBeo. The temperature dependence
o f specific heat below Tc obtained by Ott et al. shows a deviation from BCS theory and is
approximately proportional to T3. This power law dependence o f the specific heat has
been interpreted as evidence for non-conventional superconductivity with the possibly
existence o f point nodes in the gap function. The results from Mayer et al. [14] however
showed that the temperature dependence o f the specific heat cannot be described by a
simple power law (figure 1-3(b)).
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240
UBei3
Electrical Resistivity
200
L60
...............
P
[ ( L iQ c m ]
158
80
100
40
0
58
2
■ ■
40
I
80
i
120
3
1
. 1 ,—
160
200
240
280
T(K)
Figure 1-2. Temperature dependence of the electrical
resistivity of UBei3 between 1 and 300 K. The inset shows
the resistivity between 0.5 and 3 K on an expanded scale.
After Ott et al. [5]
R e p r o d u c e d with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e rm is s io n .
13
2.0
1.5
<u
o
£
1.0
V
0.5
0
5
15
10
T (K)
2
1
(U
o
0.5
£
0.2
0.1
U
J.05
0.02
0.02
0.05
1
2
T (K)
Figure 1-3. (a) Electronic specific heat of UBen for
200 mK < T < 18 K. (b) Specific heat of UBen for
T < 3 K on an expanded log-log scale. After Mayer
et al. [14]
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14
2.2.3 Upper Critical Field o f U Bei3
The temperature-dependent upper critical field, Hc2 (T), is the most extensively
studied property of UBen.
The HC2 (T) curve (figure 1-4) shows an extraordinary
behavior starting with a large initial slope at Tc ( -----45 T /K [15]), followed by inflection
points near 2 T and 6 T, finally flattening as T —> 0 at a value Hci(0) ~ 14T. points near 2
T and
6
T, finally flattening as T —> 0 at a value HC2 ( 0 ) ~ 14T. Using the dirty-limit
approximation for conventional superconductors calculated by R. Hake et al. [6 6 ], (-dH C2
/ dT)rc= (4.41xl04)py, B. Maple et al. [16] estimated (—dH C2 / dT)xc = 74 T/K, which is
in order o f magnitude agreement with the measured value of 42 T/K.
Using their
measured initial slope of the upper critical field, Maple e t al. also estimated the coherence
length £, = 142A. From the size o f the specific heat jum p at Tc U. Rauchschwalbe [67]
calculated the initial slope o f the thermodynamic critical field Hc' = (poAC / Tc)/: = 0.17
T K . From HC2 = ®o/
= 2'/’kHc [6 8 ], U. Rauchschwalbe calculated HC2 ' = 14 T K
by adopting a k value of 58. He also pointed out that L4 T K is in fact the slope o f the
H C2 curve just_below the point where AC is read off. Although explanations Although
explanations have been proposed by some authors [15,16,17], the origins o f this
anomalous shaped curve is still not clear.
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15
2.3 Collective M odes
The search for collective modes in the heavy fermion superconductors is a
different approach to trying to identify the symmetry o f the order parameter.
The
existence of collective modes with a frequency less than 2A / h is one o f the special
properties of unconventional superconductors. The collective modes in the triplet p-wave
states o f the neutral, superfluid 3He have been studied intensively, both experimentally
and theoretically [18,19] and have been clearly observed by ultrasonic attenuation
experiments [2 0 ].
The observation o f an ultrasonic attenuation peak in the heavy fermion
superconductors UBei3 [2 1 ] and UPt3 [2 2 ] has also been reported: a sharp attenuation
peak is observed just below the superconducting transition temperature Tc. This was
initially interpreted as a low-lying collective mode. It is more likely that the ultrasonic
attenuation peaks observed in these heavy fermion superconductors are not due to
collective modes [23]. One o f the more simple (and plausible) explanations [24] o f the
observed attenuation peaks is in terms o f enhanced pair-breaking and coherence effects,
which are consequences o f the usually large quasiparticle effective masses.
Collective modes may appear quite generally as a consequence o f some
continuous symmetry breaking. In the case o f the heavy fermion superconductors (with
strong spin-orbit coupling), the only continuous symmetry o f the system is the global
gauge symmetry. The so-called Goldstone mode generated by spontaneously breaking
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
16
14
12
10
H
X
■5H,/SfTi=-45T/K;
0
0.2
0.4
0.6
0.8
T (K)
Figure 1-4. Upper critical field HC2 (T)
measured by resistivity. After Thomas
et al. [15].
R e p r o d u c e d with p e r m i s s io n o f t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n proh ibited w ith o u t p e r m is s io n .
1
17
this global gauge symmetry in the superconducting state is the plasma resonance. The
plasma frequency, however, is much larger than the gap frequency and hence this mode is
not relevant to superconductivity [25].
The order parameter o f some heavy fermion superconductors may transform
according to a nontrivial representation o f the symmetry group. Some of these states
correspond to higher dimensional representations, i.e., multicomponent order parameters.
These unconventional states (corresponding to higher dimensional representations)
should support other kinds of collective oscillations. These order parameter collective
modes, involving the oscillations o f the internal structure of a Cooper pair, are expected
be observable in the heavy fermion superconductors with mode frequencies less than
twice the maximum value of the anisotropic gap function. Examples of such modes are
the ‘"squashing mode” in the B W state as well as the “flapping” and the “clapping” modes
in the AM state of the superfluid 3He [19].
Hirshfeld et al. [26] studied the contribution of the order parameter collective
modes for the BW state to the electromagnetic absorption.
The result o f their
calculations was that the collective modes would contribute significantly to the power
absorption at frequencies well below the gap edge of 2A. They suggested that these
absorption
features
should be
qualitatively observable
for all unconventional
superconductors. They also pointed out that the absorption is very sensitive to impurity
scattering which will broaden the collective modes, eventually overdamping them and
washing out the structure in the absorption spectrum.
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18
The important distinction between the BW and most o f the unconventional
superconducting states that have been suggested as possible ground states for heavy
fermion superconductors is that the BW gap does not have a node in any direction in kspace.
Hirschfeld et al. [27] therefore calculated, within the framework o f a matrix
kinetic equation [28], the power absorbed by order parameter collective modes at T =
0
for UPt3 assuming a 2D Eig representation. They concluded that, while the peak was
broader than in the pseudoisotropic case, broadening due to excited nodal quasiparticles
does not prevent the observation o f the mode.
In their subsequent paper [29], the
calculation was extended to examine the effects o f thermal broadening on the mode
feature.
Figure I-5(a) shows their calculated temperature-dependence o f the mode
frequency. The mode frequency,
Q o,
with propagation vector q =
0
is found to display
roughly the same temperature dependence as the order parameter itself with Qo = 1.19
Ao(T), where Ao is the maximum o f the gap.
temperature curves for different frequencies
shown in figure 1-5(b). The peak for the
determined by
0.5Q oo ~ 1.2A o(T).
The calculated power absorption vs.
(Q = Qoo = Q o (T =
Q = Q oo/2
0
), 0.5Q oo,
and
O.8 Q 0 0 )
is
curve occurs roughly at a temperature
Closer to the resonance, the mode evolves into to a
large broad absorption shoulder due to screening currents and nodal quasiparticles.
Figure I-5(c) shows the power absorption spectrum at two different temperatures. From
this figure we can see that the mode is well-defined, even at finite temperatures.
A model for d-pairing has also been proposed by Brusov and Brusova [30,31]
who utilized the path integral technique.
Within their model o f d-pairing for heavy
fermion superconductors, the collective mode spectrum has been calculated, including the
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19
damping terms due to pair breaking. Low-lying collective modes are also predicted to be
observable in the microwave frequency absorption spectrum.
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20
to
ci
0.00
0.25
0.50
0.75
1.00
0.8
1.0
T/To
oCM
Q_
a. o
o
0.0
0.5
T/T 0
Q_
Q_
oCM
o
T=0
o
0.0
0.5
1.0
1.5
2.0
2.5
Q/Ao(T)
Figure 1-5. Power absorption by the collective mode for UPt3 for
an Eig representation: (a) the temperature-dependence o f the q = 0
mode frequency, normalized to gap maximum; (b) normalized
power absorption vs. temperature curves for various frequencies;
(c) power absorption vs. frequency at T = 0 and T = 0.5 T showing
a well defined peak. After Hirshfeld et al. [27].
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Chapter 3
Experimental Techniques and
Instrumentation
The complex surface impedance measurements made as a part o f this work
were performed using the so-called “microwave cavity perturbation technique”. The
microwave cavity containing the sample was cooled using an Oxford dilution
refrigerator.
In this chapter we will mainly concentrate on the microwave cavity
perturbation technique and the microwave spectrometer setup used. The operation o f a
dilution refrigerator has become a standard procedure in low temperature measurements
and will not be reviewed here. The cryogenic system is discussed briefly below.
21
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22
3.1 Cryogenic System
The cryostat used was a Model TLE-400 dilution refrigerator system from
Oxford Instruments with a cooling power o f 400 p.W at 100 mK. The base temperature
depends on the experimental configuration and various base circulation conditions. For
our microwave cavity measurement setup we were able to reach a temperature o f —60
mK. The cryostat is also equipped with a superconducting magnet, which has a field
homogeneity o f 1 part in 1000 over 1 cm and a maximum field o f about 13 Tesla.
One of the important features for this cryostat is the ability to top load the
samples. This gives us the advantage of not having to warm up the system and open the
resonant cavity in order to reposition or remove the samples. When comparing cavity
signals with and without a sample, this feature also reduces most o f the factors that may
alter the prevailing condition within the cavity, other than those resulting from the sample
itself.
Pre-calibrated germanium resistors from Lake Shore and Dale 1000 RuCL thickfilm resistors were used as the primary thermometers. Some Mazushita carbon resistors
were also employed as secondary thermometers.
In the absence o f a magnetic field,
germanium resistors are the main temperature sensors, while RuCL resistors are used
when magnetic fields are applied. All o f these thermometers are either glued directly to
OFHC copper adaptors or are held via holders by using GE 7031 vamish. The resistance
o f all the major temperature sensors was determined in a 4-wire configuration, using an
AVS-46 automatic AC resistance bridge from RV-Elektroniikka. When the temperature
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23
is swept, a temperature-controller was employed which consisted of a temperature sensor,
a heater located near the sample, and a current source controlled by a personal computer.
The AC resistance bridge reads the resistance o f the sensor, which the computer converts
to the corresponding calibrated temperature. This temperature is then compared with the
programmed reference settings. The error signal was used as a proportional-integraldifferential (PID) feedback to adjust the output o f an A/D converter connected to the
heater.
3.2 Experimental Setups
Earlier studies at Northwestern involved various resonant microwave cavity
configurations such as an adjustable length cavity, and coaxial and hollow fixed-length
cavities [32]. We will describe mainly the design o f the fixed-length cavity used in this
work.
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24
3.2.1 Cavity
The fixed-length cavity used is shown in figure 1-6; it is cylindrical with an
inner diameter o f 2.2-cm and a height of 2.82-cm and was fabricated from. OFHC copper.
The cavity was constructed in two sections: i) a cylindrical bottom cup machined from a
single piece o f OFHC copper; and ii) an OFHC top plate, which has two symmetric holes
(3 mm in diameter), one for coupling microwave power into the cavity and one for
extracting a signal; each hole was 6.35 mm from the center o f the plate. This top plate
also has a hole (6.35 mm in diameter) in the center for top loading the sample into the
cavity.
The cavity was attached to an adaptor flange (figure 1-6). The adaptor, which
contains a copper tube with a flange at the end, is thermally anchored to the mixing
chamber. Each part o f the cavity assembly was sealed using an indium O ring.
The sample was top loaded into the cavity with a specially designed sample
holder. The assembly is shown in figure 1-7. The spring that pushes the main copper
piece against the inner bore of the top plate serves not only as a sample-position-lock but
also as a thermal anchor. We also adopted a “slip-stick” disengagement mechanism.
When the sample is in position, the long nylon sample-transport-rod and the copper main
piece were withdrawn slightly, disengaging the remainder of the sample holder and,
thereby, preventing a sample position change caused by thermal expansion or
contraction.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Adapter flange
Stainless steel
ring
Sample loading
Coupling holes
Top plate
Stainless steel
ring
Bottom cup
Figure 1-6. Schematic diagram of the resonant cavity.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Center
conductor
Nylon rod
N\
j
I
%
“Slip-stick”
disengagement
mechanism
Coaxial
cable
y r4
//
Spot weld
A
/7
Spring
(u)
Main copper
piece
Small-loop
antenna
<-
Samplemounting
rod
Sample
. Teflon
sleeve
(a)
(b)
Figure 1-7: (a) The schematic drawing of one of the two loop antennas
fabricated directly from a coaxial cable; (b) the assembly used for
sample top loading.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
27
The sample is glued onto the bottom end o f the sample holder with GE 7031
varnish via a piece of cigarette paper for electrical insulation. A sleeve made from Teflon
was attached via silicone grease to the main copper piece not only to protect the sample
when loading but also to prevent the sample from falling loose into the cavity which
would otherwise terminate the run; if the glue fails, we can simply lift the loading rod out
o f the refrigerator, reattached the sample and reload it. The nylon sample-transport-rod
extends all the way out o f the cryostat. The space between the nylon rod and rubber CDring seal is pumped during the experiments to prevent any air leaking past the O-ring
from collecting in the cavity when the rod is moved to adjust the position of the sample.
As a reference to the position o f the sample, we also measured the resonant
frequency and the Q at different distance of the sample from the bottom o f the cavity.
Figure 1-8 shows the resonant frequency and the Q for TEou mode at different sample
positions relative to the bottom o f the cavity. The vertical line indicates the location
where the sample is at the top o f the cavity.
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28
9516-1
9514-
T o p
o f C a v ity
951295109508-
N
X
9506-
2
950495029500949894960.9
1.0
1.2
1.3
1.4
1.5
1.6
1.7
1.5
1.6
1.7
Sample Position (in)
240000
220000
-
200000
-
180000
-
160000
-
O '
140000 120000 100000 80000
0.9
1.0
1.1
1.2
1.3
1.4
Sample Position (in)
Figure 1-8. The resonant frequency and the Q of the
cavity for the TEoi i mode at different sample positions.
The dashed line indicates the position where the sample is
at the top of the cavity.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
29
3.2.2 Procedures For Lead Electroplating on C opper
Fluboric acid (HBF4 )
Basic lead carbonate ((PbC 0 3 )2 -Pb(0 H)2 )
Water soluble glue
320 g/L
130 g/L
0.2-5 g/L
Table 1. Bath compositions for lead-plating.
I
n
m
IV
Phosphoric acid (H 3 PO 4 )
n-Butyl alcohol
350mL
90 mL
Sodium Hydroxide (NaOH)
Sodium Carbonate (NasCCb)
Sodium Pyrophosphate CN&tPzCb)
Sodium Metasilicate Anhydrous (Na 2 SiC)3 )
12g/L
llg/L
18g/L
18g/L
Fluboric acid (HBF 4 )
Distilled water
Sodium cyanide (NaCN)
50mL
200mL
25g/L
Table 2. Compositions o f solutions for cleaning.
All inner surfaces of the cavity, including the sample-mounting rod (to be
discussed shortly - see figure 1-7), were electroplated with Pb employing the procedure
[33] described as follows: The compositions o f the solution for the plating bath are listed
in Table 1. Solutions used for surface preparation are listed in Table 2. The copper
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30
surface is prepared before electroplating in the following sequence: 1) Mechanically
polish the surface followed by degreasing with organic solvent;
2
) electropolish the
surface in solution I at current density o f —15-290 mA/cm 2 for 5-10 minutes
(electrocleaning); 3) soak in solution II for —5 minutes
(alkali cleaning); 4) dip in
solution EQ for -1 minute (acid dip); 5) dip in solution IV for -1 minute (cyanide dip); 6 )
rinse in distilled water thoroughly. Three different pieces, involving the inner surfaces o f
the cavity, are plated separately. Pure lead anodes with an anode/cathode ratio o f 1:1 are
used in the electroplating. According to the Faraday’s law of electrolysis, the quantity o f
metal deposited, W, is equal to
W=
rt a\
(1-4)
I-t
A
x—
96500 z
where I is the current passed in the circuit, t is the time of deposition, z is the valency,
and A is the atomic weight.
The thickness o f deposition, T, is obtained from
T = W / (S-d), where S is the plated area and d is the density o f the material to be plated.
From the formula given above, the thickness o f lead plated on the inner surface o f our
cavity is —6 pm.
This coating with superconducting Pb resulted in the cavity quality factor Q o f
the order o f 103 at temperatures below 7.19 K (the Tc o f Pb).
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31
3.2.3 The Antenna and Transmission Lines
Microwaves were coupled into and out o f the resonator through two semi-rigid
Cu-Ni 50 Q coaxial transmission lines, each terminated in a small-loop antenna.
These coaxial cables, which have an outer diameter o f 0.085 inch and are about 7
feet long, extended from the top o f the cavity out o f the cryostat and were vacuum sealed
at the top of the cryostat by rubber O-rings. Extending the coaxial cables outside the
cryostat enables us to adjust the orientation of the antenna loops as well as the insertion
depth into the coupling holes. We chose coax as our microwave transmission line since it
transmits a broadband o f frequencies and is reasonably flexible while rigid enough to
rotate. The Cu-Ni center conductor provides low electrical loss and acceptable thermal
conduction. The outer conductor o f the coaxial cables is thermally anchored at several
points by using spring loaded Be-Cu finger stock. To secure the orientation and position
o f the antennas, the outer conductor of the coaxial cables near the cavity end was
threaded; matching threads were also cut in the top flange of the cavity. In this way not
only can the orientation and depth o f the antenna into the coupling holes be adjusted, but
also the required stability against thermal contraction (caused by a changing He bath
level) is achieved.
The magnetic field lines in the loop antenna “look into” the resonator through
small circular channels ending in symmetrically placed coupling holes. Two small loops,
one serving as a transmitting antenna, one serving as a receiving antenna (th e
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32
transmission mode), were fabricated directly from the center conductors o f the coaxial
cables. The loops are spot welded onto the outer conductor o f the coaxial cable, as
shown in figure 1-7.
3.3 Tuning Procedures
To locate a particular resonance o f the cavity we used the FM sweeping mode
o f our microwave synthesizer, as shown in figure 1-9. The microwave synthesizer used
was a model HP83623A manufactured by Hewlett & Packard. The synthesizer operates
in the frequency range from 10 MHz to 20 GHz. When working at frequencies higher
than 20 GHz, an MX2M260400 passive frequency doubler from Miteq was used. With
this frequency doubler and corresponding microwave amplifiers we were able to cover
the spectral range from
6
GHz to 40 GHz in our measurements. In the FM sweeping
function mode, the synthesizer generates a FM sweeping signal, which is coupled into the
cavity. The center frequency and span of the signal can be controlled directly from the
synthesizer.
Saw-tooth signals, which were also generated by the synthesizer, were
applied to the x-input o f an oscilloscope. The output signal from the cavity is applied to
one of two broadband amplifiers: one was used for lower frequencies from 6-18 GHz
(AFS4-06001-800 from Miteq, 25 dB gain), and the other for higher frequencies from 18-
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
33
Amplifier
Detector
Cavity
Cryostat
iDoubler
Oscilloscope
Sweep
Out
Microwave Synthesizer
OUT
Figure 1-9. Block diagram o f the tuning arrangement using the FM
sweeping mode. Also shown is a plot o f resonance curve observed with
the oscilloscope.
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34
40 GHz (JS4-18004000-30 from Miteq, 25 dB gain). The signal then went through a
microwave diode detector and was applied to the y-input o f the oscilloscope. Figure 1-9
also shows a typical resonance curve at a microwave power o f approximately 0 dBm
observed from the oscilloscope. This procedure was used either to categorize various
resonance modes o f the cavity or to adjust the antennas to optimize the coupling
conditions for a specific resonance mode (i.e., to achieve critical coupling).
3.4 Cavity Perturbation Technique
To extract the complex surface impedance o f a sample from microwave cavity
resonance measurements, the so-called “microwave cavity perturbation technique”
[34,35] was employed. Basically, this technique involves measuring the change o f some
characteristic responses o f the cavity caused by introducing a sample into the cavity. The
characteristic responses o f the cavity we measured in this work are the half-maximum
bandwidth, T, and central frequency, fo, of the resonance by using the “lock-in FM
method”.
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35
3.4.1 Principles o f the M icrowave Cavity Perturbation
Technique
When investigating the response of a highly conducting specimen to an
electromagnetic excitation at microwave frequencies, the parameter usually studied is the
complex surface impedance, Zs = R* —/Xs, where Rs is the surface resistance and Xs is the
surface reactance. In our measurements a small sample was introduced into the cavity
through the top-loading hole in the top plate of the cavity. This minimizes the frequency
shift due to the field volume excluded by the introduction of the conducting sample. The
changes in the characteristics of the cavity, particularly the quality factor Q and the
resonant frequency, fo, due to the introduction of the sample were obtained using an FM
detection method, which will be described in the next subsection.
The field in a resonant lossy cavity, following the removal o f an initial excitation,
is a damped oscillation [4],
E(f) = E 0e ie e-,co*'
(1-5)
where E is the electric field; Q =
quality factor of the cavity;
co r
co r x
(energy stored in the cavity / power loss), is the
is the central resonant frequency.
The complex frequency is then defined as co = <
dr - Hjcor / 2Q).
The complex frequency change, 5co, and the change of the time-averaged energy
stored in the cavity, <5(U), caused by the introduction of the sample, are related by
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36
U
[34]. The subscript 0 represents the unloaded cavity characteristics. The
6)0
power loss, however, is proportional to the surface impedance change, <5ZS. Therefore,
the variation o f the cavity characteristics and the complex surface impedance are related
by
8co
con
iO5ZS =■
(1- 6)
where £ is a resonator constant, which involves the shape, size, and mode o f the cavity.
In terms of Q, f, Rs, and Xs, equation (1-6) can be expressed as
f [ ( X , - X , „ ) - / ( R , - R , 0)]
f
f
ft - H
fso
S°
2Q sO
2QS
f sO - i1-
(1-7)
s0
2Q sO
sO
V
QS Q sO
sO
1+
l + i2Q
-
4Q so y
For large Q values, the second term in the second parentheses o f the numerator and the
second term in the parentheses o f the denominator can be neglected. We then have
f[(x.- X j - / ( R S- R j ]
f-
' f l
2 Qs
L,
Q^,
0 - 8)
Therefore the surface resistance of the sample is related to the measured cavity
characteristic through a change of the inverse o f Q’s; while changes in the surface
reactance are obtained from frequency shifts.
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37
3.4.2 The FM Detection M ethod
From the previous subsection we know that dissipation in the cavity results in a
damped oscillation.
By taking the Fourier transform o f equation (1-5), the power
absorption spectrum can be expressed as
|E H 2 « ----------
l—
----rr •
O 9)
(co -a>R)2 + —
v
U q
The absorption spectrum then has a Lorenzian line shape, as shown in figure 1-9. The
half-maximum bandwidth, T, is equal to
cor
/ Q.
We measured both the Q and the characteristic frequency in the transmission
mode by using an FM detection method, in which a narrow-band FM signal was applied
while measuring the power transmitted through the cavity at the resonance.
The
frequency modulated signal (with modulation frequency a>mand modulation width Aco «
T), which was imposed on the carrier signal (with frequency coc), induces an amplitude
modulated (AM) output signal from the cavity, as illustrated in figure I-10. The AM
signal has frequencies corresponding to multiples o f com. When coc differs significantly
from
co r
and Aco «
cor
/ Q, the transmitted AM signal consists largely o f the fundamental
harmonic (com) component only with minimal second harmonic (2 com) component. When
coc is close to
cor
the transmitted AM signal involves both com and 2com components; in the
special case when coc = c o r , the output signal consists only of the 2comcomponent. Note
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38
that the fundamental AM transmitted signals when the carrier frequency goes from one
side o f the resonance to the other are 180° out of phase (figure 1-10).
The block diagram o f the electronic setup used in this work is illustrated in
figure 1-11.
In these measurements the microwave synthesizer was operated in the
FM/CW mode. The FM modulation for the synthesizer was provided by one o f the lockin amplifiers (an EG&G 5210 in this setup) with the modulation ratio set to 1MHz/volt.
Before being fed into the cavity, the FM modulated signal from the microwave
synthesizer was attenuated in order not to heat the sample or saturate the amplifiers at the
receiving end (the 1 dB output compression point o f the amplifiers is 10 dBm and 8 dBm,
respectively).
The amplitude modulated transmission signal from the cavity was
amplified and passed through a diode detector into the inputs o f the two lock-in
amplifiers (EG&G models 5210, and 5209). One of the lock-ins was used to measure the
o)m AM output signal, while the other was used for the 2com AM output signal. An
oscilloscope was used to observe the trace of the AM transmission signal from the signal
monitor output of one of the lock-in amplifiers. The reference from the EG&G 5210 was
also used to synchronize the EG&G 5209. The in-phase component from the output o f
the EG&G 5209(1 com) was digitized by an A/D converter. Both o f the lock-in amplifiers
and the microwave synthesizer communicated with a personal computer through a GPEB.
The Q of the cavity was obtained as follows.
As the frequency o f the
synthesizer was swept through the resonance, the computer recorded the output o f the
lock-in operating at the first harmonic along with the corresponding carrier frequency.
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AM
Signs
(a)
(b)
Figure I-10. (a) The Lorenzian
Also shown schematically are
resulting output AM signals for
The derivative o f the Lorenzian
from the 1 com lock-in amplifier.
resonance absorption line shape.
the modulation signal and the
different carrier frequencies, (b)
line shape as would be observed
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40
PC
CTocIc j j
EG&G 5210 (2o>.)
Microwa re Synthesizer
SIC HON
Doubler
Amplifier
K > Detector
-^ h
Attenuator
I—
1
SCOPE
EG&G 5209 (lco
TTL
Cryostat
Figure 1-11. The electronics for the FM spectrometer.
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41
Near the resonant frequency the lock-in output will be proportional to the derivative o f a
Lorenzian and the Q value is obtained by fitting the data to this form.
To keep the microwave spectrometer locked to the cavity as the temperature
was swept, a feedback loop was incooperated. In order to lock onto the resonance, one
must constantly m in im ize the difference between coc and
o jr .
This can be achieved by
using the com lock-in output as an error signal which is fed to the computer. The PID
signal generated by the computer constantly adjusts the synthesizer carrier frequency so
as to zero the com lock-in output. The computer then records the temperature, resonant
frequency, and 2 com lock-in output amplitude, as a function o f the temperature.
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C hapter 4
Results and Discussions
In order to confirm the microwave cavity methodology used, an experiment on
Zn was performed in previous work [32]. The results were in reasonable agreement with
the Mattis-Bardeen formula [36]. Measurements were also performed on UPt3 [37] and
UBei 3 [38]. No clear evidence of absorption by collective modes was observed. These
null results may arise from overdamping caused by quasiparticles scattering at the sample
surface (a pair-breaker for an unconventional superconductor).
We present here the results of recent surface impedance measurements on a
very high quality UBei3 single crystal by the frequency modulation technique employing
a lead-plated cavity, as described in the previous chapter. We will also describe some
general aspects of our data analysis procedures and assumptions.
42
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43
4.1 Samples
The sample used was a very high quality single crystal o f UBeu with an
approximately rectangular shape o f dimensions ~ 4.5 x 3.6 x 1.3 mm3 with a flat and
shiny surface.
This sample was prepared by J. Smith (his sample #3781) from Los
Alamos National Lab and was made by the following procedure. High purity U, Be, and
Al in the atomic ratio 1 : 15 : 174 were loaded into an outgassed BeO crucible. An oven
containing the crucible was heated to 1200 °C in flowing helium gas. It was then cooled
through the aluminum freezing point over a period of 300 hours. Cubic single crystals o f
UBen with natural facets [100] were revealed by removing the aluminum in a
concentrated NaOH solution. The surface was then lightly etched in dilute sulfuric acid.
Tc of this sample was ~ 0.905 K.
4.2 Data Analysis
We examined primarily the TEoip
modes in this work. A sketch o f the field
distribution o f the TE012 mode is shown in
Figure 1-12. A sketch o f the TEon
cylindrical resonant cavity mode. The
dashed lines are magnetic field; • and
X represent the electric fields [adapted
from Reintjes and Coate, p. 608
(1952)1.
figure 1-12. These modes coupled well
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44
with our inductive antenna loops and so are easily distinguished from, neighboring modes
by adjusting the antenna loops. The frequencies corresponding to TEou, TE 0 1 2 , TE 013 ,
and TE 014 modes for the cylindrical cavity configuration we used are 17.22, 19.58, 23.00,
and 27.09 GHz, respectively. The field distributions and configuration for these four
modes are approximately the same at the center o f the top plate. The TEoip modes and
the TMnp modes are degenerate in an ideal cylindrical resonant cavity. The degeneracy
o f these two modes is lifted by the nonsymmetrical distribution resulting from the
antenna coupling holes at the top plate, so that the mode interference is minimized and
was not an issue.
The resonant signal for modes with p > 4 could not be reliably
measured probably due to large power losses in the transmission coaxial lines.
With the microwave cavity perturbation technique described in section 3.4, the
data directly recorded are the resonant frequency, fa, and the amplitude of the 2 com signal,
A 203, both as a function o f temperature. From the discussion in previous chapter, the
amplitude A2 0 is proportional to Q4.
To convert A 20) to the corresponding Q, the
microwave frequency, f c , was swept around the resonance, fa, at two temperatures: one
at the temperature just above Tc, one at the lowest T. The amplitude o f the com signal,
Ac, is recorded as a function of frequency. Figure 1-13 shows a typical spectrum o f the
resonance, A*, as a function of f c. Also shown in the figure is a curve fitted to the
derivative of a Lorenzian line shape using formula:
A* = ax(£-b)/((£—b)2 +((b/2xQ)2 )2 +c,
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45
where a, b, c, and Q are the fitting parameters (here the Q value is found to be 34471.62
and b, indicating the resonance frequency, is 9502.788 MHz in this fitting). The Q value
associated with the A2 0 signal for these two temperatures was then determined.
To
convert the Aim signals into Q values for intermediate temperatures we interpolated using
the following expression
A2c3 = a Q 4 + b
(MO)
The surface resistance and the surface reactance are determined using equation
(1-8) once the Q values have been obtained from equation (1-10). Both components o f the
surface impedance are normalized to unity above Tc and zero at T —> 0.
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46
a = -0.04141117
b = 9502.788
c = 0.1433712
Q = 34471.62
5
3
Formula: A«, - a*(/e-b )/((_£ -b)A2
+ « b/2*01A21A2 H c
1
( a.u.)
1
•3
•5
9502 .2 1 0
9502.455
9 5 02.700
9 5 0 2.945
9 5 03.190
f c (MHz)
Figure 1-13. A typical fit to the derivative of a Lorenzian curve (Q =
34471.62 in this case). Open circles are data points. Solid line is the fit.
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47
4.3 Results and Discussions
The normalized surface resistance of each o f the TEoip (p = 1, 2, 3, and 4)
modes is presented in figure 1-14. Each curve is basically flat above Tc, which implies
that the temperature dependence o f the surface resistance is very weak in the normal state
at low temperatures.
The solid arrows indicate anomalies observed in the surface
resistance curves which we identified as collective mode resonances. Successive curves
have been shifted upward for clarity in the figure. The results are quite reproducible.
Here the actual procedure for normalization and scaling is not critical since we
will primarily be concerned with the relative positions o f these anomalies with respect to
temperature and to frequency. The solid arrows in the figure point to the anomalous
peaks observed. The dashed arrow shows the position o f the peak that is predicted but
not clearly observed, which will be discussed later.
To determine more precisely the position, T*, of each peak, the monotonic
background was subtracted from each curve. The procedure is described as follows. The
17.22 GHz curve was fitted with the Mattis-Bardeen formula, assuming the London limit.
The only fitting parameter in this process is
c = A (0)/kBTc,
(I-11)
where A(0) is the value o f the gap function at T = 0, kB is the Boltzmann constant, and Tc
is the transition temperature. Figure 1-15 shows the fit to the 17.22 GHz data. The fitting
parameter c for this curve is found to be 1.45. We then use the same value o f c to
generate the monotonic backgrounds for the other frequencies. The anomalous peaks are
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48
1.4
27.09 f23.00i19.58 17.22(GHz) .
1.2
1.0
Rc/ R
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Temperature (K)
Figure 1-14. Normalized surface resistance as a function
o f temperature. The arrows point to the positions o f the
anomalous peak(s) for each cavity mode. The dashed
arrow shows the position of a predicted peak which is not
clearly observed.
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49
revealed clearly as shown in figure 1-16 after the subtraction o f the monotonic
backgrounds.
Introducing a normalized temperature t* s T* / Tc, the peak maxima for 19.58
GHz and 23 GHz can be readily read from figure 1-15; they occur at t* = 0.79 and 0.69,
respectively. At the highest frequency, 27.09 GHz, two anomalies are observed: one
anomalous peak is clearly seen at higher temperatures at a t* o f 0.74; we associate a
second peak with the broad plateau located at a lower temperature with a t* o f 0.28. The
determination o f the position of
for the plateau feature will be discussed shortly.
Before subtraction o f the background, the anomalous peak for the 17.22 GHz curve is
nearly obscured by noise near Tc. After subtraction, a broad feature is seen with a /* o f ~
0.85. This is indicated by the position o f the dashed arrow in figure 1-14.
Each surface resistance peak is accompanied by an “S”-shape surface reactance
anomaly, as expected for a resonant phenomena.
In figure 1-17 is shown both the
normalized resistance and reactance for a single mode, TEou- Similar features are also
observed for lower frequency modes. Though not as pronounced as in TEou, these
anomalies are still well-defined.
The “S”-shape anomaly in the surface reactance
indicates a transition from an “inductive" regime to a “capacitive” regime upon warming.
This is what one would expect when passing through the resonant frequency o f a (series)
resonator from high to low frequencies. The peak maximum, T* for a resistance peak is
identified by the corresponding “zero-crossing” point o f the “S”-shaped reactance
anomaly (indicated by the downward arrow in figure 1-17). The value o f t* (0.28) for the
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50
1.0
0.8
TE0U (17.22 GHz)
Measured
Fit to the Mattis-Bardeen theory
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
T /T
0.8
1.0
C
Figure 1-15. Surface resistance curve fitted with the
Mattis-Bardeen theory for the TEou mode. The fitting
parameter c is found to be 1.45 for this case.
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51
0.18
27.09 GHz
0.16
0.14
0.12
0.10
23 GHz
19.58 GHz
0.08
-Fit
R/R
0.06
0.04
0.02
0.00
0.02
-
-0.04
17.22 GHz
-0.06
-0.08
-
0.10
-
0.12
0.0
0.2
0.4
0.6
0.8
1.0
T/Tc
Figure 1-16. The normalized surface resistance curves o f TEoip
modes (p = 1, 2, 3, and 4) vs. the reduced temperature after
subtraction o f the backgrounds. The anomalous peaks can be
easily observed.
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52
plateau feature at frequency 27.09 GHz was determined in this way as the midpoint o f the
corresponding reactance “S”.
Our interpretation o f the anomalous resistance peaks appearing at 19.58 GHz,
23.00 GHz, and the broad plateau at 27.09 GHz is that they arise from absorption by a
single collective mode with a temperature dependent resonant frequency
C 1 q (T ).
With
this picture in mind, the temperature dependent resonant frequency displays a similar
behavior with that o f the BCS gap function A(t). The high temperature resistance peak
seen at 27.09 GHz (represented by the hallow square in figure 1-18) is then presumably
indicative of a separate, higher frequency collective mode.
In analogy with collective mode behavior in superfluid 3He, we may make the
assumption that the temperature dependent collective mode frequency is proportional to
the gap function by the following relation,
hQ0(t) * bA(t) ,
( 1- 12)
where h is the Plank’s constant and b is a proportionality constant, which we assume to
be temperature independent. We therefore expect to see an absorption peak at t* with a
resonant frequency f(t*) given by hf(t*) « bA(t* ), or,
(1-13)
where ka is the Boltzmann constant, 8(t*) = A (t*)/A (0) is the gap function normalized to
unity at T - 0, and c is the scaling factor defined by equation (1-11), which gives the
value o f the gap function at zero temperature. The normalized gap function 5(t) can be
calculated numerically from the usual BCS equation (1-3). The factor be was used as a
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53
1. 2
1.00
1.0
TE014(~ 27 GHz)
0.8
0.95
,C
0 .4
0.90
0.2
0.85
0.0
0.0
0.2
0 .4
0 .6
0.8
1.0
Temperature (K)
Figure 1-17. The normalized surface resistance and
reactance measured at 27.09 GHz.
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54
fitting parameter to adjust the peaks to coincide with the
o f the normalized resonant frequency,
C2o(t*) / £2o(0),
to
8 (t)
8(t)
curve at t* . The resulting fit
curve is shown in figure I-
18. The open circle in the figure is the predicted position o f the peak at frequency 17.22
GHz but is obscured in the resistance curve. The value o f be is found to be 1.442, and the
collective resonant frequency at zero temperature is estimated to be at 27.19 GHz.
Calculations on a /7-wave ABM-like state [27] yield a collective mode with
energy hcocoii ~ 1.2 Ao, where
Ao
is the maximum value o f the gap over the Fermi surface.
Comparison with equation (1-12) gives b ~ 1.2. Combining this value o f b and the value
be ~ 1.442 fit from our data we obtain c —1.2. The BCS theory for the isotropic state in
the weak-coupling limit gives c = 1.76. For an unconventional superconductor, however,
one should not be surprised to find a significant deviation o f c from this value.
In a specific heat investigation, Ott et al. [39] analyzed their data in terms o f an
ABM state involving a single parameter. The value they obtained for c was —1.65 in the
weak coupling limit and —1.865 in the strong coupling limit. A naive adoption o f the
Mattis-Bardeen formula to our 17.22 GHz, as described above, yields c = 1.45. This
result does not account for anisotropy o f the gap. Including the effect o f the nodes would
increase the value o f c [40]. Note also that the Mattis-Bardeen formula employed here is
derived in the weak coupling limit. Recent point-contact spectroscopy measurements on
UBen [41] suggest c > 3.35 and support the argument for strong coupling effects in this
material. Higher c values imply lower resonant frequencies for the collective modes
relative to the energy gap. The large c value o f 3.35 would imply a rather low-lying
collective mode with energy hQo « 0.43A.
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55
1 .0
27.09 GHz
Q o (t)/Q o (0 )
0.8
23.00 GHz
-
19.58 G H z
(17.22 GHz)
0.6
0.4
0.2
0.0
0.0
0.2
0 .4
0.6
0.8
1.0
T /T C
Figure 1-18. Proposed temperature dependence of the
collective mode frequency, assuming it to be
proportional to the gap function. The solid circles
indicate the resistance peaks observed. The open
circle is the predicted peak position at 17.22 GHz.
The hollow square is the high temperature peak
observed at 27.09 GHz.
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56
The damping o f the order parameter collective resonance depends on the gap
structure. The pair-breaking caused by the existence o f nodes in the gap will lead to a
line width o f the resonance following a power law in temperature. It is also expected that
the collective resonance will be damped by the effect o f impurities and other defects.
The theory is not well-developed at this time, but a reduction o f the damping at
temperatures T « Tc has been suggested [42]. This is qualitatively consistent with our
observations.
The width o f the observed absorption peak in our resistance measurements can be
taken as a measure o f the damping strength. Each peak was fitted with a Lorenzian line
shape with the center frequency defined by equation (1-13); the peak width (in the
frequency domain), will be inversely proportional to an effective relaxation time x. A
physical interpretation o f t requires knowledge o f the actual damping mechanisms, but it
should depend strongly on the concentration o f impurities and defects, and therefore on
the normal state carrier lifetime. The relaxation times are calculated to be: x = 220 x 10~
l2, 130 x 10‘12, and 70 x 10'12 s at
fo
= 27.09, 23.00, and 19.58 GHz, respectively. For
fQ
= 17.22 GHz, the estimated x is ~ 10 x 10'12 s. Here the relaxation time is found to
decrease dramatically as the measurement frequency f o is decreased. Figure 1-19 shows a
plot o f the inverse of the relaxation time x against temperature.
Due to a lack of
sufficient data points at low temperatures, it is unclear whether it is a power law or
exponential function of temperature. Furthermore, we find that if the relaxation times are
decreased by an order o f magnitude, the calculated resonance peaks would be smeared
beyond recognition. This also points out the importance o f high quality crystals.
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57
J__________|__________ t__________|__________ |__________L
0 .1 0
1 7 .2 2 G H z
-
•
1 / t (xlO 12 sec'
0.08 ~
0.06 ~
0.04 -
0.02
1 9 .5 8 G H z
~
2 3 G
2 7 .0 9 G H z
H
z
#
*
0.7
0.8
•
0.00
-
0.2
0.3
0.4
0.5
0.6
0.9
t*
Figure 1-19. A plot o f the line width vs. the position of
the resonance peak for different frequencies.
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C hapter 5
Introduction
In the March o f 19 8 1 G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel [43,44] at
the
IBM Zurich Research Laboratory combined the concepts o f scanning, vacuum
tunneling, and point probing to create the first high-resolution surface microscopy,
scanning tunneling microscopy (STM). One of the most important features o f the STM is
that it gives local information on the surface properties o f materials. Also, because the
energy of the tunneling electrons is low, the interaction between the probe and the sample
is nondestructive, combined with the three-dimensional character o f the imaging, the
STM has proven to be a very powerful and unique tool in nanometer-scale science and
technology.
With careful vibration isolation and tunneling tip preparation, atomic
resolution can be achieved.
STM can be used as a very powerful tool in the study o f superconductivity. It can
provide information on the quasiparticle density of states and give a spatial image of the
surface. There has been much interest in STM spectroscopic and topographic studies at
cryogenic temperatures in the last decade. Hess et al. [45,46] developed a low
58
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59
temperature STM to study the layered superconductor NbSe 2 . This was followed by
STM studies o f other novel superconductors, such as the high-Tc and heavy fermion
superconductors [4,5,6].
However, obtaining high quality tunneling data on these
materials remains problematic.
The design of a high quality STM working at cryogenic temperatures is
challenging and there have been several different approaches [47-50].
Factors which
must be considered are: the tip-to-sample approach mechanism, the vibration-isolating
strategy, noise isolation, stability o f the electronic control, and the method o f
refrigeration (commonly evaporative cooling o f 3He).
The low temperature scanning tunneling microscope (LTSTM) developed by our
group was designed with both spectroscopic and topographic studies o f superconductors
at millikelvin temperatures in mind. In this thesis we present the design o f our LTSTM
which can be mounted on one o f our dilution refrigerator cryostats. The system employs
a novel and effective vibration-isolating mechanism, and fits within a 1.68-inch diameter
cylindrical space inside the radiation shield o f our dilution refrigerator. The use o f a
unique piezoceramic-cross greatly extends the scan range at low temperatures, making it
possible to study the surface physical properties o f a material at both large and small
scales.
This part of the thesis is organized as follows.
In chapter 6, we will briefly
discuss the basic operating principles of STM and the theory o f the associated
spectroscopy.
In chapter 7, we will describe the setup o f our LTSTM, including the
cryogenic system, the coarse approach mechanism at low temperatures, the methods used
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60
to improve the images, the electronics, and the control system. STM scanning images
will be presented as a part of the discussion.
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C hapter 6
Basic Theory and Operating Principles of
STM
6.1 Basic Theory
6.1.1 Elastic Tunneling Through a One-dimensional
Rectangular Potential Barrier
Tunneling phenomena can be treated by two different theoretical approaches.
One starts with the time-independent Schoedinger equation, constructs the solutions both
inside and outside a potential barrier, and matches the boundary conditions at an
appropriate point.
The other approach starts with the time-dependent Schoedinger
equation and uses Fermi’s golden rules. We will start with the first approach for the case
61
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62
o f an electron tunneling through a 1-D rectangular barrier.
In elementary quantum mechanics, one solves the problem o f an electron
on a rectangular barrier with height
im p in g i n g
Vb
and width
d.
The barrier transmission coefficient, T, is defined as
T = ^ r = \D\2,
J,
(fi-1)
where j t is the transmitted current density, y", is the incoming current density, and | D \2 is
the transmission probability. The transmission probability can be determined by solving
the Schoedinger’s equation subject to boundary conditions, which gives
. ,
D=e
where
k
ktc
------------ 7—.— r~->------ Tx------7— r ,
2£fccosh(fa/) —i{ k ~ — K ~ ) s m h ( K d )
2
( 11 - 2 )
= [2m( Vb —£)]'/2 /
Therefore,
1+ (A:2 +Kr2)" /(4 ^ 2/c2)sinh2 (fo/)
For a strongly attenuating barrier, i.e. when
Kd
»1, we may approximate equation
(II-2) as
16At2x 2
-e
_lKd
(fi-4)
( k '- + K '-y-
Thus the transmission coefficient, and hence the tunneling current, for a strongly
attenuating barrier behaves qualitatively as
/ oc e~lKd
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(II-5)
63
From equation (II-5) one sees that the tunneling current is very sensitive to the width o f a
potential barrier. For this reason, STM is very sensitive to the tip geometry, the local
surface properties, and the tip-surface spacing; this in turn results in a high sensitivity to
the surface topography.
Lateral variations in the tunneling probability, e.g., between
oxide and metallic regions, add further contrast.
6.1.2 Tunneling in More General Cases
The WKB approximation can be applied to the tunneling problem for a potential
barrier V(z) o f arbitrary shape. The probability that an electron can penetrate the barrier
is expressed as
(h-6)
where sj and s? are the classical turning points o f the barrier [51].
When considering tunneling between two metals, the electron distribution
function in the two metals has to be taken into account; in addition, to sustain a current, a
bias voltage must be applied. A one-dimensional barrier is an appropriate model in the
case of planar tunnel junctions. However, when replacing one of the planar metals with
a sharp tip geometry, as in STM, a three-dimensional potential barrier must be
considered. It becomes awkward to apply the wave-matching method used in the
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64
previous subsection to more complicated tunneling cases. To address this problem, we
apply perturbation theory to the time-dependent Schoedinger equation.
In 1961 Bardeen [51] proposed a strategy for treating tunneling problems. He
introduced a ‘Transfer Hamiltonian”, which describes electron tunneling from one
electrode to another. In Bardeen’s formalism, the tunneling current in an STM can be
evaluated according to
(E-7)
where f(E) is the Fermi distribution function, V is the bias voltage, M^v is the tunneling
matrix element between states t//u,
with energies EMand Ev corresponding to the two
metals (sample and tip), and the delta function describes the conservation of energy for
elastic tunneling. This equation is the starting point for Tersoff et el. [53] and Lang’s
[54] theories of STM imaging. In the limit o f small voltages and low temperatures, the
tunneling current can be approximated as
The problem remaining is to find the tunneling matrix elements, M ^ . Within Bardeen’s
formalism, the tunneling matrix elements can be determined by a surface integral,
n2
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(n-9)
65
The tip states can be expanded in terms o f spherical harmonics as
(n-10)
l,m
where K/(fcp) is the spherical modified Bessel functions o f the second kind. Here
k
=
(2mcp) l/2/ h ; cp is the work function; p = |r - r 0| , and r0 is the center o f the apex atom of
the tip. The leading term, the s-wave o f the tip, is then given by
X s = C K 0Y 00= C — e-Kp.
(E -ll)
Kp
Note that the Green’s function, G (r-ij)) = e~'clr"b' /47r|r
which satisfies the equation
(v 2-*c2) c ( r - ^ ) = -<5(r-^), can be written in terms o f the zeroth order spherical
modified Bessel function as G ( r - ^ ) = -^ -K 0(fcp). Therefore, the s-wave tip state can be
4 7T
expressed in terms of the Green’s function as
* ,(r) = — G (r-f„ ).
fC
(n-12)
Substituting equation (11-12) into (EL-9), the s-wave tunneling matrix element becomes
Km
^ r.)v
(n-13)
'
From equation (11-13) we see that the matrix element is simply proportional to the
amplitude o f the sample states i/V at the position r0 of the probe. Equation (H-8) can
qualitatively be expressed in the form
1
- E F) = p(vQ, E F) .
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(TI-14)
66
From equation(II-14), we see that STM, ideally, is a measure o f the local density
o f states o f the sample surface at the position of the tip.
6.1.3 Scanning Tunneling
Superconducting State
Spectroscopy
in
the
Scanning tunneling spectroscopy (STS) involves measuring the current-voltage
tunneling characteristics, and can give us information on the local electronic density of
states of the sample (as illustrated in equation II-14).
The superconducting critical
temperature o f tungsten, which is usually used as the probe in STM, is about 12 mK [55],
Therefore, when using STS to study superconductors at low temperatures, one has
electrons tunneling between a normal-metallic tip and the quasiparticle states o f the
superconductor.
The dynamic conductance at a particular location r can be expressed by
^ f c £ l o c - ] | y v( e ,x ) |/ f r | * f0 ? - e ^ ,x - r X | «5(e>7,(fr- e V )
(11- 15)
x 4 f { s - e V ) - f { e ) ] ds
d(eV )
where H j is the transfer Hamiltonian, r is the in-plane coordinate of the tip, x is the
coordinate at the sample su rfa ce ,/(e) is the Fermi distribution function, and s is the
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67
reduced energy o f the states. At low temperatures, d\f(s - eV)—
f(s)] /d(e V) behaves like a
delta function with a sharp peak at s = e V . Furthermore, the moveable tip with in-plane
coordinates r results in a tip wave function o f the form %, (x - r) in the overlap term
|(^s(x)|/7r |Z(x —f))|-, and acts as a spatial, delta-fimction probe on the sample surface at
position r . From a similar argument as in the previous subsection, in the limit o f low
temperatures and for an ideal tip, the dynamic conductance is then a measure o f
|i//^(eF,f)|X(eF) .
6.2 Principles o f STM Operation
6.2.1 STM Im aging
The basic idea o f STM operation is quite simple. A very fine tip is used to
probe the sample by bringing it to within a few angstroms o f the surface. With an applied
bias voltage between the tip and the sample, a tunneling current is established. This
position dependent tunneling current is amplified and either recorded by a computer or
used as a feedback signal, as will be described shortly. The same computer produces
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68
appropriate voltages which scan, in a raster-like fashion, the tip over the surface o f the
sample using a suitable "motor”; usually a piezoelectric driver is employed either in a
single-tube or in a tripod configuration. In the constant current mode, a feedback loop
retracts or elongates the piezo to keep the position o f the tip tunneling current constant at
a preset value. The voltage applied to the piezo is a measure o f the local vertical position
of the tip and this data is acquired by a computer along with the in-plain coordinate o f
the tip, and displayed as a space image (3D display) or as a gray-scale image (2D)
involving the pixel intensity.
An alternative mode o f operation is with the feedback turned off. The tip then
scans over the surface o f interest, and the position dependent tunneling current is
recorded. This operational mode is called the “constant height mode”. Small features of
the surface are then reflected as fluctuations of the tip current.
The constant height mode requires a relatively smooth surface and care must be
taken in aligning the tip position relative to the sample surface. For most of the cases, the
constant current mode is preferred.
6.2.2 Operational Principles of STS
To obtain the position dependence of the dynamic conductance dl / dV, a small
AC modulation voltage, VAc> is added onto the DC bias voltage, V. This result in an AC
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69
component o f the tunneling current response and that is proportional to the dynamic
conductance. The AC current response can be measured using a lock-in amplifier. There
are several steps then need to be accomplished in each data acquisition cycle.
procedures are as follows.
The
Initially, at a specific in-plane coordinate position, the
feedback loop is activated. A large bias voltage, Vp, is applied so that the spacing
between the tip and the sample can be set. After the tip-to-sample spacing is set, the
feedback loop is then deactivated.
Different bias voltages are then applied and the
‘dynamic conductance’ is recorded. In this way one obtains dl/dV versus V at a specific
position. After each cycle, the feedback loop is reactivated and the bias voltage is set to
VP again. The tip is then moved to a new position on the sample and same procedures are
repeated. Continuing this process we obtain the bias voltage dependence o f the dynamic
conductance as a function o f position on the sample surface. The ‘dynamic conductance’
is normalized to a reference value, which is the dynamic conductance at some bias
voltage,Vn, which is much larger than the gap in the superconductor.
With the principles of operations stated above, two STS data acquisition modes
can be performed with our TopoMetrix instrument: Current Imaging Tunneling
Spectroscopy (CITS), where the bias voltage is varied between steps, and Difference
Imaging Tunneling Spectroscopy (DITS), where the bias voltage remains constant and
the Z height is varied between steps.
Each measurement may be preceded by a step
change in voltage (height) for CITS (DITS). The measurements at a single point are
stored as several “layers” o f data.
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70
There is a more direct way that can be used to image the flux lattice in a type-II
superconductor, without using the time-consuming sample-hold procedure mentioned
above. Here the applied bias voltage is reduced to a value comparable to the relevant
superconducting energy gap and the whole area is scanned in the “constant current mode”
with the feedback loop activated. The tip will retract above relatively higher conductance
areas. This leads to the simple expectation that a superconducting-like spectrum will be
observed between vortices while a metal-like spectrum will be observed in the core areas
because o f the suppression o f the gap in these regions. The flux pattern is then revealed
as an STM gray-scale image. Note that this operation is applicable to samples with a
relatively smooth surface. Figure (II-l) shows the Abrikosov flux lattice observed in an
STM image of 2H-NbSe2 by Hess et al. [6] (Tc of NbSei is 7.2 K). This image was taken
at bias voltage o f 1.3 mV and a magnetic field of 1 Tesla at a temperature o f 1.8 K.
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71
6000A
Figure H-l. The Abrikosov flux lattice image
of NbSej at 1.8 K. The bias voltage is 1.3
mV. The 1 T field induced a vortex spacing of
489 A. This image is taken from H. F. Hess et
al. [45].
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Chapter 7
LTSTM System
The LTSTM block diagram is illustrated in figure II-2. The STM housing is
attached to the mixing chamber of a dilution refrigerator via a specially-designed
vibration-isolating mechanism. A novel long piezoceramic cross was used as a driver to
control the spacing between the tip and the sample. At room temperatures the tip was
pulled away from the sample surface by applying a voltage to the long piezo cross. This
serves the purpose of preventing the tip from crashing on the sample surface due to
thermal contraction when the system is cooled from room temperature. A bias voltage
between the tip and the sample was applied by a battery-driven power supply. At low
temperatures the tip and sample were moved close to each other to establish a tunneling
current by the same long piezo cross.
The tunneling current signal was transmitted through a well-shielded lowtemperature coaxial cable out of the dilution refrigerator. A preamplifier was built to
amplify the tunneling current. From the discussion in the previous chapter, the tunneling
current varies exponentially with the spacing between the tip and the sample. Therefore,
the log of the tunneling current was evaluated with a computer in order to achieve a linear
72
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73
output voltage-displacement behavior. A control system then processed the signal and
generated driver voltages that were applied to the scanners. In the '‘constant current
mode”, a feedback loop with a PED feature was activated. The tunneling current was
compared with a preset current in the computer program and the error signal was
amplified and applied to a piezoelectric ceramic driver to maintain a constant tip-sample
spacing. The piezos were also used as x and y scanners and were driven by a set o f
coupled ramp voltages which rastered the tip across the surface o f sample. A very large
lateral range can be achieved.
The data set consisting o f the error signals and the
positions of the tip were processed by a computer and the topographic image of the
sample surface was then generated.
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PR EAM P
COMPUTER
ELECTRONIC
CONTROL
UNIT
D IL U S IO N
FRIDGE
STM
II v
OPAMP
TMX1000
STAGE
Figure II-2. Block diagram of our LTSTM.
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7.1 Cryogenic System
The system used to cool our LTSTM involved a cryostat constructed at
Northwestern which employs a commercial dilution refrigerator. The dilution stages,
including the still, sintered silver heat exchangers, cold plate (gold plated), impedances,
and mixing chamber (with a gold plated bottom plate for mounting samples), was a
model MK-400 DR purchased from Janis Research Company, Inc., which has a
refrigeration capacity o f 400 pW at 100 mK.
The cryostat is also equipped with a
superconducting magnet, which has a maximum field of about 6 Tesla.
All pumps were placed in a separate pumping room, where a layer of soundproof
acoustic board covered the walls. This pumping room is located on a different floor from
the experimental setup. The pumping lines, connecting the pumps and the rest o f the
cryogenic system are buried inside concrete walls which are a part o f the building. This
provides excellent vibration isolation between the pumps and the cryostat. The cryostat
also involves additional plumbing and a home-made gas handling system.
One o f the important features for this cryostat is that it can easily be adapted to
various other experiments.
One of the other experiments that can be performed is
ultrasonic measurements in which the sample may be rotated relative to the magnetic
field, which we briefly describe. An extension stage, containing four copper rods and a
copper plate, is attached to the bottom o f the mixing chamber. To insure good thermal
contact, all the copper used was OFHC. A sample was mounted inside a copper cell,
which could be rotated via a pair of bevel gears. The orientation was controlled by
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76
rotating by a long shaft which extended out o f the cryostat. This shaft is actually made
up o f several short pieces of stainless steel tubing connected by miniature universal
joints. With a minimal modification o f the cryostat, the setup can be easily changed to
accommodate the LTSTM stage, simply by replacing the ultrasonic stage b y the LTSTM
stage. The design o f our LTSTM will be described shortly.
Pre-calibrated CR50 germanium resistors from Cryo Cal and D ale 1000 RuCL
thick-film resistors were used as the primary thermometers. The MK-400 DR dilution
refrigerator also comes with a still heater, a still capacitance level gauge, a still resistance
thermometer (Speer 100), a mixing chamber heater and a resistance thermometer. Some
Matzshita 100-Q carbon resistors, which were repeatedly cycled between room
temperature and liquid-nitrogen temperature before using, were also employed as
secondary thermometers. All of the thermometers and heaters were positioned within
copper holders and thermal contact was made using GE 7031 varnish, or Apiezon
vacuum grease. The resistance of the major temperature sensors was determined in a
four-wire configuration, using an AVS-45 automatic resistance bridge. A temperaturecontroller was employed which consisted of a temperature sensor, a heater located near
the sample and a current source controlled by a personal computer operating in a PID
mode.
All electrical wires, which were connected on thermometers and heaters, were
twisted in pairs.
We also employed home-made low-temperature coaxial cables to
transmit electrical signals to and from the cold parts of the cryostat. These coaxial cables
were designed with a low heat leak and a good electrical conductivity in mind. The outer
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77
conductor o f the coaxial cable was made by forcing out the flux core from a commercial
Pb-Sn solder (which is a superconductor with Tc ~7.2K). This was accomplished by
putting several 100 to 150 cm long pieces o f commercial, water-soluble organic core
solder in a glass beaker filled with water and heating it to a temperature between 120 and
160 °C. One end o f the solder was pressurized with about 30 psi o f gas. It took several
hours for the flux core to flow out. Afterwards the solder capillaries were flushed a few
times with solvent. A piece o f 5 mil. superconducting NbTi wire was used as the center
conductor o f the coaxial cable. It was then filled with silicone (100,000 cst. viscosity).
Indium was used for connecting the outer conductor o f the coaxial cable, with the
soldering iron temperature set below 300 °F.
The coaxial cables and electrical wires were thermally anchored at different
points having successively lower
temperatures. This was achieved by
winding a considerable length of the
cable or wire tightly around a copper
post which was then varnished for
thermal contact. Figure 11-3 shows
Figure H-3. A sketch o f the
thermal anchoring post.
such a thermal anchoring post. The
upper section o f the post was threaded
to form a groove for winding and the
lower part was threaded for screwing onto the anchoring points.
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78
7.2
STM Scanners
7.2.1 Piezoceramic Cross
One o f the special features o f our LTSTM design was the use o f a long
"piezoceramic cross" as a scanner. This piezoceramic cross was obtained from Ragspace
Corporation and has a “cross-like” cross section in the plane perpendicular to its long axis
(see figure II-4(a) and (b)).
The direction o f polarization o f each o f the legs o f the cross is defined by a vector
pointing from the positive electrode toward the negative electrode as indicated in figure
Et-4(b). The four quadrants of the cross are coated with aluminum to form the electrodes.
When an electric field is applied parallel to the polarization direction, a strain was
developed which, in turn, causes the cross to deform. The piezoelectric constants are
defined by the strain developed by unit applied electric field, and which depend on the
orientation to the poling axis, are expressed as dy, where i and j denote the components
o f the axes: 1 corresponds to the x-axis, 2 corresponds to the y-axis, and 3 corresponds to
the z-axis. Conventionally, the direction
o f the polarization is defined as the 3 axis. The first subscript, i, identifies the i^
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79
+
m l 'm
+
Direction o f
Polarization
(b)
Electrodes
(a)
Figure II-4. Diagram o f the piezoceramic cross: (a) geometry
of the piezoceramic cross, (b) cross-sectional view. The arrows
indicate the direction o f polarization.
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80
component o f the applied electric fields and the second subscript, j, identifies the j*
component o f the response. The piezoceramic cross we used was machined from PKR7M ferroelectric and has piezoelectric constants d3 i = —3.50 A /V and d 33 = 8 A /V . The
dimensions of the cross section are 0.3-inch by 0.3-inch and 3.5 inches long, and it has
displacements of 800 n m /V in x and y directions and 69.4 n m /V in the z direction at
room temperatures. This results in a scan range about one order o f magnitude larger than
the commonly used PZT-5H tube, which will be described in the next subsection.
The cross has a breakdown electric field of 1.5 K V /m m . The room temperature
resistance of the device was 300 MQ and the Curie temperature is 175°C. The lowest
resonant frequency is 15 kHz. The hysteretic behavior of the cross was also studied:
figure H-5 shows the hysteresis of the cross in the z-direction over a range o f 300 volts.
We operated the cross in the range —400 to 800 volts.
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Extension
20
10
B
3
0
-1 0
-2 0
- 1 5 0
- 1 0 0
- 5 0
0
50
100
15(
Voltage (V)
Figure H-5. Hysteresis of the piezoceramic cross in the z-direction.
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82
7.2.2 Piezoceramic Tube
For the short-range scanner, we used a piezoceramic tube from Morgan Matroc,
Inc.. This piezo tube is machined from PZT-5H, which has the lowest Curie temperature
o f the PZT-5 family with Tcune = 195°C. The piezo constants for the tube are d3 i = —2.73
A / V and d33 = 5.93 A / V at room temperatures. Figure H-6 shows the change o f the
piezo constant, d 3 i, in the temperature range from 400 K to 4 K. Here we see that d 3 i at
liquid He temperature is approximately one sixth o f its room temperature value. The
resistivity o f this material is greater than 1011 Cl at room temperatures.
The lowest
resonant frequency is ~64.3 KHz. The operating voltage is limited to about 500 volts.
A schematic drawing o f the piezo tube is shown in figure H-7. It has a 0.25-inch
outer diameter, 0.024-inch wall thickness, and is 1-inch long, which gives displacements
o f 15 n m /V in Z, and 19.2 n m /V in x and y at room temperatures. The outer and inner
walls are silver coated electrodes. The direction o f the polarization points inward (the
direction o f the arrow in figure H-7), i.e. the inner wall poled negative.
The outer
electrode is separated into four equal quadrants. This is achieved by etching four strips in
the electrodes with nitric acid with the rest o f the area coated with a layer o f finger nail
polish. This strip electrode configuration o f the tube enables it to be scanned in all three
directions.
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- 0 .4
- 0 .3
-
0.2
-
0.1
0
-2 7 3
-2 0 0
-
100
100
200
300
Temperature (°C)
Figure II-6. The temperature dependence of dsi
for a PZT-5H piezoceramic tube.
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84
0.25"
-H
0.024'1
Direction of
polarization
Electrodes
Figure H-7. The piezo tube used in our LTSTM. Also
shown are the dimensions o f the tube. The arrow
pointing from the outer wall to the inner wall shows
the direction of the polarization.
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85
7.2.3 Repolarization o f the Piezos
Periodically, the piezoelectric ceramics need to be repolarized due to the
application o f excessive electric fields, high temperatures approaching the Curie point, or
applied high mechanical stress.
In addition, the polarization o f the piezo will decay
approximate exponentially with time and also with thermal cycling. To repolarize the
piezos, high voltages (1.2 kV for the piezoceramic cross, 800 V for the piezoceramic
tube) were applied to the electrodes o f the piezo for a few hours. This was carried out in
an evacuated glass tube with the ceramic heated by an infrared light to the desired
temperature.
7.3 Configuration of the LTSTM
In a previous configuration o f the LTSTM [56], a short piezo ceramic cross was
mounted vertically under a long piezoceramic cross as shown in figure U-8.
In this
configuration, the tip was mounted perpendicular to the axis o f the cross and the plane o f
the sample was mounted vertically. Here the long cross controlled the XY scan as well
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86
Long piezo cross
(X and Y scanner,
coarse approach)
Tip
Short piezo cross
(Z, coarse, and
feedback)
Sample holder
Sample
Figure II-8. Configuration o f two piezo crosses. Here the short
cross was connected vertically under the long cross. The tip was
mounted horizontally and perpendicular to the sample surface,
which was mounted vertically.
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87
as the tip-sample coarse approach; the short cross was part o f the feedback circuit which
controlled the z-position o f the tip. There are many advantages to this design.
The
majority o f the thermal contraction occurs parallel to the vertical axis (i.e., in the plane o f
the sample) which greatly reduced the chance o f a tip crash during cooling.
Large
vertical features can be imaged since the Z-range is greatly increased (the maximum z
range is ~1.1 mm if both o f the crosses were operated in the feedback mode, while the
maximum z is ~ 0.27 mm when only the short cross was used); a large scanning area can
also be achieved since the x and y directions have a very large range (the maximum
•y
scanning range in this configuration is ~ 1100 x 95.2 pm-). In the same sense, the coarse
approach mechanism has a very large range. With this configuration the tip does not
have to be manually moved too close to the sample, as with most other designs, which
further reduced the chance o f crashing the tip.
This also provided a mechanism to
compensate for thermal contraction during cooling.
Figure LI-9 shows the image o f a "raised-square" silicon grid coated with Au.
Here an "S"-shape non-linearity of the image is clearly seen. This was probably caused
by cross-talk between the X and Y scanners.
Creep and hysteretic behavior o f the
piezoceramic cross may also result in this distortion of the image. Another cause o f the
non-linearity could be an asymmetry o f the cross itself, i.e., a non-linear response for
fields opposing the polarization direction o f the cross. We also found that all four plates
o f the cross did not have the same thicknesses; even the thickness in a given plate was not
very uniform.
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88
Figure II-9. An image o f a "raised-square"
sample. The range is about 35 x 35 pm-.
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89
To solve this non-linearity problem, one method is to correct the non-linearity
electronically.
We built an operational amplifier which provides non-linear drive
voltages to the scanners, which will be described shortly in the section on electronics.
Another method is to use computer software to correct the image after acquisition
(this method has not been used to date), by measuring a known sample and calculating
the necessary display parameters. The same parameters could then used to display an
unknown specimen image. There are other drawbacks to the above configuration. The
geometry o f the configuration is more sensitive to external vibrations, which can easily
cause overshooting o f the scanner and may result not only in a blurred image but also a
crashed tip.
Here we present a new configuration o f the piezo actuators (see figure II-10). A
long piezoceramic cross is mounted vertical to the STM housing. Figure 11-11 shows a
schematic drawing o f the LTSTM housing.
It is 8 inches long and the m axim um
diameter at the bottom part is 1.0625 inches. The housing is made o f stainless steel for
structural stiffness.
At the free end of the cross an adaptor was mounted which extended horizontally
out of the LTSTM housing through window #3 (see figure 11-11). The sample was
mounted vertically on the adapter via a machineable ceramic (macor) holder. A bundle
o f thin copper wires was mounted near the sample and thermally anchored to the mixing
chamber. A l"-long piezoceramic tube was mounted on a tube holder via an insulating
macor disk (see figure 11-10); this tube extends horizontally across the LTSTM housing
through window #3. The vertical part o f the tube holder was connected to a horizontal
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90
Adapter
Long
piezo cross
Sample
Machineable
ceramic
Short
piezo tube
Tunneling
Tube holder
Base holder
Microtranslator
Figure 11-10. A schematic drawing of the new
configuration o f the piezo actuators. The base
holder is clamped in the hole of the housing
shown in figure 11-11.
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91
Hole for the holder of
the long piezo cross
<-
Window #1
<-
Window #2
Window #3
Window #4
Figure H-l 1. The stainless steel LTSTM housing; it is 8” long,
and the maximum diameter at the bottom part is 1.0625”.
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92
section which was routed into the LTSTM housing through window #4. Windows #1
and #2 were designed to allow access when connecting the electrical leads to the cross.
The base holder (see figure 11-10) was clamped in the hole at the bottom o f the LTSTM
housing.
The horizontal section o f the tube holder was coupled to the base holder via a
micro-positioner manufactured by Charles Supper Co.. The base has dimensions 0.5-inch
x
0.5-inch and the height is 0.125-inch; it can translate in both o f the horizontal
directions. Each rotation of the thread results in a 0.01-inch linear displacement and the
m axim um
displacement is 0.1-inch in each direction. This micro- positioner served two
purposes: it was used to position the tip relative to the sample, and it could be used to
adjust the horizontal position o f the tip on the surface o f the sample.
We designate each quadrant o f both o f the scanners as shown in figure 11-12. By
applying appropriate voltages to each o f these quadrants, the scanners can execute
different displacements. The long cross was used to control the Z-course approach and
the position on the sample surface to be scanned by the small piezo. The XB electrode
was grounded. The sample can be brought close to or moved away from the tip by
applying a voltage to XD. The sample can be positioned relative to the tip by applying
voltages to XA and XC. The scanning and feedback operations are controlled by the
short tube scanner. Electrodes TC and TD were grounded. When a saw-tooth voltage is
applied to TB, the tip scans over the sample surface in the X-direction. With a ramp
voltage applied to TA, a scan in the Y direction is executed. The feedback voltage
(controlling the tip-sample separation) was applied to the inner wall of the tube scanner.
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Cross-sectional view
XA
XB
XD
XC
Sample
(a)
TA
Piezo tube
1
Q
H
DO
TC
Y
Figure 11-12. Quadrant designations on (a) the piezoceramic
cross, and (b) the piezoceramic tube.
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94
The combination o f the two piezos results in a very large scanning area o f ~ 968
pm x 85 pm which is composed o f ~ 6.68 pm x 6 pm blocks. This configuration also
permits a coarse approach range o f ~ 960 pm and a feedback range Z o f ~ 6 pm.
This new configuration o f the piezos results in greater structural stiffness of the
LTSTM, which reduced the noise dramatically relative to the previous configuration.
This design is also more compact and easily fits into the radiation shield o f our dilution
refrigerator.
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95
7.4
Vibration-isolating System
For a low temperature STM, vibration isolation is much more problematic than
with a room temperature instrument.
efficient vibration-isolating strategy.
To solve this problem, we developed a very
Figure 11-13 shows a schematic drawing of the
system.
We employed an eddy current damping mechanism in conjunction with two sets
o f springs. The damping system involved a thick-walled copper tube and a magnet which
was placed inside the tube.
The copper tube was mounted on the mixing chamber
through four copper rods. The magnet used was a cylindrical neodymium iron boride
magnet supplied by Arbor Scientific Inc.. Two magnetic steel disks were stuck to the top
and bottom surfaces o f this magnet and the extension springs were attached to them. One
set o f the springs was connected to a copper plate extended from the mixing chamber and
the other set was connected to the STM housing clamp.
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1
Extension spring
Steel disk
Magnet
Copper tube
Steel disk
Extension spring
STM housing
clamp
Figure H-13. A schematic drawing o f the vibration
isolation system.
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97
7.5 Preparation o f STM Tips
The size, shape, and chemical composition o f the tip play an important role in
STM measurements [57]. The materials favored for obtaining a sharp STM tip have been
tungsten (W), for its great stiffness, and (90:10) platinum iridium alloy(Pt-Ir), for its great
chemical inertness.
A Pt-Ir tip is usually obtained by simply cutting a piece o f Pt-Ir wire at an angle.
Though some procedures for electrochemical etching o f Pt-Ir have been proposed [58,
59], we found that a sharp tip was not easily obtained due to the inertness o f this alloy.
On the other hand, we found a very sharp tip was easy to obtain by electrochemical
etching of a piece of tungsten wire.
To prepare a sharp W tip a lamellae drop-off technique [60] was employed. In
this approach a piece o f W wire (0.25 mm diameter, 1-2 cm long) is inserted through a
ring electrode (0.7 cm diameter) constructed o f stainless steel wire (see figure 11-14). The
ring, which served as the cathode, was dipped into 2M NaOH solution (the electrolyte)
causing a liquid drop to form within the ring. The W wire (passing through the center o f
the ring) served as the anode. During the etching a DC voltage o f ~ 4.5 V was applied
between the anode and cathode [61]; the initial contact is to the top o f the W wire. An
ammeter was used to monitor the current. The current decreases dramatically just before
the tip drops off. At this point, we interrupt the etching process, rotate the wire 180°,
attach the current source to the other end of the W wire (now the top), and
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|i | i
0
Clamp
S.S. ring
with 2M NaOH
solution lamella
Glass beaker
With distilled water inside.
Soaked cotton is at the bottom
Figure H-14. A drawing o f the apparatus used for
etching W tips.
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99
resume the etching process. After some time the solution etches through the wire, and
the lower part falls away leaving a sharp tip with a short taper. This tip is then placed in
a glass beaker which was filled with distilled water which dissolves away any remaining
acid.
Figure H-15. A scan showing two different images.
Not only the shape o f the image but also the heights
o f these two areas are different.
After the electrochemical etching, the tip will be coated with a thin layer of
tungsten oxide [62]; it is thought that this oxide is one o f the primary causes o f instability
in W STM tips. In a scan o f raised squares (see figure 11-15), an abrupt change was
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100
observed.
This was possibly caused by a change o f the tip properties at a vertical
coordinate o f 10.75 pm.
The oxide layer can be removed by a 10 to 30 seconds exposure o f the tip to
concentrated hydrofluoric acid [63]. We found that more stable tips were produced by
this procedure.
7.6 Electronics
Noise and drift are the primary factors that have to be carefully considered when
designing the electronics for controlling an STM device. Although today's state-of-theart electronic components have very low noise and little drift over a long period of time
and over a wide range o f temperature, careful circuit layout and shielding are still critical
if stable performance o f the system is to be achieved. The electronic control tasks and
system monitoring were mainly achieved by using a commercially available electronic
control unit (ECU), the model TMX2000 from TopoMetrix Inc. which has 16-bit DACs.
A high impedance preamplifier is required to detect and amplify the small
tunneling currents. We built two preamplification stages for our LTSTM: one for use
with TMX1000 to TopoMetrix Explore Stage (see figure II-16 (a)), and one which could
be directly connected to ECU (see figure II-16 (b). The tunneling current sensing was
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101
Cl
C2
T u n n e lin g
c u r r e n t
L2V
T u n n e lin g
c u r r e n t
r e tu r n
C3
O f f s e t
R I = 1 M
£ 2 ,
R 2 =
R 4 = 1 0 0 Q ,
C l = 5 p F ,
2 . 6 1 k O ,
R 5 = 5 0 Q ,
C 2 = 2 2 0 p F ,
R 3 = 1 0 0 Q ,
R 6 = 5 0 Q ,
C 3 = 0 .0 1 p F .
CL
-I5V
15V
15V
—1 5 V
R l= 1 0 0 k Q
.
R
2 = l k a .
R 3 = l M
O
. C l= O .O O lu F .
Figure 11-16. Circuits diagram o f tunneling current
preamplifier.
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102
achieved using an AD549 FET input operational amplifier (op-amp), which has an ultra
low input bias current (60 fA. max), low noise (~ 4p.Vp.p at 0.1-10Hz), and low offset drift
(~ 5p.V /°C ). The tunneling current is sensed by the 1MQ resistor (in a T-network, see
figure II-16(a)). An adjustable, battery-charged bias-voltage was applied to the sampie;
<t
this resulted in a current-to-voltage gain o f 10 for the preamplifier.
0 pun
7.43 tun
14.85 urn
Figure 11-18. STM scan o f the
raised square sample.
The maximum scanning voltages from the ECU are ± 200 V. In order to achieve
the highest performance of our system (note that our piezoceramic cross can operate in
the range from —400 V to +800 V), homemade high voltage operational amplifiers were
built to drive the piezos. Figure 11-17 shows one of the high-voltage operational
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
SOOV
-W l
CL
=r -
A ll th e
r e s is ta n c e
a r e
in
R l = R 2 = R 3 = R 4 = l0 k ,
R 9 = 7 0 0 k ,
R
R 1 0 = 1 4 k ,
l 4 = R l 5 = R 1 6 =
R 1
1 O O k ,
C l= C 2 = 2 0 0 p F .
D
T r a n s is to r = M
l2 0 0 4
J H
u n it Q
.
R 5 = R 6 = 1 6 0 k ,
l= 3 0 k ,
R 7 = 1 0 M
R l 2 = 2 0 0 k ,
,
R 8 = 1 .2 M
,
R 1 3 = 3 .4 k ,
R 1 7 = 8 0 k .
io d e = lN
n p n
8 2 5 A
,
6 V
z e n e r
d io d e .
p o w e r tr a n s is to r .
Figure H-17(a). HV op amp circuits.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
+/- 10V
HV
SO k
V\AA
ECU
i/p
VWV
80 k
X/Y
+ HV
Figure II- 17(b). HV op amp circuits.
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105
am plifier
w ithin
circuits. Amplifiers A and C (see figure H-17(a)) reduced the input voltages to
the operational range o f the commercial OP27 op-amps. The OP 27 is a low noise
(80 nVp_p, 0.1-10Hz), low drift (0.2 pV/°C), high speed (2.8V/ps), operational amplifier.
Amplifiers B and D are the amplification stages which use high voltage power transistors.
These HV op-amps allow us to obtain the maximum scan range from the piezos.
Figure H-18 shows a scan o f the raised squares. We see from the picture that a
nonlinear response is observed.
In order to compensate for the nonlinearity, we have modified the HV op-amps so
that we could add or subtract a portion of the different control voltages. The circuits of
the modified HV op-amps are shown in figure 11-19. The driving voltages from ECU are
applied to the input. Here a and b are parameters that can be adjusted manually (from a
20-tum potentiometer) resulting in “hybrid” output voltages being applied to the piezos.
Figure II-20(a) and (b) show the resulting raised-squares image, and demonstrate the
effectiveness of the approach, as compared to figure H-18.
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106
SO. 6 k
+/—10V
-WWr-
SO. 6 k
-vww-
X+/-aY
3 0. b k
8 0 .6 k
+/-10V
HV
SO. 6 k
•WM—
> Y+/-bX
I00k %
|—l^Wr
SO. 6 k
-WM-
Figure 11-19. Electronic circuit o f the modified HV op
amps.
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Figure II-20a. Image o f raised
squares.
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108
p 778.79 nm
L 389.39 nm
L 0 nm
15.3S|un
Figure II-20b. 3-D image o f raised
squares.
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Reference
1. H. kamerlingh Onnes, Akad. Van Wetenschappen (Amsterdam) 14, 113 (1991).
2. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phs. Rev. 108, 1175 (1957).
3. S. Adenwalla, S.-W. Linn, Q.-Z. Ran, Z. Zhao, J. B. Ketterson, J. A. Sauls, L.
Taillefer, D. Hinks, M. Levy, and B. EC Sarma, Phys. Rev. Lett. 65, 2298 (1990).
4. F. Steglich, J. Arts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer,
Phys. Rev. Lett. 43,1892 (1979).
5. H. R. Ott, H. Rodigier, Z. Fisk, and J. L. Smith, Phys. Rev. Lett. 50, 1595 (1983).
6. G. R. Stewart, Z. Fisk, J. O. Willis, and J. L. Smith, Phys. Rev. Lett. 52, 679 (1984).
7. D. Rainer, Phys. Scrip. T23, 113 (1988).
8. P. Fulde, J. Keller, and G. Zwicknagl, Solid State Physics, Vol. 42 (Academic Press,
San Diego, 1988), p.l.
9. R. A. Fisher, S. Kim, B. F. Woodfield, N. E. Phillips, L. Teillefer, K. Hasselbach, J.
Flouquet, A. L. Giorgi, and J. L. Smith, Phys. Rev. Lett. 62,328 (1989).
10. N. W. Ashcroft and N. D. Mermin, Table 2.3, Solid State Physics (Saunders College,
Philadelphia, 1976).
11. L. N. Cooper, Phys. Rev. 104, 1189 (1956).
12. J. L. Smith, Z. Fisk, J. O. Willis, A. L. Giorgi, R. B. Roof, and H. R. Ott, Physica
135B, 3 (1985)
13. G. R. Stewart, Z. Fisk, J. L. Smith, H. R. Ott, and F. M. Mueller, in proceedings of
the 17th International Conference on Low Temperature Physics (LT17), Karlsruhe,
1984, edited by U. Eckem et al. ( North-Holland, Amsterdam, 1984), p. 321.
14. H. M. Mayer, U. Rauchschwalbe, C. D. Bredl, F. Steglich, H. Rietschel, H. Schmidt,
H. Wiihl, and J. Beuers, Phys. Rev. B 33, 3168 (1986).
109
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
110
15. F. Thomas, B. Wand, T. Ltihmann, P. Gegenwart, G. R. Stewart, F. Steglich, J. P.
Brison, A. Buzdin, L. Glemot, and J. Flouquet, J. Low Temp. Phys. 102, 117 (1996).
16. M. B. Maple, J. W. Chen, S. E. Lambert, Z. Fisk, J. L. Smith, H. R. Ott, J. S. Brooks,
and M. J. Naughton, Phys. Rev. Lett. 54, 477 (1985).
17. L. Glemot, J. P. Brison, J. Flouquet, and A. I. Buzdin, physica B 259-261, 629
(1999).
18. Reviewed by M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).
19. Reviewd by B. K. Sarma, J. B. Ketterson, S. Adenwalla, and Z. Zhao, "Sound
Propagation and Collective Modes in SuperfLuid 3He” in Physical Acoustics Vol. XX,
Moises Levy Ed., (Academic Press 1992).
20. D. B. Mast, B. K. Sarma, J. R. Owers-Bradley, I. D. Calder, J. B. Ketterson, and W.
P. Halperin, Phys. Rev. Lett. 45, 266 (1980).
21. B. Golding, D. J. Bishop, B. Batlogg, and W. H. Haemmerle, Z. Fisk, J. L. Smith, and
H. R. Ott, Phys. Rev. Lett. 55, 2479 (1985).
22. V. Muller, D. Maurer, E. W. Scheidt, Ch. Roth, K. Liiders, E. Bucher, and H. E.
Bommel, Solid State Commun. 57, 319 (1986).
23. For a discussion o f the orgin o f the untrasonic attenuation peak observed in the heavy
fermions superconductors, see section “Ultrasonic Attenuation Due To Domain
Walls” in Ref. 13.
24. L. Coffey, Phys Rev. B 40, 715 (1989).
25. P. Wolfle, J. Low Temp. Phys. 95, 191 (1994).
26. P. J. Hirschfeld, P. Wolfle, J. A. Sauls, D. Einzel, and W. O. Putikka, Phys. Rev. B
40, 6695 (1989).
27. P. J. Hirschfeld, W. O. Putikka, and P. Wolfle, Phys. Rev. Lett. 69, 1447 (1992).
28. P. Wolfle, J. Low Temp. Phys. 22, 157 (1976).
29. P. J. Hirschfeld, and W. O. Putikka, Physica B 194-196, 2021 (1994).
30. P. N. Brusov, N. P. Brusova, Physica B 194-196, 1479 (1994).
31. P. N. Brusov, N. Kulik, P. Brusov, Physica B 259-261, 496 (1999).
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ill
32. C.-T. Lin, Ph.D. Dissertation, Northwestern University (1999).
33. F. A. Lowenheim, Electroplating, (McGraw-Hill Co. 1978).
34. O. Klein, S. Donovan, M. Dressel, and G. Griiner, Int. J. Infrared and Millimeter
Wave 14, 2423 (1993).
35. S. Donovan, O. Klein, M. Dressel K. Holczer, and G. Griiner, Int. J. Infrared and
Millimeter Wave 14, 2459 (1993).
36. D. C. Mattis and J. Bardeen, Phys. Rev. 111,412 (1958).
37. S.-W. Lin, C. T. Lin, D. Wu, B. K. Sarma, G. R. Stewart, and J. B. Ketterson, J. Low
Temp. Phys. 101, 629 (1995).
38. S.-W. Lin, C. T. Lin, B. K. Sarma, G. R. Stewart, and J. B. Ketterson, Phys. Lett. A
217, 161 (1996).
39. H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, and J. L. Smith, Phys. Rev. Lett.
52, 1915 (1984).
40. D. Einzel, P. J. Hirshfeld, F. Gross, B. S. Chandrasekhar, K. Andres, H. R. Ott, J.
Beuers, Z. Fisk, and J. L. Smith, Phys. Rev. Lett. 56, 2513 (1986).
41. Ch. Walti, H. R. Ott, Z. Fisk, J. L. Smith, Phys. Rev. Lett. 84, 5616 (2000).
42. P.H. Frings, B. Renker and C. Vettier, J. Magn. Magn. Mat. 63 & 64, 202 (1987).
43. G. Binnig, H. Rohrer, Ch. Gerber, E. Weibel, Appl. Phys. Lett. 40, 178 (1982).
44. G. Binnig, H. Rohrer, Ch. Gerber, E. Weibel, Phys. Rev. Lett. 49, 57 (1982).
45. H. F. Hess, R. B. Robinson, and J. V. Waszczak, Physica B 169, 422 (1991).
46. H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, Jr. and J. V. Waszczak, Phys.
Rev. Lett 62, 214 (1989).
47. D. N. Davydov, R. Deltour, N. Horii, V. A. Timofeev, and A. S. Grokholski, Rev.
Sci. Instru. 64, 3153 (1993).
48. H. P. Rust, J. Buisset, E. K. Schweizer, L Cramer, Rev. Sci. Instru. 68, 129 (1997).
49. D. Wehnes, J. Meier, J. Classen, C. Enss, Appl. Phys. A 66, s41 (1998).
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
112
50. S. H. Pan, E. W. Hudson, J. C. Davis, Rev. Sci. Instru. 70, 1459 (1999).
51. L. I. Schiff, Quantum Mechanics 3rd Ed, New York (McGraw-Hill Co. 1971).
52. J. Bardeen, Phys. Rev. Lett. 6, 57 (1961).
53. J. Tersoff, and D. R. Hamann, Phys. Rev. B 31, 805 (1985).
54. N. D. Lang, Phys. Rev Lett. 56, 1164 (1986).
55. C. Kittel, “ Introduction to Solid State Physics'', 6th ed., T a b l e p .320, John Wiley &
Sons Inc., New York (1986).
56. J. A. Helfrich, J. B. Ketterson, S. Adenwalla, V. Sakehenko, G. Zhitomirsky, Rev.
Sci. Inst.
57. M. Jobin, R. Emch, F. Zenhausem, S. Steinemann, and P. Descouts, J. Vac. Sci.
Technol. B 9, 1263 (1991).
58. A. J. Nam, A. Teren, T. A. Lusby, and A. J. Melmed, J. Vac. Sci. Technol. B 13,
1556(1995).
59. L. Libioulle, Y. Houbion, and J.-M. Gilles, J. Vac. Sci. Technol. B 13, 1325 (1995).
60. M. Klein and G. Schwitzgebel, Rev. Sci. Instrum. 68(8), 3099 (1997).
61. J. P. Ibe, J Vac. Sci. Technol. A 8, 8570 (1990).
62. D. K. Biegelson, F. A. Ponce, J. C. Tromontana, and S. M. Koch, Appl. Phys. Lett.
50, 696 (1987).
63. L. A. Hockett, S. E. Creager, Rev. Sci. Instrum. 64 (1), 263 (1992).
64. A. Hiess, M. Bonnet, P. Burlet, E. Ressouche, J.-P. Sanchez, J. C. Waerenborgh, S.
Zwimer, F. Wastin, J. Rebizant, G. H. Lander, and J. L. Smith, Phys. Rev. Lett. 77,
3917(1996).
65. E. Knetsch, Ph. D. Dissertation, University o f Leiden, The Netherlands (1992).
66. R. R. Hake, Appl. Phys. Lett. 10, 186 (1967).
67. U. Rauchschwalbe, Physica B 147, 1 (1987).
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
113
68. T. Vaan Duzer and C. W. Turner, “Principles o f Superconductive Devices and
Circuits”, p 317, Haddon Craftsmen Inc., London (1981).
69. L. P. Gor’kov, Soviet Phys. JETP 10, 998 (1960).
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
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