Settingχ =−1 in Eq. (3.2) (as is expected in the Meissner phase of a superconduc-
tor), leads to an expression forD:
D= 1+χeff
χeff
, (3.3)
whereχeffis the measured gradient of the initial magnetization against field.
3.2.2
Vibrating Sample Magnetometry
Measurements of magnetization as a function of field were performed using an Oxford Instruments Vibrating Sample Magnetometer (VSM). A drawback of SQUID magnetometry is that the sample must be moved through the pick-up coils after each temperature and field has been stabilized, such that the magnetization does not change appreciably over the course of the measurement. This can make measurements of magnetization as a function of field time-consuming, as the magnetic field must be stabilized at each value before measuring the magnetization. Use of a VSM overcomes this, by holding the sample in a fixed position between two pick-up coils. An oscillator is used to stimulate vertical motion of the sample through a small distance within the coils, with a typical frequency of 60 Hz. The change of flux due to the motion of the magnetized sample induces a voltage in the coils, which can be converted into a magnetization based on calibration measurements performed with a sample of known susceptibility and mass. The VSM used in this work can apply
magnetic fields up to 12 T, and operates between temperatures of 300 K and 1.5 K
with the aid of a Rootes pump.
3.3
Resistivity
Resistivity was measured using a Quantum Design Physical Properties Measurement System (PPMS), with the alternating-current transport (ACT) option installed. The system consists of a sample probe mounted in a large helium bath, surrounded by a
nitrogen jacket. A temperature range between 1.8 K and 400 K was attainable, and a
magnetic induction up to 9 T could also be applied.
The four-terminal method was used to make AC transport measurements. As depicted in Fig. 3.3, two outer wires are used to supply a current to a sample, and two separate inner wires are used to determine the potential difference across the sample. The advantage of this method is that the voltage leads draw very little current, which means that the resistance of the leads and the contacts can be ignored. This allows
3.4. Heat capacity 34
I- I+
V+ V+
L
Fig. 3.3 Schematic diagram of a sample configured for a four-probe
resistivity measurement, whereLis the distance between the two voltage
wires.
very accurate values for the voltage,V, developed across the sample in response to
an input current,I, to be determined. The resistivity,ρ is then calculated using the
equation
ρ=VA
IL. (3.4)
Here, A is the cross-sectional area of the sample, and L is the separation of the
voltage wire contacts. These values are measured using Vernier callipers before preparing the sample for ACT measurements.
Typically, samples were cut into bar shapes with a uniform cross-sectional area,
and four silver wires 0.05 mm in diameter were attached using DuPont 4929N silver
paste. The current leads generate an electric field over the length of the bar shaped sample, and so it is necessary to attach the voltage wires in line with the current wires to properly measure the associated potential difference. It is also important that the current and voltage wires all have distinct contact points, otherwise contact resistance is no longer negated and can affect the voltage measurement. The wired sample was then affixed to a sample puck using GE varnish, and the wires were soldered to the appropriate connectors. The puck was then installed in the sample chamber using a loading rod.
3.4
Heat capacity
Heat capacity measurements were performed using thermal-relaxation calorimetry
using a Quantum Design PPMS. Temperatures as low as 0.4 K were attainable using
a3He insert, and measurements were performed in a magnetic induction of up to
9 T. Samples were prepared with a polished face, which was then mounted on a sapphire sample stage (chosen for its high thermal conductivity) using Apiezon N
3.4. Heat capacity 35 Heat sink, T0 Platform, Tp P Sample, Ts K1 K2 Cp Cs
Fig. 3.4 Heat flow diagram of the heat capacity setup. The heater provides
a powerPto the sample platform, and the subsequent evolution of the
platform temperature is monitored. Ideally, the sample has a strong
thermal coupling to the platform, such thatTp≈Ts.
or H grease to ensure good thermal contact. The sample stage has a heater and thermometer attached to the underside, and is suspended on the sample puck by wires attached to a copper heat-sink that is held at a constant temperature. As well as providing electrical connection for the puck components, the wires allow heat to conduct between the heat sink and the isolated sample stage.
The heat-flow diagram for the measurement is shown in Fig. 3.4. Here we denote
the unknown heat capacity of the sample byCs and the combined heat capacity
of the sample mount, including the thermal grease, as an addenda heat capacity
Ca. The thermal conductivity between sample and platform isK2, and the thermal
conductivity between platform and heat sink isK1. Also, we denote the temperatures
of the heat sink, platform, and sample asT0,Tp, andTs, respectively. If there is very
good thermal conductivity between the sample and the platform (i.e. K2≫K1), then
Tp≈Ts, and the heat-balance equation for the system is [40]
P= (Ca+Cs)dTp
dt +K1(Tp−T0), (3.5)
wherePis the power of the heater. Upon applying a power to the heater, the platform
heats up by an amount∆T =P/K1. Turning off the heater then allows the sample