Matthew Crowley

MSE 4034

Department of Materials Science & Engineering

Virginia Tech

12/14/2014

The Thermodynamics of the Heavy Firmion Superconductor CeIrSi3

Introduction

The goal of this paper is to describe the superconducting behavior of the recently

discovered and highly unique Heavy Firmion Superconductor CeIrSi3, and how this behavior

relates to its thermodynamics. Since the initial discovery of superconductivity in 1911, scientists

have been racing to discover superconductors with higher and higher operating temperatures

because of the tremendous promise that superconductors have to offer. This paper will first

explain the origin of superconductivity of materials, and then use a recently published peer

edited article written by team of Japanese scientists lead by Naoyuki Tateiwa at the Advanced

Science Research Center in Japan to explain superconductivity as it pertains to the CeIrSi3

compound. Lastly, this paper will briefly go over some of the current and future applications of

superconductors in society, and the benefits they have to offer.

Theory and Background (2-3 pages)

Before presenting the theory behind superconductivity in materials, it will be useful to

first review the concept of electrical resistance, as it pertains to normally conducting materials.

Let us analyze the interior of a copper wire as current is being passed through it. Figure 1 shows

a diagram of this situation, where the wire is connected to a negative voltage source (left) and a

positive voltage source on the other side (right). Naturally, the electrons will be accelerated by

this applied electric field and tend to drift towards the positive terminal on the. In order to drift

from left to right, they must weave their way through a lattice of positively charges copper ions.

Occasionally, they will collide with these positively charged ions. Upon collision with a copper

ion, some amount of kinetic energy from the moving electron is transferred into vibrational

energy of the copper ion in the metal lattice. This energy exchange is essentially the electrical

resistance of the copper wire. The lattice vibrational energy given off is also known as phonon

energy, and the more phonon energy that the copper lattice receives, the hotter the wire gets.

Here the electrons are simultaneously attracted to the positively charged copper ions at the same

time as they are repelled from each other due to their like Coulombic charges. In summary, the

kinetic energy from moving electrons gets transferred to phonon energy, or heat. Since electrical

resistance gives of wasted energy as heat (assuming there was no good reason

Figure 1: Electrons in a Copper Wire [1].

to heat the wire in the first place), there is an electrical power loss during the process. This is

given according to the commonly known Ohm’s law relationship, P=I2R, where P= power, I=

current, and R= resistance. Because of this, it is intuitive to understand why scientists and

engineers might desire a conductive material with minimal resistance.

Superconductivity is the term that describes a material which has zero resistance to an

electrical current passing through it. This phenomenon was first discovered in 1911 by

Kamerlingh Onnes at the University of Leiden in Holland [2]. Holland was studying the effects

of electrical resistance vs temperature when he observed that when a sample of mercury is

cooled to a temperature below 4.2 K, its resistivity vanishes and the material behaves as a

superconductor, exhibiting zero resistance to electrical current flow. The temperature at which

the solid phase metal transitioned to a superconducting material was termed the critical

temperature, or TC. At the time, there was very little insight about the scientific theory behind the

phenomenon, but after decades of research, in 1957, John Bardeen, Leon Cooper, and Robert

Schrieffer proposed a theory describing the origin of superconductivity in terms of the quantum

mechanics at play [2]. This theory was coined “BCS Theory”, (due to the order of their last

names). BCS theory proposed that when the temperature is sufficiently low, (T<Tc, or in other

terms T/Tc <1), electrons can actually pair indirectly through interactions with the metal lattice.

Consider a situation in where electrons in a conductive metal sample are being drifted

due to an electric field, such as that shown in Figure 1. As a negatively charged electron drifts by

a positively charged ion in the metal lattice there will be a coulomb attraction between the two

opposite charges. As a result, the positively charged ion will be drawn towards the electron as it

passes by. This causes a distortion in the lattice, and as a consequence a vibration of the ion

lattice. At room temperature conditions, there is adequate thermal energy in the atmosphere to

randomize any induced lattice distortion, but below the critical temperature of the material, this

is no longer the case. The position of the positively charged ion over will now oscillate back and

forth over time in the direction of the passing electron. Now suppose a second traveling in the

opposite direction as the original one. As the positively charged metal ion recovers from its

initial distortion, which we will call the +x direction, it will oscillate back in the –x direction.

The second electron now fees a net attractive force towards the positive ion due its slight

displacement from equilibrium position. The second electron will also move towards the positive

ion, just as the first one did. Figure 2 depicts this interaction between the metal lattice and two

different electrons. The two electrons have now interacted indirectly through the vibrations of the

metal lattice. This interaction can be referred to as an electron-phonon interaction for short. At

sufficiently low temperatures (T<Tc) this indirect attraction between the two electrons is able to

overcome the Coulombic repulsion they experience due to close proximity of like charges, and

the two electrons are in effect bound together [2].

Figure 2: Cooper Pairing Mechanism [2]

The electron couple is known as a Cooper Pair. The net electron spin and linear momentum a

Cooper Pair is 0, and all the paired electrons can be described collectively (for quantum

mechanical purposes) by one coherent wave function, ψ. The standard weak coupling BCS

theory predicts that a universal discontinuity in the electronic heat capacity of a material

undergoing a normal to superconducting phase transition is given by equation 1 below [3].

∆ Cγ S TC

= 127 ζ (3)

1.43 [Eq. 1]

In this equation: ∆ C is the change in electronic heat capacity, ζ is the Riemann Zeta Function

(commonly used in probability statistics and physics), γS is the Sommerfield constant pertinent to

the normal phase, which can be expressed in terms of the electron density of states (DOS) per

spin on the Fermi Surface (FS) of the conducting material. For example, if the electron DOS=0,

the Sommerfield constant is given by equation 2 below.

N(0): γS=2 π2 N (0)/3 [Eq. 2]

Although the BCS theory has been successful in describing traditional superconductors

(simple metals), there have been a number of superconductors developed in recent history that

have proven to have more complex mechanisms at play than simple Cooper Pairing as proposed

by the BCS theory. In fact, scientists are still unveiling entirely new realms of possible origins of

superconductivity in complex intermetallic compounds with one of the elements having occupied

electron states in the f band. These types of superconductors are known as Heavy Firmion

Superconductors. Heavy Firmion Superconductors have distinct properties when it comes to

electrical conductivity because the electrons from the f band hybridize with the normal

conduction electrons [4]. As a result the effective mass of these conduction electrons (which is a

function of quantum physics beyond the scope of this paper) is increased a few hundred fold that

of a regular free electron. Studies have shown that the forces binding superconducting electron

pairs in these complex Heavy Firmion compounds are extremely strong compared to that

predicted by BCS theory, which was originally modeled for simple metal superconductors [4].

These unusually strong binding forces between superconducting electron pairs are also known as

“strong coupling” forces. Hence, it becomes apparent that crystal lattice properties play a key

role in the superconductivity of certain materials.

Another key discovery in the more recent history of superconductor research was the

pressure dependence of superconducting materials. A recent multi institutional research project

funded by the National Science Foundation came up with several substantial conclusions about

the pressure dependence of superconductors [5]. In short, these conclusions can be summarized

in twofold:

1) Most intermetallic compounds (such as Heavy Fermion Superconductors) exhibit lattice

instabilities.

2) The application of external pressure to these intermetallic superconductors can drive the

compounds towards or away from lattice instabilities by varying the parameters that

determine their superconducting properties. These properties include (the electronic DOS at

the Fermi energy, N (EF), the coupling constants of electrons and phonons, and the

characteristic phonon frequency.

The electronic DOS at the Fermi energy is significant because it is these electrons that contribute

to electrical conduction. This study suggests that pressure can be used to “fine tune” the

properties of intermetallic superconductors, such as the Tc [5].

The specific superconducting material that will be examined in this paper is the pressure

induced CeIrSi3 Heavy Firmion Superconductor. This tri elemental superconductor is among the

most complex in existence today, and belongs to the Heavy Firmion class due to the electron

configuration of Iridium: [Xe] 4f14 5d7 6s2. Note that this compound has electrons in its f band.

This complex lattice structure forms in the antiferromagnetic domain, which is marked by the

opposite magnetic ordering of adjacent atoms as depicted in Figure 3. In the presence of an

external magnetic field the antiferromagnetic material will tend to have a small magnetization

effect, M, and in the direction of the applied field to the magnetic moments of A atoms being

greater than that of a B atom. However, antiferromagnetism can only occur at temperatures

below a critical temperature known as the Neel Temperature (TN) [2]. Above the Neel

Figure 3: Magnetic Ordering in an Antiferromagnetic material [2]

Temperature an antiferromagnetic material will become paramagnetic, which is basically a state

of random magnetic ordering. For example, oxygen gas is paramagnetic. Paramagnetic materials

have a very weak positive magnetic susceptibility, meaning that they in essence just “go with the

flow” of an externally applied magnetic field, not drastically enhancing or fighting against it. It

follows that at the Neel transition temperature, there must be some sort of magnetic fluctuation

due to the realignment of magnetic dipole moments. A team of Japanese scientists lead by

Naoyuki Tateiwa at the Advanced Science Research Center in Japan set out to study the effects

of superconductivity in the CeIrSi3 Heavy Firmion pressure induced superconductor with varying

temperature and pressure [6].

Salient Results (2 pages)

Naoyuki Tateiwa and his team of scientists used several commonly known concepts to

quantitatively measure the superconducting properties of the intermetallic CeIrSi3 compound.

They did this by measuring critical points where the compound transitioned from a normally

conducting material to a superconductor. This is the point at which the resistivity of the

compound drops drastically from some finite value to zero as mentioned in the previous section.

They denoted with superconducting transition temperature as TSC. The electronic heat capacity

of the material (Cac) was also measured, keeping in mind the common understanding in

thermodynamics that the heat capacity of materials changes discontinuously at phase transitions.

In other words, they expected to see jumps in the Cac values near the TSC points. The electrical

heat capacity and resistivity were measured in the same run for the same sample. Additionally,

the critical pressure (PC) or the antiferromagnetic state of the material was determined based off

of the measured TN and TSC values during the experiment. Figure 4 (a) shows the pressure phase

diagram of the CeIrSi3 superconductor. At ambient pressure, CeIrSi3 is an antiferromagnet with a

Neel Temperature, TN = 5.5 K. As the pressure was increased, the Neel Temperature decreased,

until it vanished completely at a critical pressure of Pc = 2.25 GPa. Superconductivity in the

sample was observed over a wide range of pressure values in the sample, from approximately

1.3-3.5 GPa. The superconducting transition temperature of the material showed a maximum of

1.6K around 2.6 GPa of pressure. Keeping in mind that the critical pressure of the material PC =

2.25 GPa, heat capacity (Cac) and electrical resistivity (ρ) measurements were taken at four

different values of applied pressure ranging from below the Pc to above it. Figure 5 shows the

temperature dependencies of heat capacity and resistivity at 1.99 GPa and 2.19 GPa of pressure.

Cac is plotted in red and ρ in blue. At P = 1.99 GPa, there is a clear jump in heat capacity and a

kink in the resistivity at the Neel Temperature TN = 2.95K. The sharp drop in resistivity indicates

a superconducting transition at TSC = 1K. What is most interesting here is that there is no jump in

heat capacity at the superconductor transition temperature, TSC. This goes against the theory that

phase changes in a material are marked by a discontinuity in heat capacity. This means that there

was in fact, no bulk superconductivity

occurring in the compound at these lower temperatures approaching 0K [6]. At P = 2.19 GPa, the

heat capacity curve shows a more complex jump consisting of two localized peaks. These peaks

correspond to the antiferromagnetic and superconducting transition temperatures, respectively.

The local peak in heat capacity on the lower temperature side occurs at a value approximately

equate to the TSC =1.4K, where resistivity plummets to zero. Figure 5 shows the

dependencies of heat capacity and resistivity at two select temperatures above the

antiferromagnetic state critical pressure PC = 2.25 GPa. At both selected pressures (P = 2.30 GPa,

and P = 2.58 GPa) there one clear jump in the heat capacity of the material, and it occurs exactly

at the superconducting transition temperature, as expected going into the experiment. This makes

sense because above the Pc the material becomes paramagnetic, and paramagnetic materials are

simple and don’t play a significant role in the presence of a magnetic field (which in this case is

generated by the movement of charge, i.e. current through the superconductor).

At the highest value of pressure, where P = 2.58 GPa, the values of Tsc based on the clear

anomalies in the resistivity and heat capacity curves are TSC = 1.62K and TSC = 2.59K

respectively, in other words they were in strong agreement. These values were included with

markers In Figure 1 where TSC values obtained from heat capacity anomalies were indicated with

squares, and those obtained from anomalies in the resistivity curve were indicated by circles. The

heat capacity jump in the form of ∆ Cac

Cac (T ac) is 3.4 at P = 2.30 GPa, and 5.7 at P= 2.5 GPa. In this

form, the heat capacity jump has arbitrary units; where ∆Cac is the jump of the heat capacity at

Tsc, and Cac(Tsc) is the value of the heat capacity just above superconducting transition

temperature that corresponds to γ*Tsc. This equation is now in the form of equation 1 from the

previous section. As shown via calculation in the previous section, BCS theory predicts that this

jump in heat capacity = 1.43. The values obtained by this experiment (3.4, and 5.7) for the heat

capacity jump are substantially greater than those that BCS theory would predict. The large jump

of the heat capacity at Tsc was also observed in CeCoIn5 and UBe13 where the values of

∆C/(γ*Tsc) are 4.5 and 2.7, respectively. Both of these compounds also have occupied electron

states in the f band of their electron configurations, thus they can also be regarded as Heavy

Firmion Superconductors with strong coupling, just like CeIrSi3. The jump in heat capacity over

the transition temperature, ∆ Cac

Cac (T ac), of 5.7 in CeIrSi3 is the largest value among all known

superconductors. The pressure dependence of this jump in ∆ Cac

Cac (T ac) is shown directly in Figure 1

(b). From the figure, the strong coupling phenomena in CeIrSi3 is evident at a pressure of

approximately 2.5 GPa. Additionally, the increment of the heat capacity change, ∆ Cac

Cac (T ac),

suggests that the superconducting electron coupling parameter increases with pressure. It

becomes apparent that in the unique CeIrSi3 crystal, increasing the pressure favors the

superconducting state of the material. Figure 1 (c) shows the superconducting transition width as

a function of the resistivity in terms of ∆ T SC

T SC. The transition width decreases with increasing

pressure and goes to zero asymptotically above the critical pressure PC = 2.25 GPa.

Applications

The potential role of superconductors in society are immense. Considering that the

phenomenon of superconductivity was only discovered about a century ago, and that the first

“high temperature” superconductor is still less than thirty years old, it is safe to say that we are

just getting started discovering the full potential of superconductors. Gains in science and

technology tend to grow exponentially. For example, new techniques for creating higher

pressures have been developed over the last couple of decades pushing the limits of static

pressure generation to a few hundred GPa [5]. As this experiment showed, being able to

manipulate the externally applied pressure of a system over an extreme range of values can have

profound effects on the parameters of an intermetallic compound that effect its

superconductivity. As it stands right now, even the highest temperature superconductors are still

only functional about as high as 130K. This means the use of a cryogenic fluid is required to

induce superconductivity in even the highest temperature superconductors. However, with the

discovery of new superconducting compounds and new ways to experiment and “fine tune”

them, it is reasonable to expect enormous progress to be gained in our lifetimes towards

achieving superconductivity at more reasonable conditions. Some of the current applications of

high temperature superconductors (with Tc > 30K) include: medical imaging systems, magnetic

shielding devices, and superconducting quantum interference devices (SQUIDS). The SQUID

magnetometer may be the most sensitive device known to man. Due to the vast superiority of

superconducting quantum interference devices over standard magnetometers when it comes to

sensitivity, SQUID’s are currently being used in the field of bio-magnetic imagery to detect

minute signals given off in the human brain such as seizure activity. Moving forward into the

future, one field that high temperature superconductors could revolutionize is power

transmission. As mentioned in this paper, zero resistance means zero loss of electrical power.

Imagine a power transmission cable with 100% efficiency. The AmpaCity project in Germany

has already implemented the world first superconducting power line in the city of Essen [7]. This

power line is capable of holding five times the power as standard power lines, and with zero loss

in efficiency. This is projected to save the city billions of dollars in the long run. A diagram of

what this cable looks like is shown in Figure 7. The obvious downside to this style of cable is

Figure 7: Super Cable in Essen, Germany

that it requires the cryogenic fluid liquid nitrogen. Regardless, the upshot is enormous here.

Discussion and Conclusion (1 page)

Clearly the superconducting properties of the CeIrSi3 superconductor are complex and

dependent on a multitude of parameters that all interact with each other. Some of these

parameters include: The externally applied pressure, the temperature, the magnetic state of the

material, the density of electron states at the Fermi level, and the electron coupling parameter.

Perhaps the most interesting point of discussion from this experiment is the relationship between

superconductivity and antiferromagnetism in the CeIrSi3 superconductor. Comparing the results

from Figures 1 (b) and (c) the conclusion can be drawn that these two factors are actually

competing with each other. In other words, antiferromagnetism is deconstructive towards

superconductivity in the compound. Considering the co-existence of the superconductivity and

antiferromagnetism in the CeIrSi3 superconductor, Tateiwa and his team proposed two

possibilities.

1) Both states co-exist only in a small pressure region close to Pc.

2) Both states do not co-exist, and the superconductivity exists non-homogeneously in the

antiferromagnetic state below the PC. This can be interpreted as the antiferromagnetic field

penetrating the sample in certain regions as the pressure is lowered below the Pc value, which

causes disruption of Cooper Pairs, thus destroying superconductivity.

Considering the second possibility, the pressure dependence of the transition region width, ∆ T SC

T SC

, and the relatively gradual increase of the heat capacity jump, ∆ Cac

Cac (T ac), around the critical

pressure can be interpreted as the increment of the superconducting volume fraction of the

material. For further research on the co-existence of superconductivity and antiferromagnetism,

microscopic experiments such as Nuclear Magnetic Resonance are necessary [6].

Discoveries such as those found by Tateiwa and his team in this experiment are the

stepping stones towards the next generation in superconductivity. There is no doubt about it;

superconductors will revolutionize society within the lifetime of the younger generation.

References:

1) http://electronics.stackexchange.com/questions/78673/why-does-the-thickness-of-a-wire- affect-resistance

2) Kasap, S.O. , Principles of Electronic Materials and Devices. Susanne Jeans3 ed , Vol . ;.2006.

3) Gabovich, A , Thermodynamics of Superconductors with Charge Density Waves , Journal of Physics: Condensed Matter , vol , no 2, p. –

4) Mizukami, Y , "Extremely strong coupling superconductivity of heavy-electrons in two-dimensions". RetrievedDecember , 2014 Available: http://phys.org/news/2011-10-extremely-strong-coupling-superconductivity-heavy-electrons.html

5) Lorenz, B , "HIGH PRESSURE EFFECTS ON SUPERCONDUCTIVITY". RetrievedDecember , 2014 Available: http://arxiv.org/ftp/cond-mat/papers/0410/0410367.pdf

6) Tateiwa, N , "Large heat capacity jump at the superconducting transiton temperature in the noncentrosymmetric superconductor CeIrSi3 under high pressure". RetrievedDecember , 2014 Available: http://iopscience.iop.org/1742-6596/121/5/052001/pdf/1742-6596_121_5_052001.pdf

7) Templeton , G , "World’s first superconducting power line paves the way for billions of dollars in savings, more nuclear power stations". RetrievedJanuary , 2014 Available: http://www.extremetech.com/extreme/182278-the-worlds-first-superconducting-power-line-paves-the-way-for-billions-of-dollars-in-savings