Uniaxial stress measurements using the CS100 cryostrain cell

Jelle Lorenz

July 28, 2020

Abstract

The CS100 is a temperature-compensated apparatus for applying tunable uniaxial

strain to test samples. Doing so often causes substantial changes to the electronic struc-

ture of superconducting materials. Measuring sample resistance and AC magnetic suscep-

tibility at the superconducting transition temperature with varying uniaxial stress can be

an excellent way to probe phase diagrams of interesting materials. The cryostrain cell is

tested extensively on various materials and a miniature AC magnetic susceptibility coil is

crafted. This thesis together with the CS100 datasheet can be utilized as a comprehensive

guide to uniaxial stress measurements with the CS100 cryostrain cell.

Studentnumber 10454993

Supervisor Dr. Anne de Visser

Examinator Dr. Jorik van de Groep

Course Master’s Project

Study Advanced Matter and Energy Physics

Contents

1 Introduction 3

2 Theory and Context 5

2.1 Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 β-PdBi2 and SrxBi2Se3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 The Strain Cell 18

3.1 Function and operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Sample mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Accuracy, resolution and strain inhomogeneity . . . . . . . . . . . . . . . . . 22

4 The Experiment 25

4.1 Measuring sample resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Measuring sample AC magnetic susceptibility . . . . . . . . . . . . . . . . . . 28

5 Results and Discussion 30

5.1 Resistance measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 AC susceptibility measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusion 34

7 Acknowledgements 35

2

1 Introduction

The response of a material to uniaxial stress can reveal much of its electronic properties.

This is because uniaxial stress directly affects nearest-neighbor overlap integrals between

atomic sites. Due to its anisotropic nature, the induced changes to the electronic structure

of materials are typically much larger than those induced by equal hydrostatic pressure.

Additionally, the directional nature of uniaxial stress allows for the comparison of responses to

different lattice distortions. All things considered this makes uniaxial stress an excellent way

to probe phase diagrams of interesting materials, such as Sr2RuO4, β-PdBi2 and SrxBi2Se3.

Uniaxial stress has already been successful in tuning superconducting transition temper-

atures (Tc) in the past. Here are a few examples: The Tc of La1.64Eu0.2Sr0.16CuO4 almost

doubles when a modest pressure is applied along a [1 1 0] crystal direction, but [1 0 0] pres-

sures have a much smaller effect [1]. Some iron pnictide superconductors are very sensitive

to [1 1 0], but not to [1 0 0], pressures [2]. Compressing the a axis of the Sr2RuO4 lattice by

about 0.6% drives the Tc through a significant maximum of a factor 2.3 higher than that of

the unstrained material [3].

The aim of this research has been to gain a good understanding of the functionality

and potential of the Razorbill CS100 cryostrain cell (shown in Fig. 1), together with the

development of a miniature AC magnetic susceptibility coil. Phase diagrams can then be

probed using both electrical resistivity and AC susceptibility measurements. In this thesis

first the necessary theory and context is presented, followed by a detailed description of the

strain cell aimed to serve as a comprehensive guide. Then, the experiments performed and

the results obtained are laid out. The thesis ends with a summarizing conclusion and the

next steps to be taken in the research.

3

Figure 1: The Razorbill CS100 cryostrain cell. Details provided in section 3. Figure by Razorbill.

4

2 Theory and Context

The terms stress (uniaxial force per unit surface) and strain (proportional deformation) are

used interchangeably throughout this thesis. This is because the goal is to perform the

measurements in the linear elastic region where their relationship is governed by Hooke’s law,

see Fig. 2. In this region, the stress is proportional to the strain with the Young’s modulus

(stiffness) of the material as slope. Here the material only undergoes elastic deformation. It is

however important to keep in mind that the applied strain will never be perfectly uniaxial and

that the control over the strain will never be perfectly rigid. Samples have nonzero Poisson’s

ratios, so strain applied along their length will in turn induce strains along their width and

thickness. The epoxy used to mount the samples will also deform slightly, taking up some of

the applied displacement.

Figure 2: The schematic stress-strain curve typical of a low carbon steel as example. Measurements

are performed in the elastic region before the yield strength.

In previous uniaxial pressure measurements an apparatus would be used to set the pressure

at room temperature by turning a bolt. In situ adjustability has also been achieved, by using

helium-filled bellows to apply the force, see Fig. 3 [4, 5, 6].

5

Figure 3: A schematic of the uniaxial pressure cell used in research conducted in 1992. Fig. adopted

from Welp et al. [4]

Iterations of strain tuning devices using piezoelectric stacks have ultimately led to the design

of excellent measurement tools, such as the CS100 cryostrain cell. Its high precision, compen-

sation of thermal contraction, large achievable strains, wide temperature ranges, potential for

high strain homogeneity and in situ tunability have opened up the possibility to take a closer

look at the phase diagrams and anisotropy of interesting materials.

Broken symmetries are the most fundamental organizing principle for phase transitions.

For example, C4 rotational symmetry can be broken using uniaxial stress, as shown in Fig. 4.

6

Figure 4: Schematic representations of strains experienced by a tetragonal material while held under

(a) hydrostatic pressure, and (b) uniaxial stress applied along the [1 0 0] direction. Black arrows

indicate stress. The strain tensor, shown on the right side of the symbolic equations in (a) and (b), is

derived by dividing the stress tensor by the stiffness. The white lines in panel (a) and (b) indicate the

orientation of the tetragonal crystal axes. In both cases, the strain tensor experienced by the material

is decomposed into irreducible representations of the crystal symmetry. Panel (c) illustrates in-plane

deformations as well as the associated preserved symmetries (white line). While symmetric A1g strain

(hydrostatic pressure) preserves C4 rotational symmetry (white arrow) as well as vertical, horizontal

and diagonal mirror planes (white dash dotted lines), antisymmetric B1g and B2g strains lower the

primary rotational symmetry to C2 and break diagonal and vertical mirror planes, respectively. Fig.

adopted from Ikeda et al. [7].

2.1 Sr2RuO4

One material that has already been extensively studied by utilizing a state-of-the-art uniaxial

strain apparatus is Sr2RuO4. Its crystal structure is shown in Fig. 5. It is an unconventional

superconductor: a kind of superconductor that differs from a conventional superconductor

in the symmetry of its Cooper pair wave function, also known as the superconducting order

7

parameter. Conventional superconductors are well described by a so-called s-wave order pa-

rameter which implies isotropic attractive forces between electrons in all spatial directions.

High-Tc superconductors for example, a kind of unconventional superconductor, have a dx2−y2-

wave order paramater, implying a strong directional dependence of their electron-electron

interactions (see Fig. 6). Unconventional superconductors can have both even (spin-singlet

superconductivity) and odd (spin-triplet superconductivity) parity in the order parameter,

and are sensitive to the presence of disorder [8, 9]. They can be utilized to study the collective

physics of interacting electrons and the mechanisms by which the condensation from the nor-

mal metallic state to the superconducting state occurs. Sr2RuO4 in particular is an interesting

and popular unconventional superconductor because it is the most sensitive to disorder of all

that are known [10]. This sensitivity enables it to be studied in special ways, such as via the

de Haas-van Alphen effect: in order to measure this effect disorder should be very small, a low

scattering rate allows electrons to repeatedly make orbits over extremal Fermi surface cross

sections [11]. Two decades of work [12, 13, 14, 15] has resulted in a detailed understanding of

its quasi-two-dimensional Fermi surface topography and the effective masses of the Landau

Fermi liquid quasiparticles, but the superconductor order parameter is still not known with

certainty. Many experiments concerning this order parameter have answered questions, but

raised new ones in the process [16, 17, 18, 19]. With its Fermi surfaces known with accu-

racy and precision [11], and it showing good Fermi liquid behavior in the normal state [20],

gaining full understanding of Sr2RuO4 is an important benchmark in the field of unconven-

tional superconductivity. With the idea to lift the degeneracy of the superconducting order

parameter by strain, uniaxial stress has the potential for tuning the electronic structure of

Sr2RuO4 without introducing additional disorder and destroying the superconductivity. This

is contrary to other methods of perturbing the underlying electronic structure with the aim

to explore its effects on the superconductivity, such as chemical substitution [21, 22, 23].

8

Figure 5: The crystal structure of Sr2RuO4. Fig. adopted from Bergemann et al. [11]

Figure 6: Schematic 2D examples of an s-wave and a dx2−y2-wave superconducting gap. The momen-

tum space representation shows the regions of the Brillouin zone having a positive (+) or negative (-)

phase. Red and blue dotted circles indicate large and small Fermi surfaces, respectively. Image made

by Inna Vishik.

Research performed by Steppke et al. has demonstrated the existence of a well-defined

peak in Tc at 3.4 K, at about 0.6% compression (see Fig. 7) [3]. These results differ significantly

from those obtained through hydrostatic instead of uniaxial pressure: hydrostatic pressure

is known experimentally to decrease the Tc of Sr2RuO4 (see Fig. 8). The results from the

Steppke paper are complemented by density functional theory to give evidence that the peak

likely coincides with a Lifshitz transition: a change in the shape or filling of the Fermi surface

such that the way it connects in momentum (k) space changes, or disappears altogether [24].

In Fig. 9 the calculated Fermi surfaces of unstrained and strained Sr2RuO4 are shown. It

points out the Van Hove point: the point where the Lifshitz transition occurs and where the

9

Fermi surface connects to the 2D zone boundary. Lastly, the results are also complemented

by weak-coupling calculations to gain insight into the effect of the large strains on possible

superconducting order parameters of Sr2RuO4.

Figure 7: The Tc of three samples of Sr2RuO4 against strain along their lengths. Negative values of

εxx denote compression. Fig. adopted from Steppke et al. [3]

Figure 8: The temperature dependence of the sample resistivity of Sr2RuO4 around the Tc under

several hydrostatic pressures. Fig. adopted from Shirakawa et al. [25]

10

Figure 9: The Fermi surfaces of (A) unstrained and (B) strained Sr2RuO4, calculated using density

functional theory and colored by the Fermi velocity. The three surfaces are labeled α β and γ. On the

right is a cross-section through kz = 0. Dashed lines indicate the zone of an isolated RuO2 sheet. The

Van Hove point is shown: the point where the Lifshitz transition occurs and where the Fermi surface

connects to the 2D zone boundary. Fig. adopted from Steppke et al. [3]

11

2.2 β-PdBi2 and SrxBi2Se3

Two other unconventional superconductors are β-PdBi2 and SrxBi2Se3. They are interesting

because of their link to topological superconductors, a branch of unconventional superconduc-

tivity that has gained popularity over the last decade due to the possibility of them hosting

Majorana fermions [26, 27, 28]. β-PdBi2 is a material with a centrosymmetric tetragonal

crystal structure shown in Fig. 10 [29], which is reported to host spin-polarized topological

surface states that co-exist with superconductivity [30, 31].

Figure 10: The crystal structure of β-PdBi2. Fig. adopted from Sakano et al. [30]

The Little-Parks effect, periodical oscillations of the Tc as a result of collective quantum

behavior of superconducting electrons, is used by Li et al. [32] to explore the nature of

superconductivity in β-PdBi2. The results (Fig. 11) show that the flux quantization becomes

(n+ 1/2)Φ0: the oscillation of Tc as a function of the applied flux is shifted by a phase of π.

These findings imply that the superconductivity of β-PdBi2 originates from an unconventional

pairing symmetry consistent with spin-triplet pairing.

12

Figure 11: The Little-Parks effect of the β-PdBi2 ring device used in the experiment of Li et al.

[32] as a function of the perpendicularly applied magnetic field. The sample was held at a constant

temperature of 2.5 K. The x-axis is displayed in units of the oscillation period 30 Oe, in agreement

with the expected magnetic flux quantum for the device geometry. The black dashed lines denote the

applied magnetic flux of Φ = (n+ 1/2)Φ0, which corresponds to the oscillation minima. The red-dot

dashed line denotes the zero external field. Fig. adopted from Li et al.

13

SrxBi2Se3 is derived from the topological insulator Bi2Se3 [33, 34] and has a trigonal

crystalline symmetry (Fig. 12).

Figure 12: (a) The crystal structure of SrxBi2Se3. The blue particle is the dopant Sr ion and sits

in the van der Waals gap between quintuple layers of Se (red) and Bi (green) ions. (b) Its threefold

symmetric basal plane. (c) Its superconducting gap structure, breaking threefold crystal symmetry

and giving rise to a state with twofold symmetry. Fig. adopted from Smylie et al. [38]

Unconventional superconductivity in SrxBi2Se3 has been tested by probing the upper critical

field, Bc2 [39]. For a standard BCS layered superconductor one expects Bc2 to be isotropic

when the external magnetic field is rotated in the plane of the layers. In the research of Pan et

al. however, field-angle dependent magnetotransport measurements reveal a large anisotropy

of Bc2 when the magnet field is rotated in the basal plane (Fig. 13). This anisotropy is

two-fold, while six-fold is anticipated. The rotational symmetry breaking of Bc2 indicates

unconventional superconductivity with odd-parity spin-triplet Cooper pairs. Topologically

superconducting states have been proposed because the superconductivity induced by the ion

intercalation [35, 36, 37] occurs in its topologically non-trivial bands [40, 41].

14

Figure 13: The angular variation of Bc2 in the basal plane for Sr0.15Bi2Se3 at T = 0.3 K and T = 2 K.

The solid black line represents Bc2(θ) for an anisotropic effective mass model with two-fold symmetry.

Fig. adopted from Pan et al. [39]

Research performed by Kostylev et al. ties in with that of Pan et al., reporting for the

first time control of the nematic superconductivity in SrxBi2Se3 (see Fig. 14) [42]. In nematic

superconductivity the superconducting gap amplitude spontaneously lifts the rotational sym-

metry of the lattice. Strain tuning then allows one to choose which of the (in this case three)

domains to populate. Kostylev et al. also measured the dependence of Tc on the applied

strain (Fig. 15). Their findings show a decreasing trend of Tc with compressive strain, but

the overall change is less than 1%. They argue the change in Tc may be due to a change in

the density of states, as reported in a hydrostatic-pressure study of Sr0.06Bi2Se3 [43].

15

Figure 14: Color polar plot of magnetoresistance for the external magnetic field H parallel to the ab

plane of SrxBi2Se3 measured at no strain (a) and -1.19% (compressive) strain (b) with 250 µA applied

current and 2.2 K. The light-green regions extending along ±30 and ±150 in (a) indicate existence

of nematic subdomains, which substantially disappears under applied strain (b). The contours are

drawn from 0.5 mΩ to 7.5 mΩ in steps of 1 mΩ. In (c) a table is shown of the six possible nematic

superconducting states that can exist in the sample as domains. Xn and Yn (n = 0,1,2) domains

exhibit ∆4x and ∆4y states with large upper critical field along one of the a axes (φab = (60n)) and

a* axes (φab = (90 + 60n), respectively, as indicated with the red arrows. The crystal structure of

the ab plane of Bi2Se3 is shown with the schematic superconducting wave function in its center. The

thickness of the blue crescent depicts the superconducting gap amplitude. Fig. adopted from Kostylev

et al. [42]

16

Figure 15: The dependence of the superconducting critical temperature Tc on the applied strain. The

numbers and lines indicate the order of the measurements. Dotted lines indicate that the measurement

sequence number increases by more than 1. The blue and red data points indicate the cases that the

measurement was performed after a decrease and increase in applied strain, respectively. The relative

difference is defined as the change in Tc from the zero-strain Tc. Fig. adopted from Kostylev et al.

[42]

Using uniaxial stress to further study these two materials, as has been done with Sr2RuO4,

may prove to reveal valuable information about their electronic transport properties and

topological superconductivity as a whole.

17

3 The Strain Cell

3.1 Function and operation

The apparatus used in this research is the Razorbill CS100 cryostrain cell. It is a small cell

designed to utilize piezoelectric stacks to apply tunable uniaxial stress to test samples, all

within highly constrained sample spaces such as inside a Physical Property Measurement

System (PPMS). Fig. 16 schematically shows the cell’s principle of operation. The piezoelec-

tric elements lengthen along their poled direction when cooled. Since experiments are done

at temperatures as low as millikelvins, the cell is designed in a symmetric way that cancels

out the thermal expansion of the piezoelectric stacks.

Figure 16: The principle of operation of the cryostrain cell. Displacements are not to scale.

18

The stacks are much longer than the sample in order to provide the option for large

sample strains. The uniaxial stress the sample experiences is the applied displacement ∆L

divided by the sample length. The displacement is measured using a capacitive position sensor

integrated into the apparatus, under the sample plates. The measured capacitance can be

inserted into the calibration formula (Eq. 1, provided by Razorbill, specific for each cell) to

get the displacement:

∆L =εA

C − 0.1 · 10−12− d0, (1)

with ∆L the sample displacement, C the measured capacitance, ε = 8.854 · 10−12 F/m the

vacuum permittivity, A = 5.53 · 10−6 m2 the area of each of the two capacitor plates and

d0 = 4.42 · 10−5 m the distance between the capacitor plates after performing a zeroing

procedure. Keep in mind that A and d0 are parameters specific for each cell and that d0

can change slightly over the lifetime of the cell. Eq. 1 is plotted in Fig. 17, showing the

displacement in µm against the capacitance in pF for the specific strain cell used in this

research.

Figure 17: The capacitance curve for the specific strain cell (CS100 #128) used in this research,

relating measured capacitance to sample displacement. A positive increase in position means tension.

Important to note is that the uniaxial stress applied to the sample by the piezoelectric

stacks is slightly hysteretic. Fig. 18 shows the relation between measured capacitance and

applied voltage. Here, the total applied voltage is defined as

∆V = Vi − Vo, (2)

with Vi and Vo the applied voltage on the inner and outer stacks, respectively. Make sure

there is enough time between setting the voltages and starting the measurement (between five

19

and ten minutes, depending on the jump in voltage). This will allow the capacitance drift to

catch up and will minimize the effect. The effect will become more substantial as the strain

cell ages with use.

Figure 18: The relation between measured capacitance and applied voltage. Note the hysteretic effect:

measurements started at 0 applied voltage, went to positive 150V, then to negative 150V and back to

0 again. Ten minutes was taken between each measurement to account for the capacitance drift. The

capacitance bridge used is the Andeen-Hagerling 2700A.

20

3.2 Sample mounting

Fig. 19 shows an exploded view of the sample mounting components.

Figure 19: An exploded view of the sample mounting components.

Before mounting the bar-shaped sample onto the strain cell it is recommended to secure

the device in place using a clamp. Make sure to clamp the main body, not the bridge. The

following steps refer to the matching numbers in Fig. 20. Step 1 shows the application of

the sample plate guides with the appropriate screws. Two pieces of double-sided tape are

applied in the center to hold the bottom sample plates in place. These sample plates, shown

in step 2, are placed on the cell with a distance inbetween them depending on the length of

the sample. Keep in mind that the sample is required to be long enough in order for it to be

reached through the small center holes of the sample plates. Typical sample dimensions in

mm are 2 long, 0.4 wide and 0.2 thick. In step 3 the sample is applied using slow-hardening

blue Araldite glue (12 hr) at the tips of the plates. Note the sufficient length of the sample.

The sample plates and the sample are not allowed to touch directly because of shortcircuiting.

In order to reliably prevent this from happening, short pieces of insulated 70 µm diameter

copper wire are used as spacers between the sample and the plates. Not yet waiting for the

glue to harden, step 4 shows the securing of the sample in place by one of the two top sample

plates. Glue has been applied between the sample, a new piece of copper wire and the top

sample plate. The spacer ring that goes between the bottom and top sample plates is also

pictured. Its thickness depends on the thickness of the sample and pieces of spacer wire used,

including about 10 to 20 µm leeway. Great care must go into screwing in the top sample

plates, as sample materials are often very fragile and tend to break easily. Step 5 shows the

21

sample properly secured with all sample plates in place. After twelve hours the glue has

hardened and the sample plate guides can be removed.

Figure 20: The sample mounting procedure shown in five steps, from the application of the sample

plate guides to both of the top sample plates being secured with screws.

3.3 Accuracy, resolution and strain inhomogeneity

Once mounted, a number of factors affect the accuracy of the applied strain to the sample,

some do so more significantly than others. Resolution is typically limited by the noise of

the capacitance bridge. Since this is less than one nanometer when generally straining in

the order of micrometers, this shouldn’t be the limiting factor. Accuracy is affected by the

stiffness of the used epoxy, the dielectric properties of the surrounding gas, rapid cooling or

heating of surrounding temperature, the deformation of the device by a stiff sample and the

contraction of the capacitor plates. Of these factors, the last two are likely to be the most

severe, and should always be corrected for.

22

Fortunately, this is easily done. Regarding the deformation factor: for a sample with

spring constant ks mounted a distance hs above the top of the apparatus, the capacitor will

over-read the applied displacement as follows:

∆Lmeas∆Lreal

= 1 +kskτ

(hs + ha)(hs + hc), (3)

where kτ , ha and hc are strain cell specific constants provided by Razorbill in the datasheet

[44]. For a typical sample with a spring constant of 5 · 106 N/m mounted 1mm above the top

of the device, ∆Lmeas/∆Lreal = 1.084, see Fig. 21.

Figure 21: The value of displacement measured from the capacitance sensor relative to the true

displacement of the sample plotted against the sample spring constant ks. The different lines represent

different heights of the sample above the apparatus (hs).

23

As for the capacitor plate contraction factor, this is a result of the fact that the strain cell

chassis and sample plates are all titanium. This means that there is a thermal contraction at

very low temperatures, although minor (Fig. 22). The capacitance will vary as

C

C300K= 1 + 2δx(T ), (4)

where δx(T ) is the integral thermal expansion of titanium between T and 300K. At 4K,

C/C300K is 0.968.

Figure 22: Graph of the linear thermal expansion of titanium. At 4K the linear dimensions are reduced

by 0.15% compared with their length at 300K.

Lastly, an important factor to mention here is the hysteretic effect shown in Fig. 18. This

effect is hard to correct for, but allowing the capacitance drift to catch up as mentioned before

should minimize it.

24

4 The Experiment

4.1 Measuring sample resistance

In order to perform measurements of the electrical resistance of the sample, current and volt-

age contacts must be applied. The sample guides have made room for a small hub providing a

connection from the strain cell to the measurement system, in this research the PPMS. Pieces

of double-sided tape have again been applied, this time in five places. They serve to keep the

approximately 20 µm diameter gold wire in place. These gold wires are pictured in Fig. 23.

Fig. 24 shows an enhanced top and side view of the finished contact application procedure.

The gold wires are connected to four locations on the sample so four-terminal sensing can

be performed for increased measurement accuracy. Contact between the gold wires and the

sample is made using silver glue diluted with ethanol at approximately a 50/50 ratio. Take

note that the fluidity of this mixture varies greatly over time as it rapidly dries. Crucial is

that neither the gold wires nor the silver glue touches anything but the sample as to prevent

shortcircuiting. Minimal usage of the silver glue mixture is preferred so control over the ori-

entation of the electrical circuit through the sample is kept to an acceptable degree; keep in

mind that orientation and strain homogeneity matter for measurement accuracy. However,

make the contact too small and it may disconnect as it thermally contracts due to the large

temperature changes inside the cryostat. Connect the other ends of the gold wires to the hub

with a soldering iron, with the voltage contacts connected to the inner two solder blobs and

the current contacts connected to the outer two. Test the circuits with a multimeter (in our

case the Fluke 179 True RMS Multimeter) for shortcircuits and to gauge contact quality.

25

Figure 23: A finished strain cell set-up for measuring sample resistance using four-terminal sensing.

Figure 24: An enhanced top (a) and side (b) view of the finished strain cell set-up for measuring

sample resistance using four-terminal sensing.

26

The cell can now be inserted into the PPMS probe (Fig. 25).

Figure 25: The probe used to insert the strain cell into the PPMS. Probe wiring connects the strain

cell and the contact hub to the external devices.

After connecting the capacitance wires, power supply wires and contact hub plug, the

probe can be inserted into the PPMS. It is then connected to the Razorbill RP100 power

supply and the Andeen-Hagerling 2700A capacitance bridge. These devices are directly con-

nected to the PC, while the PPMS is wirelessly connected. Having control over all devices

from a single PC is favourable, as temperature changes, piezoelectric stack voltage changes

and measurement timings can all be performed in proper order using a single program. Alter-

natively, the RP100 power supply program supplied by Razorbill can be used to change stack

voltages manually. Because the response of the stacks to a certain voltage varies with tem-

perature, recommended maximum and minimum voltages also vary (Fig. 26). Keep in mind

that there is a risk of causing dielectric breakdown or de-poling of the piezoelectric stacks at

voltages exceeding recommended values. Because the PPMS tends to spontaneously warm

up after being at very low temperatures for extended periods, a safety feature was built into

the measurement program on the PC.

27

Figure 26: Recommended Vmax and Vmin for various temperatures.

Capacitance is logged and used to determine displacement as described in the previous

section. Resistance measurements and temperature logging are done by the PPMS.

4.2 Measuring sample AC magnetic susceptibility

Another way to measure the superconducting transition is through the use of a microscopic

AC susceptibility coil. In order to make the coil, a small piece of paper was wrapped around

a toothpick secured in place with a mini-vise. Aftering securing the paper with tape and

one end of the 30 µm diameter copper wire to the vise, the wire was wrapped around the

paper on the toothpick for a (secondary) pick-up coil of approximately 7 windings and 700

µm diameter designed to measure changes in magnetic susceptibility of the sample. The coil

was fastened to the paper using GE varnish. After the varnish had dried, a new coil was

wrapped around the secondary one in similar fashion and with an equal amount of windings.

This new primary coil generates an AC magnetic field, needed by the secondary coil, when

a current is applied. After the varnish had hardened completely, the tape and so the paper

was removed from the toothpick. Excess paper was cut off and the coil on a small piece of

paper remained. The finished coil is attached to the clamped sample using Apiezon grease

(Fig. 27).

28

Figure 27: A finished strain cell set-up for measuring sample AC magnetic susceptibility using a

microscopic coil.

The primary coil is connected to the reference signal of a lock-in amplifier as source

through a wirefuser and a 5 kΩ resistance elemenent. This resistance provides tunability of

the applied current in the preferred regime. The secondary coil is connected to the input of

the lock-in amplifier through a separate amplifier providing gain of a factor 1000. The lock-

in amplifier provides accurate phase-sensitive measurements of both the real and imaginary

parts of small changes in magnetic susceptibility of the sample. A LabView program on the

PC is used to coordinate measurement parameters while logging AC susceptibility.

29

5 Results and Discussion

5.1 Resistance measurements

For all resistance measurements the Electrical Transport Option (ETO) mode of the PPMS

was used. ETO leverages a digital lock-in technique to measure resistance in a traditional

four-terminal sensing configuration. The dependence of the resistance of β-PdBi2 on the

temperature is shown in Fig. 28. The different curves show the superconducting transition

for different values of uniaxial stress. Positive values correspond to tension and negative values

to compression. A measuring current of 70µA was used. Higher currents yield more precise

measurements, but risk the sample getting heated up, reducing measurement accuracy. Note

the peak in resistance around 5K; this peak is an artifact of the measurement arising from

specific contact geometry and is not related to electrical transport properties of the sample

itself. In repeated measurements this peak before the fall of the resistance has not been

reproduced. The transition temperature corresponds well to that found in previous research,

close to 5K [45].

30

Figure 28: The dependence of the resistance of β-PdBi2 on the temperature. The different curves

show the superconducting transition for different values of uniaxial stress. Positive values correspond

to tension and negative values to compression.

The dependence of the resistance of Sr0.15Bi2Se3 on the temperature is shown in Fig. 29.

The difference in resistance for each curve can possibly be the consequence of a change of the

geometrical factor A/l of the sample, essentially changing the cross section for the current

by perhaps microscopic cracks emerging in the sample. However, until having done repeated

measurements the possibility that this is an effect caused by the difference in uniaxial stress

cannot be entirely ruled out. It is challenging because Sr0.15Bi2Se3 is extremely brittle and

samples break easily during sample mounting. The inset of Fig. 29 shows three uniaxial

tension curves laid over each other, compensating for the consistent resistance differences.

The transition temperature corresponds quite well to that found in previous research, around

2.8K [39, 42]. The very small differences in Tc with varying uniaxial stress matches the

findings in Kostylev et al. [42]. However, it follows the opposite trend of a decrease in Tc

with tensile strain. This may be due to differing orientation, or because the experiments by

Kostylev et al. have been performed on SrxBi2Se3 with x = 0.06 instead of x = 0.15.

31

Figure 29: The dependence of the resistance of Sr0.15Bi2Se3 on the temperature. The different curves

show the superconducting transition for different values of uniaxial tension. Inset: the three curves

compensated for their consistent resistance differences.

As shown in previous research mentioned earlier in this thesis the crystallographic orien-

tation of the sample material is an important parameter in probing the phase diagram. In the

case of SrxBi2Se3 for example, uniaxial stress along the a or the a* direction is expected to

have different effects. The orientation is identified by the use of a camera that uses Laue back

reflection. Unfortunately, due to a defect Laue camera and the inability to have it repaired

due to the COVID-19 crisis, the orientation of the samples could not be determined. In future

measurements knowledge of the orientation can be used to identify a cause of variation in

measurement data.

5.2 AC susceptibility measurements

The dependence of the in-phase AC magnetic susceptibility of Sr0.15Bi2Se3 on the tempera-

ture is shown in Fig. 30. The black curve shows the superconducting transition, while the

red curve does not: the strength of the external magnetic field generated by the PPMS ex-

ceeds the critical field of Sr0.15Bi2Se3. The superconducting transition is shown by the black

32

curve in the way of a relatively large change in AC magnetic susceptibility. This is because

superconductors in very small fields exhibit the Meissner effect, expelling magnetic flux. Con-

sequently the magnetic flux in the pick-up coil on the surface changes, causing the measured

effect.

Because the external magnetic field effectively magnifies the measured AC magnetic sus-

ceptibility (including the noise), the two curves differ somewhat in order of magnitude. In

order to demonstrate the presence or absence of the superconducting transition more effec-

tively, the results have been normalized, hence the y-axis contains arbitrary units. The large

peak of the 0.3T curve around 2.6K came with a peak of similar size in the out-of-phase

AC magnetic susceptibility (the dissipation), contrary to the rest of the curve, where the

dissipation remained practically constant. This is the result of electronic transport properties

of the sample that do not affect the superconducting transition temperature and the peak

can therefore be ignored. These results show that the use of the crafted microscopic coil is

effective in identifying superconducting transitions of samples on the strain cell.

Figure 30: The dependence of the in-phase AC magnetic susceptibility of Sr0.15Bi2Se3 on the temper-

ature. For the black curve the sample experienced an external field of 0 Tesla while for the red curve

it experienced one of 0.3 Tesla.

33

6 Conclusion

As a feasibility study, a good understanding of the functionality and potential of the CS100

cryostrain cell has been obtained. A miniature AC magnetic susceptibility coil has been

crafted in order to perform new measurements. Electrical transport properties of β-PdBi2

and Sr0.15Bi2Se3 have been studied. They have shown strain effects to be very small. No

effect on the Tc has been observed, perhaps because the uniaxial stress has been applied in

arbitrary directions and not along notable crystallographic orientations. This thesis together

with the CS100 datasheet can be utilized as a comprehensive guide to uniaxial stress mea-

surements with the CS100 cryostrain cell. One possible next step is to carry out experiments

in the helium-3 and dilution refrigerator in order to probe the phase diagrams of interesting

materials such as the ferromagnetic superconductors UGe2, URhGe and UCoGe. Another is

the application of larger uniaxial stresses to samples using larger Razorbill strain cells, such

as the CS110.

34

7 Acknowledgements

I want to thank Anne de Visser for his valuable guidance throughout the entirety of this

project. I thank Ying Kai Huang for providing the β-PdBi2 and Sr0.15Bi2Se3 samples, Marc

Salis, Fumihiro Ishikawa and Huaqian Leng for valuable discussions and Ayako Ohmura for

guidance on how to craft a microscopic coil.

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